Financial Risk Management -...

Financial Risk Management

Marta Wisniewska

[email protected]

WSB Gdańsk

Financial Risk Management: In troduction Marta Wisniewska

Module Outline

Literature

Grading

FINANCIAL RISK

MANAGEMENT DERIVATIVES

PRICING OF DERIVATIVES

USE OF DERIVATIVES & RISK MANAGEMENT


PV of Cash Flows the asset is producingValue of an asset

Value of a derivative At what price there is no arbitrage?

PV Valuation: e.g. DCF, DDM

No Arbitrage Valuation


The purpose of this module is to discuss how derivaties can

be used to manage financial risk.

We will be analyzing:

the impact of using derivaties

and the pricing of derivaties.


Risk…

…something different than expected

Derivatives…

…financial assets whose value depend of value of

underlying asset (the value of an asset is derived from

value of another asset)


DATE TITLE TEXTBOOK

7.10

Introduction to Risk Management

Bonds, Interest Rates and Swaps ch 1, 4, 7

8.10

Futures and Forwards

Optionsch 3, 5-6, 9-12,

18

5.11

Value at Risk

Credit Risk ch 22, 24

3.12

Real Options

TEST ch 34


Hull, J. C. (2012): Options,

Futures and Other

Derivatives, 8th Edition,

Pearson

or earlier eddition

Available in the liberary and online:

www.witor.biz/frm

LECTURE NOTES

Financial Risk Management: In troduction Marta Wisniewska A

Grading

TEST: 100% Not multiple choice

You will need perform calculations

…and interpret the numbers (e.g. hedging)

Section 1:

Option Question

60%

Section 2:

Answer 2 out of 3 questions

40%

see Sample Exam Paper


?


1.Introduction to

Risk Management

2.Bonds, Interest

Rates and Swaps

Lecture 1:

1. Introduction to

Risk Management


Financial Markets

Financial markets facilitate the transfer of funds between

borrowers and lenders

To trade time & risk

FUNDSLENDERS BORROWERSFUNDS

HouseholdsBusinessesGovernmentsForeigners

HouseholdsBusinessesGovernmentsForeigners

DEBT

EQUITY

Money

Market

Capital

Market

Hedgers

Speculators

Arbitrageurs Arbitrage: a trading strategy that allows making profits without any risk of loss

Possible when instruments generating the same cash flow are sold at 2 different prices at 2 markets

All current methods of pricing derivatives utilize the notion of arbitrage.

Arbitrage pricing methods derive the prices of derivatives from conditions that preclude arbitrage opportunities.

Participants of the financial markets

reduce risk exposure

increase risk exposure


Risk and exposure

Why manage risk?

Types of risk

Risk management process

Risk management instruments

Misuse of derivatives


Risk and exposure

What investors care about when making the investments?

Return

Risk

Risk-return trade off: the higher the risk the higher expected rate of return


Risk and exposure

What is return (R)?

𝒐𝒓 𝑹 = 𝐥𝐧𝑫𝒕 + 𝑷𝒕

𝑷𝒕−𝟏

𝑹 =𝑫𝒕 + 𝑷𝒕 − 𝑷𝒕−𝟏

𝑷𝒕−𝟏

simple return

logarithmic return

additive properties

Income received (Dt) on an investment plus any change in the

market price (Pt – Pt-1), usually expressed as a percent of the

beginning market price of the investment.


Risk and exposureIntroduction to Risk Management

Risk and exposure

What is risk?

𝜎 =

𝑖=1

𝑛

𝑅𝑖 − 𝑅 2𝑝𝑖

In common language risk is viewed as something ‘negative’, as an “exposure to danger or hazard”.

The Chinese symbols for risk (危機) combines danger and opportunity.

In Finance Risk is something different than expected:Traditionally risk measured by standard deviation of returns (σ), and is referred to as volatility


What is risk?

Expected Return

High Variance Investment

Low Variance Investment

Probability

NO RISK

Risk and exposure

Normal distribution68, 96, 99.7 rule

+/- 1 σ , +/- 2 σ , +/- 3 σ


Risk and exposure

What is risk?

Is volatility the best measure of risk?

STD

DEV

OF

PO

RTF

OLI

O R

ETU

RN

NUMBER OF SECURITIES IN THE PORTFOLIO

TotalRisk

Unsystematic risk (Unique risk)

Systematic risk(Market risk)

Factors such as changes

in nation’s economy, tax

reforms, or a change in

the world situation.

Factors unique to a particular company

or industry. For example, the death of a

key executive or loss of a governmental

defense contract.

𝑅𝑖 = 𝑅𝐹 + 𝛽𝑖 𝑅𝑀 − 𝑅𝐹 𝛽𝑖 =𝐶𝑜𝑣 𝑅𝑀, 𝑅𝑖

𝜎𝑀2 𝜎𝑖

2 = 𝛽𝑖2𝜎𝑀

2 + 𝜎𝑟𝑒𝑠𝑖𝑑2

𝑅𝑝 =

𝑖=1

𝑛

𝑅𝑖𝑤𝑖

𝜎𝑝2 =

𝑖=1

𝑛

𝜎𝑖2𝑤𝑖

2 + 2

1≤𝑖<𝑗≤𝑛

𝑤𝑖𝑤𝑗𝜎𝑖𝜎𝑗𝜌𝑖𝑗

Capital Asset pricing model (CAPM) assumes that investors hold well diversified portfolios, thus they care only about exposure towards market risk and not the total risk.

This exposure is measure by beta (β)


Risk and exposure

. regress rbar beta sig, robust

Linear regression Number of obs = 101

F( 2, 98) = 2.54

Prob > F = 0.0838

R-squared = 0.1040

Root MSE = .00047

------------------------------------------------------------------------------

| Robust

rbar | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

beta | -.0005175 .0002296 -2.25 0.026 -.0009732 -.0000618

sig | .0327825 .0169871 1.93 0.057 -.000928 .0664929

_cons | .0002485 .0001712 1.45 0.150 -.0000912 .0005881

------------------------------------------------------------------------------

no idiosyncratic risk

Testing CAPM

Does CAPM hold?

𝑅𝑖 = 𝑅𝐹 + 𝛽𝑖 𝑅𝑀 − 𝑅𝐹

Introduction to Risk Management A

Risk and exposure

Testing CAPM

. regress rbar beta beta2, robust

Linear regression Number of obs = 101

F( 2, 98) = 2.28

Prob > F = 0.1081

R-squared = 0.0763

Root MSE = .00048

------------------------------------------------------------------------------

| Robust

rbar | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

beta | .0009477 .0004908 1.93 0.056 -.0000264 .0019217

beta2 | -.0004873 .0002336 -2.09 0.040 -.0009509 -.0000236

_cons | .0000256 .0002206 0.12 0.908 -.0004122 .0004634

------------------------------------------------------------------------------

quadratic relation

Does CAPM hold?

NO


Risk and exposure

What is risk?

Is volatility the best measure of risk?

p = (100 - X)% = 3%

VaR is a 3rd percentile of

the distribution of gain in

the value of the portfolio

in the next 5 days.

Value at Risk (VaR)

‘We are X percent sure there will not be a loss of more than VaR in the next N days’.

Focus on downside.

Example: If N = 5 and X = 97 what is the VaR?


time t0 t1 t2 … t10

Machine -350 000 0

Today you can buy a machine for £350 000. Thanks to this machine, in the next 10 years, you can produce t-shirts.Assume that you know the following about cash flows in years t0 - t10:

Sales volume: 20 000 units per year Sales price: £ 8.50 per unit Variable cost: £ 3.50 per unit Fixed costs: £ 24 875 per year

Cash Flows CF1 CF2 … CF10

Exposure Example:

Investment Appraisal: Sensitivity Analysis

Risk and exposure

Exposure: state of having no protection from

something risky

Should you buy the machine?

Is NPV positive?

Under what assumptions


time t0 t1 t2 … t10Cost of capital 15%

Machine -350 000 0

Cash Flows in years t1-t10: Sales volume: 20 000 units per year Sales price: £ 8.50 per unit Variable cost: £ 3.50 per unit Fixed costs: £ 24 875 per year


Exposure Example:


Risk and exposure

Since every year we receive constant cash flow (=CF):

Therefore to calculate the NPV the annuity formula can be used:

1st derivative of NPV with respect to chosen factor (exposure) tells us how

much NPV will change if the value of chosen factor changes by 1 unit.

𝐶𝐹 = 20 000 × 8.5 − 3.5 − 24 875 = 75 125

NPV = −350 000 +75 125

0.151 −

1

1 + 0.15 10 What happens to NPV if there is change in production costs?

𝜕𝑁𝑃𝑉

𝜕𝑉𝐶=

−20 000

0.151 −

1

1 + 0.15 10

= −133 333 × 0.752815

= −100 375.37

= 27 035

If variable cost increases by £1,the NPV decreases by £100375.37 to £-73 340, 37 (= 27 035 -100 375.37).

Approx. 12%change in VCleads tonegative NPV


time t0 t1 t2 … t10

Machine -350 000 0

Today you can buy a machine for £350 000. Thanks to this machine, in the next 10 years, you can produce t-shirts.Assume that you know the following about cash flows in years t0 - t10:

Sales volume: 20 000 units per year Sales price: £ 8.50 per unit Variable cost: £ 3.50 per unit Fixed costs: £ 24 875 per year


Exposure Example:


Risk and exposure

What are the sources of Exposure?

time

Cost Factors Income

price of cotton (in USD)USD/GBP exchange rate…


Why manage risk?

Cotton price exposure


Why manage risk?

FX exposureCurrency Risk


Why manage risk?

Interest Rate exposureInterest Rate Risk


Why manage risk?

Market exposureMarket Risk


Why manage risk?

Oil Prices exposureOil Prices Risk


Why manage risk?

Why manage risk?

Decision not to manage an expose is a speculation

Exposure to risk factors can affect:The valuation of the projectThe valuation of the assetThe value of the whole company

Yet such speculation is not necessary a bad thing!

Always consider your situation vs. your competitors and the industry standard


Types of risk

2. Event driven definition of risk

1. Diversifiability Systematic Unsystematic

Classification

Business risk Non-business risks

Financial risk Other risks

Reputational

risk

Jorion P.,(2007) Value at Risk

Regulatory,

political risk

Market risk, liquidity

risk

Credit risk

Operational risk

Business decisions

Business environment

Strategic risk

Product, marketing,

organization

Macroeconomics

Competition,

technology

These classifications are somewhere arbitrary(some risks overlap)

One company faces various types of risks, some more important

than othersWhat can he do?


Expected Return

High Variance Investment

Low Variance Investment

Probability

NO RISK


What is risk management?

Assuring an outcome

How to assure outcome?



Reduce Reduce the probability that the event will occur

Reduce the impact if the event does occur

Transfer Transfer the cost of an undesirable outcome to someone else

Avoid Completely avoid potential events thus provide a zero probability that

they will occur

Do Nothing Let the risk happen and be ready to bear the consequences

How to assure outcome?


Price Risk

What Happens to the Distribution When You Have a Floor Price at $40?

100 14040

Reduce


Oil producer


Price Risk

What Happens to the Distribution When you Establish a Short Fence from $100 to $140 ?

Via use of derivativesHow to

achieve this?

Reduce

Risk management processIntroduction to Risk Management

Production Risk

Risk: of a poor weather event causing the undesirable outcome of lower than expected yields

160 19595 wheat yields

Reduce the cost of the risk via spatial location, multiple variety selection, and other cropping practices.

Reduce


160 19595Transfer the cost of the risk via

crop insurance

Transfer

wheat yields

Production Risk

Risk: of a poor weather event causing the undesirable outcome of lower than expected yields


Financial Risk

Risk: higher interest rates causing the undesirable outcome of lower than expected cash flow

Cash FlowTransfer the risk

via fixed rate loans

Transfer


Reduce the cost of the negative impact via

lower debt financing

Financial Risk

Risk: higher interest rates causing the undesirable outcome of lower than expected cash flow

Cash Flow

Reduce



Risk management (RM) is the process by which various risk exposures are

identified,

measured, and

controlled.


Phases risk management process:

1. Identify a company’s current risk profile and set a target risk profile.

2. Achieve the target risk profile by coordinating resources and executing transactions (i.e. reduce, transfer, avoid, do nothing, or some combination)

3. Evaluate the altered risk profile.


RM process – phase 1

Decompose corporate assets and liabilities into risk pools: interest rate, foreign exchange, crude oil, etc.

Develop market scenarios and test the impact of these on the values of the risk pools and on the value of the company as a whole. This determines the company’s “value at risk”.

Develop a target risk profile, which may or may not include a complete elimination of risk.


This is the implementation phase.

Many companies centralize their risk management activities.

This allows for coordination and avoids unnecessary transactions.

Division 2

Exposed short to Polish interest

rates.

Has floating rate loan in zloty.

Division 1

Exposed long to Polish interest

rates.

Has bank account in zloty.

Net

exposure



This is the evaluation phase.

Key questions to consider:

Has the firm’s risk profile changed?

Is the current risk profile still appropriate?

What new economic and market scenarios should be considered in the next iteration?



Risk prioritization matrix

Probability of happening

Potential impact

Act if cost effective

No action required

Immediate action

Action required

Small Catastrophic

High

Low


Proximity


Risk management is an individual decision

No one "right" decision

The "right" decision depends on the characteristics of the operation and

individual decision-maker

Risk

Revenue

1

2

3


M&M



Approaches/ Actions Instruments

Eliminate/ Avoid

Transfer

Absorb/ Manage

Hedge/ Sell

Diversify

Insure

Set policy

Hold capital

derivatives


Forward/ Futures contracts A forward/futures contract is an agreement between two parties, a buyer and a seller, to exchange

an asset at a later date for a price (delivery/ futures price) agreed to in advance, when the contract is first entered into

Forwards OTC

Futures exchange traded

Options An option gives the buyer the right, but not the obligation, to buy/sell the underlying at a later date

for a price (strike or exercise price) agreed to in advance, when the contract is first entered into.

Call option: an option to buy the underlying at the strike price

Put option: an option to sell the underlying at the strike price

The option buyer pays the seller a sum of money called the option price or premium.

OTC and exchange traded.

Swaps Swap is an over-the-counter agreement to exchange cash flows in the future.


A derivative is a financial instrument whose value derives from (depends on) the value of

something else (underlying asset).


Consider a British fund manager with a portfolio of U.S. equities.

If he buys IBM shares, he is exposed to three risks:

Prices in the U.S. equity market generally.

The price of IBM stock specifically.

The dollar/sterling exchange rate.


He is bearish about:

The dollar’s medium-term prospects.

The overall U.S. stock market.

To hedge the currency risk, he could sell dollars under the terms of a forward contract

To hedge the market risk, he could short futures contracts on the S&P 500 index.

He would be left with exposure to IBM’s share price only.

EXPOSURE

Believes

Action

EXAMPLE:


Derivatives allow firms to:

Separate out the financial risks that they face.

Remove or neutralize the risk exposures they do not want.

Retain or possibly increase the risk exposures they want.

Using derivatives, firms can transfer, for a price, any undesirable risk to other parties

who either have risks that offset or want to assume that risk.

Risk management instrumentsIntroduction to Risk Management


Derivative markets have a long history.

Futures markets: date back to the Middle Ages.

Options markets: date back to 17th century Holland.

Last 35 years: extraordinary growth worldwide:

Increased market volatility.

Deregulation of markets.

Globalization of business

Derivative markets:The over-the-counter (OTC) marketThe exchanges

Derivatives and Financial Risk Management, Spring 2016 55/54M. Wisniewska


Leverage

Leverage is the ability to control large amounts of an underlying asset with a

comparatively small amount of capital.

As a result, small price changes can lead to large gains or losses.

Leverage makes derivatives:

Powerful and efficient

Potentially dangerous


EXAMPLE: LEVERAGE WITH OPTIONS

It is May.

The price of XYZ stock is £28.30.

A December call option on XYZ stock with a £29 strike price is selling for £2.80.

A speculator thinks the stock price will rise.

To make a profit, the speculator might:

Buy, say, 100 shares of XYZ stock for £2,830.

Buy 1,000 options (10 option contracts on 100 shares each) for £2,800,

(roughly the same amount of money).


Suppose the speculator is right. The stock price rises to £33 by December.

EXAMPLE: LEVERAGE WITH OPTIONS Cont


Strategy Profit

Buy the stock

Buy options

£33 − £28.3 × 100 = £470

£33 − £29 × 1000 − £2800= £1200


Suppose the speculator is wrong. The stock price falls to £27 by December.

EXAMPLE: LEVERAGE WITH OPTIONS Cont


Strategy Loss

Buy the stock

Buy options

£28.3 − £27 × 100 = £130

£2800



Entity Date Instrument Loss ($million)

Orange County, California Dec.1994 Reverse repos 1 810

margin call

Local government fund Bob Citron (county treasurer)

$7.5 million own money + $12.5 borrowed (via reverse repos) Invest money in agency notes with average maturity of 4 years Short-term funding of mind-term investment Works if rates are falling

Since Feb 1994 rates started to hike Margin calls Dec 1994 investors tried to pull out money Fund defaulted on margin payments Orange County declared bankruptcy




Metallgesellschaft, Germany Jan. 1994 Oil futures 1 580

margin call

Germany’s 14th largest industrial group 58 000 employees

US subsidiary (MG Refinig & Marketing) offered long-term contracts for oil products By 1993 180 million barrels of oil sold to be supplied over a period of 10 years Short-term futures & rolling hedge Long term exposure hedged via series of short-term contracts (3months maturity)

In 1993 oil prices fell from $20 to $15, leading to billion $ margin calls German parent company exchanges the US subsidiary management and closed the

positions at a loss Creditor stepped in with $2.4 billion rescue package Stock price dropped form 64 to 25 DM




Barings, UK Feb. 1995 Stock index futures 1 330

233 year old bank 28-year old trader Nicholas Leeson

Large exposure to the Japanese stock market (via futures) Baring’s position in Nikkei 225 added up to $7 billion Jan&Feb 1995 market fell by 15%, which lead to large losses Yet the exposure was increased 23 Feb Nicholas Lesson walked out of his job

The bank went bankrupt Nicholas went to jail (43 months), then worked as an accountant for Galway United

Football Club


Risk and exposure

Why manage risk?

Types of risk



Misuse of the derivatives


2. Bonds Interest

Rates and Swaps

Zero-coupon bonds and coupon paying bonds

The yield curve and the term structure of interest

rates

Duration, Convexity

Interest Rate Risk and Immunisation of Bonds

Portfolios

Bonds

Comparative Advantage

Swap Design

Valuation of Swaps

Various Interest Rates, Risk Free Rate Proxy

Measuring Interest Rates, Zero Rates

Swaps

Interest

Rates

Bonds, Interest Rates and Swaps

An interest rate quantifies the amount of money borrower pays the lender.

The interest rate depends on: (a) the credit risk of the borrower, the higher the risk the higher the interest rate.

Interest rates change in time.

Bonds, Interest Rates and Swaps Interest Rates

(b) the time to maturity

Various Interest Rates:

Treasury Rates interest rates paid on Treasury bills and bonds

usually assumed that the risk of default is zero

used a proxy of risk-free rate

LIBOR London Interbank Offered Rate

quoted by British Bankers’ Association

for maturities up to 12 months in all major currencies

at what rates banks make large wholesale deposits with each other

(i.e. at what rate banks loan money to other banks)

LIBID at what rates banks accept large wholesale deposits from other banks

at any time LIBOR rate > LIBID rate

Ask rate: how much I want to get from you if I deposit money with you

Bid rate: how much I want to pay to you if you deposit money with me

Repo Rates Repo or repurchase agreement

Enter a contract where securities are sold and later repurchased at a higher price,

with the difference in price creating the interest rate (called repo rate)

Overnight

Indexed Swap

Rate

Overnight indexed swap is a swap where a fixed rate for a period is exchanged for

the geometric average of the overnight rates during the period

The fixed rate is called the overnight indexed swap rate.

Defaults:Russia 1991 Mexico 1982 Argentina 2005

USA 1862 UK 1932Sweden 1812 Germany 1948Denmark 1813…and many many more

Geometric average of a, b, c : 3𝑎 ∗ 𝑏 ∗ 𝑐

Bonds, Interest Rates and Swaps Interest Rates: Various Interest Rates

Risk-free Interest Rate

The rate is used in valuation of assets (eg. CAMP, or valuation of derivatives).

time

Treasure RatesOvernight Index Swaps

(OIS)LIBOR

Believed to be at

artificially low level

because of tax and

regulatory issues

Use Eurodollar futures

and interest rate swaps

to extend the risk-free

LIBOR curve beyond

12 months

Financial crisis:

difficult to borrow

money at interbank

market

what next?

Rates used as a proxy of risk free rate in pricing derivatives

Bonds, Interest Rates and Swaps Interest Rates: Risk Free Rate Proxies

Measuring Interest Rates

If Rc is a rate of interest with continuous compounding and Rm is the equivalent rate with compounding

m times per annum and A is the amount invested for n years, then:

𝐴𝑒𝑥𝑝 𝑅𝑐𝑛 = 𝐴 1 +𝑅𝑚

𝑚

𝑚𝑛

thus: 𝑅𝑐 = 𝑚 ln 1 +𝑅𝑚

𝑚

and

𝑅𝑚 = 𝑚 𝑒𝑥𝑝𝑅𝑐

𝑚− 1

Example: If semi-annually compounded rate is 10% what is the equivalent continuously compounded rate?

m = 2 Rm = 0.1 Rc = ?

A = 2 ln 1 +0.1

2A = 0.09758

Bonds, Interest Rates and Swaps Interest Rates: Measuring Interest Rates

Zero Rates

N-year zero-coupon rate is the rate earned on investment that starts today and last for n years,

with no intermediate payments and all the interest and principal realized at the end of n years.

Example: If today you invest £100 and in 5 years time you receive back £128.40 what is the continuously compounded

zero coupon rate (R)?

A 100 𝑒𝑅∗5 = 128.4

R = 0.049996

A 5R = ln128.4

100= 0.24998

Bonds, Interest Rates and Swaps Interest Rates: Zero Rates

Definitions

A bond is a contract that commits the issuer to make a definite sequence of payments until a

specified terminal date.

Bonds are an example of fixed-income securities (like savings account) – you deposit money (`pay the price`) in

order to receive a certain stream of income (interest) at some fixed/certain dates.

The payment made each period is known as the coupon.

The amount paid at the terminal date is the maturity value (par value, face value or bond’s principal).

Notation:

Date today, t

Maturity date, T

Maturity Value, m

Time to maturity: τ = T - t

Coupon, c

Price of bond today: p

The (annual) yield from holding the bond: y

Yield is a single discount rate that applied to all cash flows of the bond gives the price of the bond equal to its market price.

Bonds, Interest Rates and Swaps Bonds: Definitions

Zero-coupon bonds

A zero-coupon bond is one that pays m at the maturity date, and nothing else (i.e. c = 0).

Since it costs p to buy the bond today, and m is received after τ time periods, it must be the case that:

Let yτ be the yield to maturity of the zero-coupon bond.

yτ is the constant annual rate of return that would be received if the bond was held until maturity.

𝑝 1 + 𝑦ττ = 𝑚

Rearranging:

𝑝 =𝑚

1 + 𝑦ττ

Negative relationship between yield to

maturity and the bond price.

Moreover:

𝑦τ =𝑚

𝑝

1τ

− 1

Restrictive monetary policy which increases

rf must increase y1, which in turn brings

about a fall in bond prices.

In this section assume annual compounding

Bonds, Interest Rates and Swaps Bonds: Zero-coupon Bonds

Note that the curve that shows p as a function of yτ is:

Negatively sloped (the higher the yield, the lower the price)

Convex from below (for successive increases in the yield, the smaller are the reductions in price).

Bonds, Interest Rates and Swaps Bonds: Zero-coupon Bonds

4𝑒−0.10469∗0.5 + 4𝑒−0.10536∗1 + 104𝑒−𝑅∗1.5 = 96

The yield curve and the term structure of interest rates

Consider the yield to maturity yτ on zero-coupon bonds with different times to maturity τ.

Bootstrapping zero rates

A curve showing the relationship between yτ and τ is known as the yield curve.

The shape of the yield curve represents the term structure of interest rates.

Zero rates can be determined from Treasury bills and coupon-bearing bonds.

Example: What is the 1.5 year zero rate (R) if 0.5 year zero rate is 10.469%; 1 year rate is 10.536% and bond that

pays coupon of 4 every 6 months and lasts for 1.5 year with par of 100, sells for 96?

4𝑒−0.10469∗0.5 + 4𝑒−0.10536∗1 + 104𝑒−𝑅∗1.5 = 96

𝑒−1.5𝑅 = 0.85196

𝑅 = −ln(0.85196)

1.5= 0.10681

Bonds, Interest Rates and Swaps Bonds: Yield Curve

Example of yield curve:

Assumptions:

Linear between bootstrapping points

Horizontal before the 1st and after the

last bootstrapping point.


The most important determinant of the shape of the yield curve is expectations of future movements of interest rates.

This dependence is summed up by the expectations hypothesis.


Example of yield curve:

Expectations hypothesis:

Assume that the market consist of only two (zero-coupon) bonds: one year to maturity (short-term) and two years to

maturity (long-term). Suppose that initially, yields on the two bonds are equal.

If the one-year yield is expected to rise next year, investors will have a preference for the one-year bonds, since they

will mature in one year, and the proceeds can be invested at one-year bonds commencing next year, at a higher rate.

This preference would cause investors to sell two-year bonds and buy one-year bonds, bringing about a fall in the price

of two-year bonds, and a rise in the price of one-year bonds.

In turn this will cause the yield on two-year bonds to rise above that of one-year bonds. The yield curve will have a

positive slope.

Therefore, theory predicts that if investors expect interest rates to rise, the yield curve will be positively sloped

Conversely, if investors expect interest rates to fall, the yield curve will be

Alternative explanations for the sign of the slope of yield curve:

(i) liquidity preference theory and

(ii) market segmentation theory.

Expectations of future

movements in the

interest rate can

therefore be deduced

from the slope of the

yield curve.

negatively sloped.


Coupon-paying Bonds

Consider a bond that promises to pay a coupon of c per year for τ years, plus the maturity value m when the bond

terminates at maturity.

This is equal to the present value of the future stream of payments arising if the bond is held to maturity:

Again price of the bond today is given by p.

𝑝 =𝑐

1 + 𝑦+

𝑐

1 + 𝑦 2 +𝑐

1 + 𝑦 3 + ⋯+𝑐 + 𝑚

1 + 𝑦 τ

The yield to maturity of this coupon-paying bond is defined as the value of y that solves the equation above.

Bonds, Interest Rates and Swaps Bonds: Coupon-paying Bonds

Par Yield

Par yield (cp) for a certain maturity bond is the coupon rate that causes the bond price (p) equal

to its par value (m).

𝑚 =𝑐𝑝

1 + 𝑦1

+𝑐𝑝

1 + 𝑦22 +

𝑐𝑝

1 + 𝑦33 + ⋯+

𝑐𝑝 + 𝑚

1 + 𝑦ττ

Bonds, Interest Rates and Swaps Bonds: Coupon-paying Bonds

Macaulay Duration

As for zero-coupon bonds, the price of a bond is a negative and convex function of its yield to maturity. For coupon-

paying bonds, the nature of this relationship is of considerable interest.

However, it is an unsatisfactory measure, because it depends on the units in which the bond is being measured.

The above represents the responsiveness of p to y.

𝑑𝑝

𝑑𝑦

We therefore use instead:

𝐷 = −1 + 𝑦

𝑝

𝑑𝑝

𝑑𝑦=

1

𝑝

1 × 𝑐

(1 + 𝑦)+

2 × 𝑐

1 + 𝑦 2 +3 × 𝑐

1 + 𝑦 3 + ⋯+τ × 𝑐 + 𝑚

1 + 𝑦 τ

MACAULAY DURATION An elasticity of price with respect to changes in yield.

We have also changed the sign so that the measure is positive.

= −𝑐

1 + 𝑦 2 −2𝑐

1 + 𝑦 3 −3𝑐

1 + 𝑦 4 − ⋯−τ 𝑐 + 𝑚

1 + 𝑦 τ+1

Y = a + bX

If X increases by 1 unit, Y

increases by b units

Bonds, Interest Rates and Swaps Bonds: Coupon-paying Bonds: Duration

𝐷 = 𝑖=1

𝑛 𝑐𝑖𝜏𝑖𝑒−𝑦𝜏𝑖

𝑝

𝑝 =𝑐

1 + 𝑦+

𝑐

1 + 𝑦 2+

𝑐

1 + 𝑦 3+ ⋯+

𝑐 + 𝑚

1 + 𝑦 τ

Example: Suppose that a bond with maturity value m = £100 pays a coupon of c = £5 for two years (τ = 2). If y = 4%,

then the price of the bond is:

and the Macauley Duration is:

𝑝 =5

1 + 0.04+

5 + 100

1 + 0.04 2= 101.886

𝐷 =1

101.886

1 × 5

(1 + 0.04)+

2 × (5 + 100)

1 + 0.04 2 =

Duration is so-called is because it is interpreted in the time dimension:

1.953

In the example, the time to maturity is τ = 2 years, and the Macauley Duration is somewhat less than 2 (i.e. ‘the average

time to payment’ is less than 2 years).

For coupon-paying bonds:

𝐷 < τ

For zero-coupon bonds:

𝐷 = τ

The higher the coupon, ceteris

paribus, the lower the value of D.Entire payment is made after

τ periods.

Duration measures how long on average the

holder of the bond needs to wait before receiving

cash payments.

D is a weighted average of the times at which

payments are received, with weights being equal

to the proportion of the bond’s total present value

provided by the cash flow at time t (=1,.., τ).


Interest Rate Risk

Buying and selling bonds is not a risk-free activity.

Macauley Duration can be used as a measure of interest rate risk.

Interest rate risk reflects the impact of Central Bank monetary policy. If the Central Bank raises the cost of borrowing,

all bond yields are likely to rise, and therefore all bond prices will fall, so holders of bonds will suffer a loss.

One type of risk is interest rate risk.

There is negative relationship between bond price p and bond yield y, that for small changes in y can be described by:

if y is expressed with compounding m times a year, or by:

if y is expressed with continuous compounding.

∆𝑝

𝑝= −

𝐷∆𝑦

1 +𝑦𝑚

∆𝑝

𝑝= −𝐷∆𝑦


Example: Bond that pays semi-annually coupon of £5 has 3 years to maturity and face value of £100. The yield to maturity is

0.12. The bond price is 94.213. What happens to the bond price if yield to maturity increases by 10 basis points (i.e. by 0.001)

and the duration is 2.653?

So the new bond price is:

Interest Rate Risk

∆𝑝 = −𝑝𝐷∆𝑦

= −94.213 ∗ 2.653 ∗ 0.001

= −0.24995

94.213 − 0.24995 = 93.9631

=A2/EXP(B2*$C$1)

=SUM(C2:C7)

=(A2*B2)/EXP($C$1*B2)

=E8/C8


𝐷 = 𝑖=1

𝑛 𝑐𝑖𝜏𝑖𝑒−𝑦𝜏𝑖

𝑝

A

Larger changeSmall change


Example: Bond that pays semi-annually coupon of £5 has 3 years to maturity and face value of £100. The yield to maturity is

0.12. The bond price is 94.213. What happens to the bond price if yield to maturity increases by 10 basis points (i.e. by 0.001)

and the duration is 2.653?

Interest Rate Risk

A

Immunisation of Bond Portfolios

Immunisation strategies (a.k.a. neutral hedge strategies) can be used to eliminate interest rate risk.

They are used by organisations that have predictable liabilities, e.g. knowing that they will be paying a client £1m in 5 years

time. The principle of immunisation is:

Example: The contract to pay the client £1m in 5 years time clearly has D = 5 years.

Choose a bond portfolio that has the same overall Macauley Duration (D) as that of the liabilities.

Duration of the portfolio is a weighted average of duration on the bonds included in the portfolio, with weights being the

proportion of portfolio being allocated to particular bond.

The company could immunise by purchasing a zero-coupon bond with time to maturity 5 years and maturity value £1m

(which has the same D of 5 years). However, this assumes that such a bond is available to be purchased.

What the firm therefore needs to do is purchase a portfolio of bonds with overall Macauley Duration of 5 years. If two

bonds are available, bond 1 with D = 3 and bond 2 with D = 6, then the firm could immunise by purchasing a portfolio

consisting of bonds 1 and 2 in the proportion 1:2, since the overall D of this portfolio would be:

1 × 3 + 2 × 6

3= 5

Bonds, Interest Rates and Swaps Bonds: Immunisation

Convexity

The duration relationship applies only to small changes in yields (it is a

linear measure).

in case of continuous compounding:

Convexity measure the curvature of how the price change as the yield change. Convexity can be measured as:

Convexity (C) helps to improve to model the relationship for larger

changes in yields.

∆𝑝

𝑝= −𝐷∆𝑦 +

1

2𝐶 ∆𝑦 2

𝐶 =1

𝑝

𝑑2𝑝

𝑑𝑝2

Bonds, Interest Rates and Swaps Bonds: Convexity A

Swap is an agreement to exchange cash flows in the future.

Most popular swaps are plain vanilla interest rate swap (where fixed rate on a given principal is exchanged for a

floating rate on the same principal) and fixed-for-fixed currency swaps.

In interest rate swap the principle is not being exchanged (thus it’s called notional principle) and at every payment

date one party remits the difference between the two payments to the other side.

Currency swap usually involves exchanging principle (both at the beginning and at the end of the swap) and

interest payments in one currency for principle and interest payments in the other currency.

Most swaps are over the counter agreements.

Bonds, Interest Rates and Swaps Swaps

Comparative advantage

Comparative advantage comes from the lack of constant spread on quotes offered on two products to two parties. One

party will have comparative advantage in one product, the other in the other.

Portugal produces both products at a lower cost.

The difference is cost between 2 countries for cloth is

10 (hours) and for wine is 40 (hours).

Portugal has comparative advantage in Wine.

England has comparative advantage in Cloth.

England produces 1 Cloth, brings it to Portugal and exchanges it for 1.125(= 1+ 10/80) Wine.

The Wine is brought back to England…it is worth 1.35 (=(1.125*120)/100) Cloth.

We are 0.35 Cloth better off.

Portugal produces 1 Wine, brings it to England and exchanges it for 1.2 (= 1+20/100) Cloth.

The Cloth is brought back to Portugal…where it is worth 1.35(=(1.2*90)/80) Wine.

We are 0.35 Wine better off.

Cloth Wine

Portugal 90 80

England 100 120

Minimum Labour Hours Required

for Production

Commodity


Who has a comparative advantage in fixed rate loan?

The difference in the spreads can lead to potential profit that could be exploited by a swap contract.

Company ALIBOR + 1%

LIBOR + 2.25%

5%

5%

Company B

+ LIBOR + 2.25%

- LIBOR + 1%

- 5 %

Total: (-) 3.75%

- LIBOR + 2.25%

+ 5%

- 5 %

Total: (-) LIBOR + 2.25%

each company 0.25% better off

For example, each party is 0.25% better off.

Total profit from swap: (2.5% - 1%) – (5% - 4%) = 1.5% - 1% = 0.5%

How much of the 0.5% each party gets? Depends on its bargaining power.



Who has a comparative advantage in fixed rate loan?

The difference in the spreads can lead to potential profit that could be exploited by a swap contract.

Company ALIBOR + 1%

LIBOR + 2.25%

5%

5%

Company B

+ LIBOR + 1.25%

- LIBOR + 1%

- 4 %

Total: (-) 3.75%

- LIBOR + 1.25%

+ 4%

- 5 %

Total: (-) LIBOR + 2.25%

each company 0.25% better off

Total profit from swap: (2.5% - 1%) – (5% - 4%) = 1.5% - 1% = 0.5%

How much of the 0.5% each party gets? Depends on its bargaining power.



alternative design

4%

LIBOR+1.25%

Company ALIBOR + 1%

LIBOR + 2.30%

5%

5%

+ LIBOR + 2.20%

- LIBOR + 1%

- 5 %

Total: 3.8%

- LIBOR + 2.30%

+ 5%

- 5 %

Total: LIBOR + 2.3%

Total: 0.1%

Financial intermediary netting out 0.1% and each company gets only 0.2%.

Company B

5%

LIBOR + 2.20%

INTERMEDIARY



Valuation of Swaps

When swap is first initiated it is worth zero. The value of swap changes with time.

Plain vanilla interest rate swap can be perceived as a difference between two bonds.

There are two valuation approaches of swaps:

(1) in terms of bond price or

(2) in terms of Forward Rate Agreements.

Therefore to the floating-rate payer, a swap can be seen as having a long position in a fixed-rate bond and a short

position in a floating rate bond. The value of the swap is determined by:

Currency swap can be valued as a difference between two bonds (in two different currencies D-domestic

currency, F-foreign currency, S0 spot exchange rate) that were converted to common currency:

The above mentioned swaps can be also valued as the sum of the Forward Rate Agreements, where each FRA comes

from the exchange of cash-flows during the life of the swap and its maturity.

𝑉𝑠𝑤𝑎𝑝 = 𝐵𝑓𝑖𝑥 − 𝐵𝑓𝑙𝑒𝑥

𝑉𝑠𝑤𝑎𝑝 = 𝐵𝐷 − 𝑆0𝐵𝐹


Zero-coupon bonds and coupon paying bonds

The yield curve and the term structure of interest

rates

Duration, Convexity

Interest Rate Risk and Immunisation of Bonds

Portfolios

Bonds

Comparative Advantage

Swap Design

Valuation of Swaps

Various Interest Rates, Risk Free Rate Proxy

Measuring Interest Rates, Zero Rates

Swaps

Interest

Rates

Bonds, Interest Rates and Swaps

EXERCISE

3. Futures and

Forwards

4. Options

Lecture 2:

3. Futures and

Forwards

Definitions

Payoffs from Forwards Contract

Forward Prices

Valuation of Forward Contracts

Forward Rates

Hedging

Forwards and Futures

Forward and future contracts are agreements to buy or sell an asset at a certain future time (the

maturity date) for a certain price (the delivery price).

They can be contrasted with a spot contract, which is an agreement to buy or sell an asset today.

One of the parties to the forward/future contract assumes a long

position and agrees to buy the asset at the future date at the

agreed price.

The other party assumes a short position and agrees to sell the

asset on the same future date at the same agreed price.

Underlying (asset): bond, stock, index, currency, commodity (gold, oil, wheat)

Forwards and Futures Definitions

http://www.goseewrite.com/wp-content/uploads/2012/12/money-exchange-rates.jpg

http://www.goseewrite.com/wp-content/uploads/2012/12/money-exchange-rates.jpg

FORWARD FUTURES

Private contract between two parties

Non standardized

Usually one specified delivery date

Settled at the end of contract

Delivery or final cash settlement usually

takes place

Some credit risk

Traded on exchange

Standardized contracts

Range of delivery dates

Settled daily

Contract is usually closed out prior to

maturity

Virtually no credit risk

Margin Account

Initial Margin

Maintenance Margin

Margin Call

Notice of intention to deliver presented by

seller to exchange


Payoffs from Forward Contract

The payoff from holding a long position in a forward contract on one unit of an asset is:

where K is the delivery price and ST is the spot price of the asset at maturity of the contract.

We are buying for K something worth ST

- K

STK

Payoff

𝑆𝑇 − 𝐾

Forwards and Futures Payoffs

The payoff from a short position in a forward contract on one unit of an asset is:

We are selling for K something worth ST

K

Payoffs from Forward Contract

𝐾 − 𝑆𝑇

Forwards and Futures Payoffs

K ST

Payoff

Short Selling

Short selling = selling an asset that is not owned.

It is done by borrowing the asset from someone who does own it, selling it, and then buying it back at a later date, and

finally returning it to the party from whom it was borrowed.

Such a trade is profitable if the price of the asset has fallen over the period between the sale and the repurchase.

BORROW & SELLBUY BACK &

RETURNPROFIT ?

Forwards and Futures Forward prices: Short Selling

Forward Prices

The easiest type of forward contract to value is one written on an asset that provides the holder with no income, such as

non-dividend paying stock or zero-coupon bond.

Consider a long forward contract to purchase a non-dividend paying stock in 3 months.

Assume that the current stock price is £40 and the 3-month risk-free interest rate is 5% per annum.

time

Suppose that the forward price is relatively high at £43.

borrow £40 at rf of 5% pa buy 1 share for £40, short forward contract to

sell 1 share in 3 mths

now in 3 mths

deliver the share and receive £43

pay back loan: 40e0.05*(3/12) = 40.50

Profit:

£43 - £40.50 = £2.50Arbitrager can lock in risk free

profit of £2.50

Assumptions: no transaction costs borrow/ lend at rf

Forwards and Futures Forward prices

time

Suppose that the forward price is relatively low at £39.

short 1 share (& receive £40) invest £40 at rf

take long position in 3 month forward contract

nowin 3 mths

£40 investment grows to:40e0.05*(3/12) = 40.50

pay £39 and take delivery of 1 share use the share to close out the short

position

Profit:

£40.50 - £39 = £1.50

Arbitrager can lock in risk free profit of £1.50

Under what circumstances do arbitrage opportunities not exist?

Arbitrage opportunities arise whenever the forward price is above £40.50or below £40.50.

Thus for there to be no arbitrage, the forward price must be exactly £40.50.

Forwards and Futures Forward prices: No income

General Formula (no income case)

Consider a forward contract on an asset that provides no income.

The current (spot) price of the asset is St, τ is the time to maturity, r is the risk-free rate, Ft is the forward price.

The relationship between the spot price and the forward price is:

If Ft > Sterτ , arbitrageurs can buy the asset and short forward contracts on the asset.

If Ft < Sterτ , they can short the asset and enter into long forward contracts on it.

𝐹𝑡 = 𝑆𝑡𝑒𝑟𝜏

Forwards and Futures Forward prices: No income

Forward market

Spot market

Buy at the market where asset is cheaper…sell at the market where it’s more expensive

EXMPLE: Consider a long forward contract to purchase a stock in 3 months that pays £1 in 1 month time.


time


borrow £39.00416 (=40 – 0.996) at rf of 5% pa for 3 months

borrow £0.996 at rf for 1 month buy 1 share for £40, short forward contract to sell 1

share in 3 mths

now in 3 mths

deliver the share and receive £42

pay back loan: 39.00416 e0.05*(3/12) = 39.49477

Profit:

£42 - £39.49477 = £2.50523

in 1 mth

receive £1 dividend

pay back short term loan: £1

PV(£1) = 1e-0.05*(1/12) = £0.996

Known income I

Forwards and Futures Forward prices: known income

Consider forwards contract on an asset that provides an income with a present value of I during the life of a forward

contract.


If Ft > (St-I)erτ , arbitrageurs can buy the asset and short forward contracts on the asset.

If Ft < (St-I)erτ , they can short the asset and enter into long forward contracts on it.

In case Ft> (St-I)erτ, we need to borrow PV of (St-I) for

period of τ, and I for period until we receive the I.

General Formula (known income I)

𝐹𝑡 = 𝑆𝑡 − 𝐼 𝑒𝑟𝜏

Forwards and Futures Forward prices: known income

Consider a long forward contract to purchase a stock in 3 months. Stock pays a dividend yield of 2%.


time


borrow £40 at rf of 5% pa buy 1 share for £40, short forward contract to sell

1 share in 3 mths

now in 3 mths

deliver the share and receive £43 receive divided:

40e0.02*(3/12) – 40 = 0.2005 pay back loan:

40e0.05*(3/12) = 40.50

Profit:

£43 - £40.50 + £0.2005 =

£2.7005

Forwards and Futures Forward prices: known yield

Known yield q

Consider forwards contract on an asset that provides a known yield q (i.e. income as percentage of the asset’s price at

the time the income is paid is known).


If Ft > Ste(r-q)τ , arbitrageurs can buy the asset and short forward contracts on the asset.

If Ft < Ste(r-q)τ , they can short the asset and enter into long forward contracts on it.

General Formula (known yield q)

𝐹𝑡 = 𝑆𝑡𝑒𝑟−𝑞 𝜏

Forwards and Futures Forward prices: known yield

Valuing Forward Contract

The value of forward contract at the time it was first entered is zero.

At the later date the value can be positive of negative.

If K is the delivery price, τ is the time to maturity, r is the risk-free rate, Ft the forward price that would apply if

contract was negotiated today, the value of the (long) forward today (f) on no income paying asset can be defined as:

The value of forward on asset that pays I income is:

𝑓 = (𝐹𝑡 − 𝐾)𝑒−𝑟τ

The value of forward on asset that pays q yield is:

𝑓 = 𝑆𝑡 − 𝐼 − 𝐾𝑒−𝑟τ

𝑓 = 𝑆𝑡𝑒−𝑞τ − 𝐾𝑒−𝑟τ

= 𝑆𝑡 −𝐾𝑒−𝑟𝜏

Forwards and Futures Valuation of Forward Contracts

Forward interest rates are interest rates implied by current zero rates for periods of time in the future.

If R1 and R2 are the zero rates for maturities T1 and T2, RF is the forward interest rate for the period between T1 and T2, then:

which shows that if the zero curve is upward sloping between T1 and T2, (i.e. R2>R1), then RF>R2.

𝑅𝐹 =𝑅2𝑇2 − 𝑅1𝑇1

𝑇2 − 𝑇1

= 𝑅2 + 𝑅2 − 𝑅1

𝑇1

𝑇2 − 𝑇1

=𝑅2𝑇2 − 𝑅2𝑇1 + 𝑅2𝑇1 − 𝑅1𝑇1

𝑇2 − 𝑇1

=𝑅2 𝑇2 − 𝑇1 + 𝑅2𝑇1 − 𝑅1𝑇1

𝑇2 − 𝑇1

R1

R2

RF

𝑒𝑅1𝑇1+𝑅𝐹 𝑇2−𝑇1 = 𝑒𝑅2𝑇2

𝑅1𝑇1 + 𝑅𝐹 𝑇2 − 𝑇1 = 𝑅2𝑇2

Forward Rates

Forwards and Futures Forward Rates

Example:

Calculate year-2 forward rate (i.e. a rate of interest for year 2 that combined with 1-year zero interest

provides the same overall interest as the 2-year zero rate), knowing that R1 = 0.03 and R2 = 0.04.

𝑅𝐹 =𝑅2𝑇2 − 𝑅1𝑇1

𝑇2 − 𝑇1

=0.04 × 2 − 0.03 × 1

2 − 1

= 0.05

Forwards and Futures Forward Rates

Hedging with Futures

Basis risk arise when (i) the asset underlying futures contract is different than the asset whose price is to be hedged.

(ii) hedge require contract to be closed out before its delivery.

The basis (b) is defined as:

𝐵𝑎𝑠𝑖𝑠 = 𝑆𝑝𝑜𝑡 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡 𝑡𝑜 𝑏𝑒 ℎ𝑒𝑑𝑔𝑒𝑑 − 𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡 𝑢𝑠𝑒𝑑

If the asset to be hedged and the asset underlying the futures is the same, the base is zero at the maturity of futures contract.

Forwards and Futures Hedging: Hedging with Futures

Cross Hedging

When asset underlying the futures contract is different to the asset whose price is being hedged the hedge is referred to

as cross hedging.

Hedge ratio is the ratio of the size of the position taken in futures contracts to the size of the exposure.

Define ΔS as the change in the spot price S during the period of time equal to the life of the hedge and ΔF change in

futures price F during a period of time equal to the life of the hedge.

ℎ∗ = 𝜌𝜎𝑆

𝜎𝐹

When both assets are the same, it is natural to use hedge ratio of 1.

When assets vary it might be optimal to use different ratio.

The hedger should choose the value of the hedge ratio that minimizes the variance of the value of the hedged position.

Minimum variance hedge ratio (h*) can be defined as:

where σS is the standard deviation of ΔS, σF is the standard deviation of ΔF, and ρ is the correlation coefficient between

ΔS and ΔF.

If ρ = 1 and σF = σS, h* = 1.

Forwards and Futures Hedging: Cross Hedging

To calculate the optimal number of contracts to be entered (N*) one should multiply the size of the position being

hedged (QA) by minimum variance hedge ratio, and divide the product by the size of one futures contract (QF):

𝑁∗ =ℎ∗𝑄𝐴

𝑄𝐹

In practise: Choose contract with closes delivery date to the exposure…but later delivery than exposure Choose contract on asset whose price is highly correlated with price of exposed asset

Forwards and Futures Hedging: Cross Hedging

Definitions

Payoffs from Forwards Contract

Forward Prices

Valuation of Forward Contracts

Forward Rates

Hedging

Forwards and Futures

EXERCISE

4. Options

European Options

Pay-off Diagrams,

Bounds of Option Prices

Options

Option Valuation

Binomial Model

Black-Scholes Model

Put-Call Parity

American Options

Dividends

Delta Hedging

Options

Top 10

Options

During 2012 477 Million equity contracts were traded on the CBOE, representing options on 47.7 billion shares of underlying stock

S&P500 :

(1) total market capitalization (USD) as of 31 Jan 2014: 16 872 585 650 000

(2) average daily volume in 2013 around 3 000 000 000 (shares)

Options

CALL OPTION PUT OPTION

A Call Option gives the holder the right to buy

the underlying asset on (or maybe before) a

certain date (the expiry date) for a certain price

(the strike price).

A Put Option gives the holder the right to sell

the underlying asset on (or maybe before) a

certain date (the expiry date) for a certain price

(the strike price).

Like forwards and futures, options can be written on a

stock, foreign exchange, market index,....

In forwards/ futures the contract creates an obligation to both parties.

The holder of the options has the right to exercise the option, whereas the person writing the option has an obligation to comply with holder decision.

The owner, or holder, of an option – who is said to adopt a long position – acquires the option by paying the option price

(premium) to the writer – who is said to adopt a short position.

If the holder of a call option chooses to exercise the option, he pays the strike price to the writer in exchange for the asset.

If the holder of a put option chooses to exercise the option, he delivers the asset to the writer, who simultaneously pays the

strike price to the holder.

European OptionsOptions

EUROPEAN OPTION AMERICAN OPTION

A security that gives its owner the right, but not the

obligation, to purchase (or sell) a specified asset for

a specified price, known as the strike price, AT some

date in the future (the expiry date).

A security that gives its owner the right, but not

the obligation, to purchase (or sell) a specified

asset for a specified price, known as the strike

price, at any time up to the expiry date.

Notations:

t is the current date T is the expiry date τ = T – t is the time to expiry.

St is the current (underlying) stock price. ST is the stock price at expiry.

K is the strike price. r is the risk-free rate of interest.

ct is the current price (or the current value) of a Call Option

pt is the current price (or the current value) of a Put Option

For the time being, we restrict attention to European Options because they are more straightforward to analyse.

Options that expire unexercised are said to die, and are worthless

European OptionsOptions

If you are the holder of a call option, you want the stock price at expiry to exceed the strike price.

K

STK

Payoff

Then, you exercise the option to buy at the strike price, and immediately sell at a profit ST - K.

If the stock price at expiry is less than the strike price, you let the option die.

Payoff diagram for a Call Option

A call option for which the

current stock price St is

above the strike price K is

said to be in the money.



below the strike price K is

said to be out of the money.



equals the strike price K is

said to be at the money.

out of the money

at the money

Long position in a Call Option

𝑚𝑎𝑥 0, 𝑆𝑇 − 𝐾

European Options: Pay-off DiagramsOptions

-K

STK

Payoff

Short position in a Call Option

Issuer of the option faces pay off or loss, thus he needs to be compensated to be willing to write the option (option premium).

European Options: Pay-off Diagrams


Options

K

STK

Payoff

Long position in a Put Option

out of the money

at the money

𝑚𝑎𝑥 0, 𝐾 − 𝑆𝑇


Payoff diagram for a Put Option

Options

- K

STK

Payoff

Short position in a Put Option



Options

A call gives the right to buy the underlying.

Thus the call payoff is always less that the value of underlying at time T, ST.

For the put the maximum value obtained at expiry is K, thus current value must be:

𝑐𝑡 ≤ 𝑆𝑡

Therefore the value of the call at time t must be less or equal to the value of the underlying:

𝑝𝑡 ≤ 𝐾𝑒𝑥𝑝(−𝑟𝜏)

European Options: Bounds of Option Prices


Options

Consider portfolio consisting of call option and a sum of money equal to Kexp(-rτ).

At T in case ST > K, this portfolio is worth the same as the underlying.

Similarly, consider portfolio consisting of put and underlying.

At T in case K > ST put is exercised and portfolio is worth K, otherwise put is not exercised and portfolio is worth ST.

Therefore at time T a portfolio of put and underlying when compared to cash currently worth Kexp(-rτ), will be worth at least

the same as the cash:

𝑐𝑡 + 𝐾𝑒𝑥𝑝(−𝑟𝜏) ≥ 𝑆𝑡

Since the cash and option produce a payoff equal to or greater than that of the underlying, thus the value of portfolio must then

equal to or greater than that of the underlying:

𝑝𝑡 + 𝑆𝑡 ≥ 𝐾𝑒𝑥𝑝(−𝑟𝜏)

Otherwise it is worth more than the underlying (i.e. K).

which implies:

𝑐𝑡 ≥ 𝑆𝑡 − 𝐾𝑒𝑥𝑝(−𝑟𝜏)

thus:

𝑝𝑡 ≥ 𝐾𝑒𝑥𝑝 −𝑟𝜏 − 𝑆𝑡

European Options: Bounds of Option Prices


Options

100

110

90

100

120

80

t=0 t=0.5 t=1

0*0.5+10*0.5=54.878

19.512

11.897

10*0.5+30*0.5=20

4.878*0.5+19.512*0.5=12.195

Consider an European put option with time to expiry of 1 year, and a strike price of 110.

The current price of the underlying is 100. Divide the time to expiry into two 6-month intervals.

Suppose that in each interval, the price can either rise by 10 or fall by 10, with equal probabilities.

The risk-free rate is 5% per annum, simply compounded.

The price movements can

be represented by a diagram

called a binomial tree.

An underlying assumption

is that the underlying price

follows a binomial process .

The value calculation proceeds backwards from T to t. Each step involves:

finding the terminal value of the option;

calculating its expected value of the option; and finally

discounting it by the risk-free rate (make sure that you use the right rate).

Risk-neutral valuation on

objective probabilities.

0

10

30

What is the value of the option?

Options Valuation: Binomial Model

Binomial Trees

Options

11.897

4.878*0.5+19.512*0.5=12.195

0*0.5+10*0.5=5

10*0.5+30*0.5=20

19.512

4.878

100

110

90

10010

1200

8030

t=0 t=0.5 t=1

0*0.5+10*0.5=5>4.878

>19.51

>11.897

10*0.5+30*0.5=20

4.878*0.5+19.512*0.5=12.195

0.4 0.6

0.4 0.6

0.4 0.6

Suppose that the probabilities of rise& fall were 40/60 instead of 50/50.

Without doing any further calculation, can you determine how the option price would change?


Binomial Trees

Options

100

110

90

100

120

80

t=0 t=0.5 t=1

10*0.5+0*0.5=54.878

0

2.3795

0*0.5+0*0.5=0

4.878*0.5+0*0.5=2.439

Now, let’s redo the question above, but assuming an European call option instead.

Suppose that the probabilities of rise & fall were 60/40 instead of 50/50.

Without doing any further calculation, can you determine how the option price would change?

0

0

10

What if we don’t know the probabilities? 1. No-Arbitrage Argument Valuation

2. Risk Neutral Valuation with Risk Neutral Probabilities

5exp(-0.05*0.5)=4.878


Binomial Trees

Options

S0

f

S0UfU

S0DfD

No Arbitrage Argument

Consider a stock whose price is S0 and option on the stock whose current price is f.

Option lasts for time T, and in that time the stock price moves to either S0U (where U > 1) or to S0D (where D < 1).

fU is option payoff if stock moved to S0U and fD option payoff is stock moved to S0D.

Consider a portfolio consisting of a long position in Δ shares and a short position in one option.

Calculate Δ that makes the portfolio riskless (i.e. portfolio has the same payoff regardless if the stock price increased or

decreased):

𝑆0𝑈∆ − 𝑓𝑈 = 𝑆0𝐷∆ − 𝑓𝐷 ∆ =𝑓𝑈 − 𝑓𝐷

𝑆0𝑈 − 𝑆0𝐷

S0UΔ - fU

S0DΔ - fD

S0Δ - f

Options Valuation: Binomial Model: No Arbitrage ArgumentOptions

For arbitrage opportunities not to exist the riskless portfolio must earn risk-free interest rate.

If r is the risk-free interest rate, then the present value of the portfolio is:

(𝑆0𝑈∆ − 𝑓𝑈)exp(−𝑟𝑇) = (𝑆0𝐷∆ − 𝑓𝐷)exp(−𝑟𝑇)

whereas the cost of creating this portfolio today is:

𝑆0∆ − 𝑓

Therefore:

𝑆0∆ − 𝑓 = (𝑆0𝑈∆ − 𝑓𝑈)exp(−𝑟𝑇)

𝑓 = 𝑆0∆(1 − 𝑈𝑒𝑥𝑝 −𝑟𝑇 ) + 𝑓𝑈exp(−𝑟𝑇)

Let’s substitute 𝑓𝑈−𝑓𝐷

𝑆0𝑈−𝑆0𝐷for Δ:

𝑓 = 𝑆0

𝑓𝑈 − 𝑓𝐷𝑆0𝑈 − 𝑆0𝐷

(1 − 𝑈𝑒𝑥𝑝 −𝑟𝑇 ) + 𝑓𝑈exp(−𝑟𝑇)


Options Valuation: Binomial Model: No Arbitrage ArgumentOptions

=𝑓𝑈 1 − 𝐷𝑒𝑥𝑝(−𝑟𝑇) + 𝑓𝐷 𝑈𝑒𝑥𝑝(−𝑟𝑇) − 1

𝑈 − 𝐷

𝑓 = 𝑆0

𝑓𝑈 − 𝑓𝐷𝑆0𝑈 − 𝑆0𝐷

(1 − 𝑈𝑒𝑥𝑝 −𝑟𝑇 ) + 𝑓𝑈exp(−𝑟𝑇)

= exp(−𝑟𝑇) 𝑝𝑓𝑈 + (1 − 𝑝)𝑓𝐷 where: 𝑝 =exp(𝑟𝑇) − 𝐷

𝑈 − 𝐷

The model allows to price an option when stock price movements are given by a one-step binominal tree, under the

assumption there are no arbitrage opportunities in the market.

=𝑓𝑈 − 𝑓𝐷 − 𝑈𝑒𝑥𝑝 −𝑟𝑇 𝑓𝑈 + 𝑈𝑒𝑥𝑝 −𝑟𝑇 𝑓𝐷 + 𝑓𝑈 exp −𝑟𝑇 𝑈 − 𝑓𝑈 exp −𝑟𝑇 𝐷

𝑈 − 𝐷

= exp(−𝑟𝑇)𝑓𝑈 exp(𝑟𝑇) − 𝐷 + 𝑓𝐷 𝑈 − exp(𝑟𝑇)

𝑈 − 𝐷

Options Valuation: Binomial Model: No Arbitrage Argument


Options

20f

22fU =1

18fD = 0

Example: Stock price today is equal to 20, and in 3 months it will be either 22 or 18.

What is a value of 3 month European call option with a strike price of 21.

The risk free rate is 12% (continuous compounding).

22∆ − 1 = 18∆ − 0

4∆ = 1

∆ = 0.25

18∆ − 0 = 18 × 0.25 = 4.5 4.5 exp −rT = 20∆ − 𝑓 4.5 exp −0.12 ×3

12= 5 − 𝑓

4.367005 = 5 − 𝑓 𝑓 = 0.632995

Step 1: Calculate Δ

Step 2: Calculate portfolio

value at horizonStep 3: Calculate portfolio value today, and thus calculate f

Options Valuation: Binomial Model: No Arbitrage Argument


Options

Risk neutrality proves very interesting since it implies that investors only care about expected returns, and not risks

associated with the investment.

Suppose there are only two assets in the economy: one risky (‘stock’) and the other riskless (‘bond’).

Risk-neutral investors will hold the stock alone – no matter how risky it is – provided that such

a stock gives a higher expected return than the bond.

If we’re willing to assume that everybody in the world is risk-neutral, then it must be the case that the returns on both

assets must be equal.

A risk-neutral world has two features that facilitate pricing derivatives:

(1) Expected return on stock (or any other instrument) is risk-free

(2) The discount rate used for the expected payoff on an option (or any other instrument) is risk-free rate.

Let 𝑝 =𝑒𝑟𝜏−𝐷

𝑈−𝐷be interpreted as the probability of an up movement in a risk-neutral world..

Thus the expected future payoff from an option in risk neutral world is:

𝑝𝑓𝑈 + (1 − 𝑝)𝑓𝐷

Options Valuation: Binomial Model: Risk Neutral Valuation

Risk Neutral Valuation

Options

Proof: Consider p as the probability of an up movement, the expected stock price E(ST) at time T:

𝐸 𝑆𝑇 = 𝑝𝑆0𝑈 + (1 − 𝑝)𝑆0𝐷

= 𝑝𝑆0𝑈 − 𝑝𝑆0𝐷 + 𝑆0𝐷

= 𝑝𝑆0(𝑈 − 𝐷) + 𝑆0𝐷

=𝑒𝑟𝜏 − 𝐷

𝑈 − 𝐷𝑆0(𝑈 − 𝐷) + 𝑆0𝐷

= 𝑆0𝑒𝑟𝜏 −𝑆0 𝐷 + 𝑆0𝐷

= 𝑆0𝑒𝑟𝜏

Thus stock price grows at risk free rate if p is the probability of an up movement.


Risk Neutral Valuation 𝑝 =𝑒𝑟𝜏−𝐷

𝑈−𝐷

Options A

Stock price today is equal to 20, and in 3 months it will be either 22 or 18.

What is a value of 3 month European call option with a strike price of 21.

Example

The risk free rate is 12% (continuous compounding).

p could be calculated as:

Thus non-arbitrage arguments and

risk-neutral valuation give the

same results.

22𝑝 + 18 1 − 𝑝 = 20𝑒0.12×312

4𝑝 = 20𝑒0.12×312 − 18 𝑝 = 0.6523

or as:

𝑝 =𝑒𝑟𝜏 − 𝐷

𝑈 − 𝐷=

𝑒0.12×312 − 0.9

1.1 − 0.9= 0.6523

thus: 𝑓 = 0.6523 × 1 + (1 − 0.6523) × 0 𝑒−0.12×312

= 0.6523𝑒−0.12×312

= 0.633

20f

22fU

18fD = 0

= 1


Risk Neutral Valuation

Options

50C

60A

40B

484

720

3220

t=0 t=1 t=2

𝑝 =𝑒0.05∗1 − 0.8

1.2 − 0.8= 0.6282

A:

0.6282*0+0.3718*4=1.4872

1.4872*exp(-0.05)=1.41668

B:

0.6282*4+0.3718*20=9.9488

9.9488*exp(-0.05)=9.463591

C: 0.6282*1.41668+0.3718*9.463591 = 4.40725

4.40725*exp(-0.05) = 4.192306

Two-Step Binominal Trees

In order to calculate the option price at the initial node of the tree, one needs to start

with calculating option price at the final nodes and then working out option price at

the earlier nodes.

Example: Consider 2-year European put option with a strike price of 52, whose stock is currently trading at 50.

There are two 1-year steps. In each step stock price can increase by 20% or decrease by 20%. The risk-free interest rate is 5%.

C

A

B

0

4

20

Is p constant in the whole tree?

Options Valuation: Binomial ModelOptions

Binomial Model → Black-Scholes Formula

n → ∞

n steps

Options Valuation: Binomial ModelOptions

The more terms the more similar BM and

BS valuation

Options Valuation: Binomial ModelOptions A

The formula for the value of a Call Option is:

And the formula for the value of a Put Option is:

𝑝𝑡 = exp −𝑟𝜏 𝐾Φ −𝑑2 − 𝑆𝑡Φ −𝑑1

𝑐𝑡 = 𝑆𝑡Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2

𝑑1 =𝑙𝑛

𝑆𝑡𝐾

+ 𝑟 +𝜎2

2𝜏

𝜎 𝜏

𝑑2 =𝑙𝑛

𝑆𝑡𝐾

+ 𝑟 −𝜎2

2𝜏

𝜎 𝜏= 𝑑1 − 𝜎 𝜏

where:

t is the current date

T is the expiry date

is the time to expiry.

St is the current (underlying) stock price.

K is the strike price.

r is the risk-free rate of interest.

σ is the volatility (σ2 is the variance of the

stock return, per unit time*)

Options Valuation: Black Scholes Model

Black-Scholes Formula

Options

Robert C. Merton

died in 1995

Long-Term Capital Management (LTCM)

The Black-Scholes formula calculates the price of European put and call options.

The model was proposed in 1973 by Fischer Black and Myron Scholes, who was later awarded the 1997 Nobel Prize in Economics

Science (joint with Robert C. Merton).

Myron ScholesFischer Black



Options

http://en.wikipedia.org/wiki/File:Fischer_Black.JPG

http://en.wikipedia.org/wiki/File:Fischer_Black.JPG

http://en.wikipedia.org/wiki/File:Myron_Scholes_2008_in_Lindau.png

http://en.wikipedia.org/wiki/File:Myron_Scholes_2008_in_Lindau.png

http://en.wikipedia.org/wiki/File:Robert_C._Merton.jpg

http://en.wikipedia.org/wiki/File:Robert_C._Merton.jpg


Options Valuation: Black Scholes ModelOptions A


At the beginning of 1998:

equity of $4.72 billion

borrowed over $124.5 billion

=>assets of around $129 billion.

debt to equity ratio of over 25 to 1

The total losses by end of 1998 were found to be $4.6 billion

$1.6 bn in swaps

$1.3 bn in equity volatility

$430 mn in Russia and other emerging markets

$371 mn in directional trades in developed countries

$286 mn in Dual-listed company pairs (such as VW, Shell)

$215 mn in yield curve arbitrage

$203 mn in S&P 500 stocks

$100 mn in junk bond arbitrage

Options Valuation: Black Scholes ModelOptions A

The original model involves the methods of stochastic calculus (continuous-time finance) – which we explored earlier

during the module. Here, we will consider a simplified version of the model.

In the Black-Scholes framework, the assumption about the evolution of the Stock price is:

𝑆𝑡+∆𝑡 = 𝑆𝑡exp 𝜇∆𝑡 + 𝜎 ∆𝑡𝑍

where Z ~ N(0,1), i.e. Z is a standard normal random variable.

0

10

20

30

40

50

60

70

0 20 40 60 80 100 120

time

S

Time series resulting from the above assumption, with µ = 0.04, σ = 0.02.



Options A

The price at expiry is:

It follows that:

𝑆𝑇 = 𝑆𝑡+τ = 𝑆𝑡exp 𝜇𝜏 + 𝜎 𝜏𝑍

𝑆𝑇

𝑆𝑡= exp 𝜇𝜏 + 𝜎 𝜏𝑍

and therefore:

𝑆𝑇

𝑆𝑡~𝐿𝑜𝑔𝑛𝑜𝑟𝑚𝑎𝑙 𝜇𝜏, 𝜎2𝜏



Options

Z ~ N(0,1)

~ N(𝜇𝜏, 𝜎2𝜏)

A

implied volatility


Working with Black-Scholes Formula

Options

Question

Suppose that the stock price at time zero is S0 = £90.The continuously compounded risk free

rate is 5%, and European call option written on S with strike price £100 and time to expiry τ

= 1 year has delta of 0.352 and trades for £2.5. Find the implied volatility of the stock to

the nearest 1%.

Φ 𝑑1 = 0.352 ⟹ 𝑑1 = −0.38 𝑐𝑡 = 𝑆𝑡Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2

2.5 = 90 ∗ 0.352 − exp −1 ∗ 0.05 ∗ 100 ∗ Φ −0.38 − 𝑠𝑖𝑔𝑚𝑎

2.5 = 31.68 − 95.123 ∗ Φ −0.38 − 𝑠𝑖𝑔𝑚𝑎

Φ −0.38 − 𝑠𝑖𝑔𝑚𝑎 = 0.306761

−0.38 − 𝑠𝑖𝑔𝑚𝑎 = −0.51

⟹ 𝑠𝑖𝑔𝑚𝑎 = 0.13 = 13%

𝑑2 = 𝑑1 − 𝜎 𝜏

1 − 0.352 = 0.648

Options Options Valuation: Black Scholes Model

put

stock

call

bond

STK

STK

0

ST

K-ST

ST

ST-K

K0

K

= =

Put-Call Parity

The put-call parity defines a relationship between the price of a call and a put – both with identical K and t.

It allows us to calculate c from p, and vice versa. The underlying assumption is that there is no arbitrage opportunities.

The parity is given by: 𝑝𝑡 + 𝑆𝑡 = 𝑐𝑡 + 𝐾𝑒−𝑟𝜏

We can prove this by considering two portfolios which always give the same payoffs at maturity:

(1) A put & a stock

(2) A call & a zero-coupon bond (or cash)

It can be shown that both portfolios give the same payoffs regardless of the terminal stock price.

ST > K ST < K

Therefore, their current values must be identical.

Put-Call ParityOptions

𝑝𝑡 + 𝑆𝑡 = 𝑐𝑡 + 𝐾𝑒−𝑟𝜏

Portfolio (1) is overpriced in relation to portfolio (2).

Buy securities in portfolio (2) and (short) sell those in portfolio (1):

Buy the call and short sell both the put and the stock.

This which will generate upfront positive cash flow that should be invested at risk free rate:

If the stock price at the expiration of the option is greater than K, the call will be exercised; if the price is less than K, the put

will be exercised. In either cases the arbitrageur will end up buying one share for K. This share can be used to close the short

position, thus the net profit is equal to:

(1) A put & a stock


𝑝𝑡 + 𝑆𝑡 − 𝑐𝑡 𝑒𝑟𝜏 − 𝐾

𝑝𝑡 + 𝑆𝑡 − 𝑐𝑡 𝑒𝑟𝜏

> 0

𝑝𝑡 + 𝑆𝑡 > 𝑐𝑡 + 𝐾𝑒−𝑟𝜏

Put-Call Parity

Put-Call Parity

Options


Portfolio (2) is overpriced in relation to portfolio (1).

Buy securities in portfolio (1) and (short) sell those in portfolio (2):

Buy the put and stock and sell the call.

To do this upfront positive cash flow will be needed, that should be borrowed at risk free rate:

If the stock price at the expiration of the option is greater than K, the call will be exercised; if the price is less than K, the put

will be exercised. In either cases the arbitrageur will end up selling one share for K. This money will be used to pay back the

loan, thus the net profit is equal to:

(1) A put & a stock


− 𝑝𝑡 − 𝑆𝑡 + 𝑐𝑡 𝑒𝑟𝜏 + 𝐾

− 𝑝𝑡 − 𝑆𝑡 + 𝑐𝑡 𝑒𝑟𝜏

> 0

𝑝𝑡 + 𝑆𝑡 < 𝑐𝑡 + 𝐾𝑒−𝑟𝜏

Put-Call Parity

Put-Call Parity

Options

Consider put-call parity

Let’s verify the put-call parity in the context of the Black-Scholes.


= 𝑝𝑡 +𝑆𝑡

= exp −𝑟𝜏 𝐾Φ −𝑑2 − 𝑆𝑡Φ −𝑑1 + 𝑆𝑡

= exp −𝑟𝜏 𝐾 1 − Φ 𝑑2 − 𝑆𝑡 1 − Φ 𝑑1 + 𝑆𝑡

= exp −𝑟𝜏 𝐾 − exp −𝑟𝜏 𝐾Φ 𝑑2 − 𝑆𝑡 + 𝑆𝑡Φ 𝑑1 + 𝑆𝑡

= exp −𝑟𝜏 𝐾 − exp −𝑟𝜏 𝐾Φ 𝑑2 + 𝑆𝑡Φ 𝑑1

= exp −𝑟𝜏 𝐾 + 𝑐𝑡

= 𝑅𝐻𝑆

𝐿𝐻𝑆

Put-Call Parity

Put-Call Parity with Black-Scholes

Options

Example: if we buy an American Call option with one year to expiry,

we can pay the strike price for the unit of the underlying at any time in the next year.

timeTt

today expiry

European Option

American Option

The right to early exercise is an advantage – the value of an American Option must be at least as high as a European option with

otherwise the same characteristics.

Value of American option can never fall below the current pay-off.

Example: if the strike is 100, and the current price of underlying is 70, the price of the put option must be at least 30.

Imagine that this condition is not met. Say the price of the option is 25.

You would buy the option for 25, and immediately exercise it, (short) selling the underlying for 100.

You would then buy it back for the current price of 70.

Your net (riskless) profit from your brief ownership of the option would be 5.

In symbols, this constraint is:

Note that there is also a constraint that the option value cannot be negative.

Why?

𝑉 ≥ 𝑚𝑎𝑥 𝐾 − 𝑆𝑡 , 0

when to exercise?

American Options are contracts that may be exercised before expiry (“Early Exercise”), whereas

European only on the expiry date

American Options

American Options

Options

There is a disadvantage:

the holder of an American Option needs to decide WHEN to Exercise;

this is not an easy decision.

0

20

40

60

80

100

120

140

160

0 100 200 300 400

The price of the underlying is 58.

If you exercise now (i.e. on day 189), your payoff is 42.

Do you exercise now, or do you wait?

0

20

40

60

80

100

120

140

160

0 100 200 300 400

It would have been better to wait until day 327, when the

price was 40, so payoff would have been 60.

But how were you to know this?

You are likely to formulate a rule:

As soon the price reaches S*, exercise the option.

S* will be called the optimal exercise point.

Example: American Put Option; Strike = 100; time to expiry 1 year. After 189 days, you are here:

American Options

American Options

Options

100

110

90

120

70

90

110

130

80

100

t=0.33t=0 t=0.67 t=1A= 5 * exp(-0.06*0.33)= 4.9

B = 20 * exp(-0.06*0.33)= 19.607

C= 2.45 * exp(-0.06*0.33)= 2.4019

D= 12.45 * exp(-0.06*0.33)= 12.265

E= 7.334 * exp(-0.06*0.33)= 7.225

We now assume that the option expires at time T, so the time to expiry is τ = T - t.

The easiest way to analyse this problem is in the context of the binomial model.

Example:

Let’s consider an American put option with time to expiry 1 year, and a strike of 100.

The current price of the underlying is 100.

Let us divide the time to expiry into three four-month intervals.

Assume that in any interval, the price can either rise by 10 or fall by 10 with equal probability.

The risk-free rate is 0.06 (continuously compounded).

B20

0A

30

10

0

0

00

At each node in the tree, we compare the

pay-off from exercising, with the

discounted expected pay-off from holding

on to the option.

Whenever the former exceeds the latter,

early exercise is rational.

C0

D

10

0

E

American Options with Expiry

American Options

American Options

Options

American Call Options

In the absence of dividends, early exercise is never rational on an American call option..

Proof:

You purchase an American call option with strike K and one year to expiry.

At some point in the next year (day t say), if the price of the underlying (St) is sufficiently far above K, you might consider

early exercise, pocketing the pay-off of St - K .

Instead of early exercise, you could do the following:

Hold on to the option, and short-sell one unit of the underlying, receiving an amount St at time t.

At expiry (T), either:

a. If ST < K : Buy the short-sold unit back at price ST.

Let the option die.

b. If ST < K : Exercise the option.

That is, pay K for a unit of the underlying.

Under (a), you receive St at t, and then lose an amount less than K at T.

Under (b), you receive St at t, and then lose an amount K at T.

Either way, this is better than exercising at t, with the pay-off St - K .

Hence, the value of an American call option is the same as the value of a European call option with the same strike and expiry date.

No such reasoning can be applied to put options. American puts are ceteris paribus more valuable than European puts.

American Options with Expiry

American Options

American Options

Options A

£ 96

£ 100

Up until now, we have been assuming that no dividends are paid on the underlying stock.

Let’s relax this assumption.

A dividend is paid to the holder of a stock on a particular date – let us call this the dividend date.

[In fact, for tax reasons, the amount by which the share value falls is slightly less than the amount of the dividend, but let us

ignore this complication.]

Example: if a share price is £100 immediately before the dividend is paid, and the dividend is £4,

we will assume that the share price will be

Immediately after the dividend date, ceteris paribus, the value of the stock will fall by an amount equal to the dividend payment

£96 on the day after the dividend date.

time

Didivend: £4

dividend date

Dividends

Dividends

Options

100

110

90

115

75

95

t = 0 t = 0.5 t=1 A= 15 * exp(-0.06*0.5)= 14.70592

B= 2.5 * exp(-0.06*0.5)= 2.450987

C= 8.57 * exp(-0.06*0.5)= 8.3167

Example: Binomial Model with 2 periods.

Consider a European call option with time to expiry one year, and strike price 90.

The current price of the underlying is 100.

Divide the time to expiry into two six-month intervals, and assume that in each interval, the price can rise by 10 or fall

by 10 with equal probability.

Further assume that a dividend of 5 is paid at dividend date five months into the life of the option.

Find the value of the option.

C

A

B

0

5

25

85

105

↓

↓The higher the dividend, the lower

the value of the call option.

Dividends

Dividends

Options

100

110

90

115

75

95

t = 0 t = 0.5 t=1

Example: Binomial Model with 2 periods.

Find the value of the put option, ceteris paribus.

B

0

A

15

0

0

85

105

↓

↓

The higher the dividend, the

higher the value of the put option.

A= 7.5 * exp(-0.06*0.5)= 7.35296

B=3.676* exp(-0.06*0.5)= 3.604323

Dividends

Dividends

Options

Case 1: Given dividends and dividend dates

If the amounts of the dividends and the dividend dates are given, a simple adjustment needs to be made:

Compute the present value of the dividend payments, discounted using the risk-free rate.

Then simply subtract this from the current stock price St.

Then apply the Black-Scholes formula in the usual way with this downward-adjusted stock price in place of St.

Example: if the current stock price is £100,

dividends of £2 will be paid after 3 months and 9 months,

the option expires in 12 months, and the risk-free rate is 0.08,

then the present value of the two dividend payments is:

We then subtract this amount from the current stock price:

We then use the Black-Scholes formula with 96.16 as the current price in place of 100.

2.0 exp −0.08 × 0.25 + 2.0 exp −0.08 × 0.75 = 3.84

100 − 3.84 = 96.16

Dividends in the Black-Scholes formula

Dividends

Dividends

Options

Case 2: Dividend given as a dividend rate

Sometimes, the dividend is given as an annual dividend rate.

The stock used in the example above paid dividends of £2 every six months, and therefore £4 each year.

Since the current stock price is £100, the dividend rate in this example is 4% (or 0.04).

Let the dividend rate be δ.

The parameter δ enters the Black-Scholes formula in the following way.

As usual, we need to define the two quantities:

The formula for the value of a Call Option is:

And the formula for the value of a Put Option is:

𝑐𝑡 = 𝑆𝑡exp(−𝛿𝜏)Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2

𝑝𝑡 = exp −𝑟𝜏 𝐾Φ −𝑑2 − 𝑆𝑡exp(−𝛿𝜏)Φ −𝑑1

𝑑1 =𝑙𝑛

𝑆𝑡𝐾

+ 𝑟 − 𝛿 +𝜎2

2𝜏

𝜎 𝜏

𝑑2 = 𝑑1 − 𝜎 𝜏

Dividends in the Black-Scholes formula

Dividends

Dividends

Options

The Greeks

A sensitivity is the change in the option value resulting from a ceteris paribus change in one of the model parameters.

The option price depends on five such parameters: τ, St, K, r, σ.

𝑐𝑡 = 𝑆𝑡Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2 𝑝𝑡 = exp −𝑟𝜏 𝐾Φ −𝑑2 − 𝑆𝑡Φ −𝑑1

𝑑1 =𝑙𝑛

𝑆𝑡𝐾

+ 𝑟 +𝜎2

2𝜏

𝜎 𝜏

𝑑2 = 𝑑1 − 𝜎 𝜏

The sensitivities are also known as the “Greeks”, and are named: delta, gamma, theta, vega, and rho.

Thy are calculated as a partial derivative of the option price/ value (V) with respect to parameter whose impact the

sensitivity is capturing.

𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦1 =𝜕𝑉𝑡

𝜕𝑃𝑎𝑟𝑎𝑚𝑒𝑡ℎ𝑒𝑟1

Delta Hedging: The GreeksOptions

Sensitivities of Black-Scholes Call Formula:

Delta is the rate of change of the option price with respect to the price of the underlying asset.

Name (symbol) Formula Sign

Delta (Δ) 𝜕𝑐𝑡𝜕𝑆𝑡

= Φ 𝑑1+

Example: Δ = 0.6: if the stock price changes by small amount the price of the option changes by 60% of that amount

Delta is the slope of curve that

relates the option price to the

price of underlying asset

Options Delta Hedging: The Greeks

Options Delta Hedging: The Greeks A

Gamma is the rate of change of the option delta with respect to the price of the underlying asset.

Example: Γ = 0.6: when stock price changes by ΔS, the delta changes by 0.6* ΔS.


Gamma (Γ) 𝜕2𝑐𝑡𝜕𝑆𝑡

2=

ϕ 𝑑1

𝜎𝑆𝑡 𝜏

+

Gamma is the second partial

derivative of the portfolio with

respect to the asset price.

It measures the curvature of the

relationship between the option

price and the stock price.



Options Delta Hedging: The Greeks A

Theta is the rate of change of the value of the option with respect to the passage of time.

To obtain the change per calendar day’ theta needs to be divided by 365.


Theta (Θ) 𝜕𝑐𝑡𝜕𝜏

= −𝑆𝑡𝜎ϕ 𝑑1

2 𝜏− 𝐾𝑟𝑒𝑥𝑝(−𝑟𝜏)Φ 𝑑2

-

Theta is usually negative because as

time passes the option tends to

become less valuable.



Vega is the rate of change of the value of the option with respect to the volatility of the underlying asset.


Vega (ν) 𝜕𝑐𝑡𝜕𝜎

= 𝑆𝑡 𝜏ϕ 𝑑1+

Example: ν = 12: 1% (0.01) increase in volatility (from 20% to 21%) increases the value of the option by

approximately 0.01 * 12= 0.12



Rho is the rate of change of the value of the option with respect to the interest rate.


Rho (ρ) 𝜕𝑐𝑡𝜕𝑟

= 𝜏𝐾𝑒𝑥𝑝(−𝑟𝜏) Φ 𝑑2+

Example: ρ = 5: 1% (0.01) increase in the risk free rate (from 5% to 6%) increases the value of the option by

approximately 0.01* 5 = 0.05





Delta (Δ) 𝜕𝑐𝑡𝜕𝑆𝑡

= Φ 𝑑1+

Gamma (Γ) 𝜕2𝑐𝑡𝜕𝑆𝑡

2=

ϕ 𝑑1

𝜎𝑆𝑡 𝜏

+

Theta (Θ) 𝜕𝑐𝑡𝜕𝜏

= −𝑆𝑡𝜎ϕ 𝑑1

2 𝜏− 𝐾𝑟𝑒𝑥𝑝(−𝑟𝜏)Φ 𝑑2

-

Vega (ν) 𝜕𝑐𝑡𝜕𝜎

= 𝑆𝑡 𝜏ϕ 𝑑1+

Rho (ρ) 𝜕𝑐𝑡𝜕𝑟

= 𝜏𝐾𝑒𝑥𝑝(−𝑟𝜏) Φ 𝑑2+

long position


Note that delta is always positive (for a Call).

The value of a Call Option always rises when the current stock price rises.

However, note that delta cannot be greater than one.

This means that the value of the option never rise by more than the rise in

the price of the underlying stock.

For a deep in-the-money call, since d1 is high, delta will be close to 1, and therefore the call price will

move penny for penny with the underlying stock.

deep in the money

Options Delta Hedging

Delta is sometimes called the hedge ratio.

Delta changes over time for two reasons:

1. The stock price (St) changes over time

2. The time to expiry (τ) falls over time (and since d1 involves τ, delta depends on τ).

Delta shows how many units of the underlying stock need to be short-sold for each call option

purchased for the position to be perfectly hedged over a short interval of time.

The position is perfectly hedged if losses made on the stock are offset by gains made on the option, or vice versa.

If you want your position to remain perfectly hedged, you will need to alter continuously the number of stocks held.

This is dynamic hedging. It can be costly.

Gamma tells us how much delta changes when the underlying price changes.

An option with a high gamma is little use for hedging, because the hedge would need to be readjusted constantly.


100

120c=20

90c=0

Example 1: Perfect Hedge: Binomial pricing model with one time-period.

c=20

c=0

The current price is 100. Between now and expiry, the price either rises to 120, or falls to 90.

The strike price of the call option is 100. If you purchase the option, you either gain 20 at expiry, or you gain 0 at expiry

How do you make sure that the amount you receive at expiry is the same, regardless of the price of the underlying?

In addition to purchasing the option, you sell w units of the underlying.

What if you do not have any units to sell? You short-sell.

The value of your portfolio at expiry is then:

Either 20 – 120w

You want these values to be the same, so:

20 – 120w = -90w

or 0 – 90w

→ w = 0.67

So, for each option that you purchase,

you need to short-sell 0.67 units of the

underlying.

This amounts to a “perfect hedge” since it

results in an outcome that is invariant to

the price of the underlying.


Delta Hedge

You use an Econometric Model (probably ARCH/GARCH) to forecast the

volatility of the underlying stock price.

If this volatility is higher than the implied volatility of the call option, you

conclude that the call option is under-priced, and you purchase it.

You paid an amount C (0.9704) for the call option. But, by your calculations, it is

worth V (1.244902), where V > C.

If the underlying stock price falls tomorrow, value of the call option falls.

Sell w units of the underlying stock.

Value of your portfolio:

How do we choose w?

In order to construct a perfect hedge, we need to ensure that the value of the portfolio is always the same whatever happens to

the price of the underlying (S), thus:

where k is some constant that does not depend on S.

Let’s differentiate both sides with respect to S.

How should you hedge against this loss?

What if you don’t have any to sell? Short-sell.

V – w*S

V – w*S = k


So, the number of units that we need to short-sell in order to create a perfect hedge is given by the option’s “delta”.

Recall that the delta of a vanilla call option is:

Δ =𝜕𝑉

𝜕𝑆= Φ 𝑑1

𝑑𝑉

𝑑𝑆− 𝑤 = 0 ⇒ 𝑤 =

𝑑𝑉

𝑑𝑆≡ ∆

where: 𝑑1 =𝑙𝑛

𝑆𝑡𝐾

+ 𝑟 +𝜎2

2𝜏

𝜎 𝜏This is delta-hedging.

A problem is that delta changes when S changes.

This means that every time the price of the underlying changes, the portfolio needs to be “re-hedged” in order to

maintain the fixed value of the portfolio.

Such re-hedging involves either further short-selling of the underlying (if S has risen) or buying back units of the

underlying (if S has fallen).

How important it is to re-hedge depends on the responsiveness of delta to changes in S. This responsiveness is given

by “gamma”:

An option with a high gamma is little use for hedging, because the hedge would need to be readjusted constantly.

Γ =𝜕2𝑐𝑡𝜕𝑆𝑡

2=

ϕ 𝑑1

𝜎𝑆𝑡 𝜏


Delta-hedged portfolio:

You are protected against small movements in the prices of underlying asset

As the price of underlying asset changes the delta changes as well

=> you need to re-hedge

=> dynamic hedging

Re-hedging usually done once a day

Gamma-hedged portfolio:

Between rebalancing at the trading times, Delta will drift away from zero as the underlying asset prices move.

If the portfolio is Gamma-hedged at the discrete trading times then the amount of such drift will be small

(comparable to the square of the change in underlying price).

You are protected against larger movements in the prices of underlying asset

Since underlying asset has Gamma = 0 ,

=> position in some other instrument that is non linearly depended on underlying asset needs to be taken;

=> this will affect delta, thus position in the underlying needs to be adjusted accordingly

Goal: Portfolio Delta = 0

Goal: Portfolio Gamma = 0

BTW: what is delta of a stock/

underlying asset?

1


Vega-hedged portfolio:

The underlying volatilities used in hedging calculations are all estimates.

If these are incorrect then delta hedging may be incorrect, consequently it is appropriate to attempt to immunise a

portfolio against (small) errors in volatility estimates.

Just as in delta hedging, achieving a portfolio Vega of zero achieves this.

You are protected against miss-specification of the volatility

Since underlying asset has Vega = 0,

=> position in some other instrument that is non linearly depended on underlying asset needs to be taken;

In order for portfolio to be both Vega and Gamma neutral position in 2 different instruments needs to be taken

Goal: Portfolio Vega = 0


Example:

Consider a delta neutral portfolio, with Gamma of -5000 and Vega of -8000.

Option 1 has Delta = 0.6, Gamma = 0.5 and Vega = 2;

Option 2 has Delta = 0.5, Gamma = 0.8 and Vega = 1.2.

To make portfolio Gamma and Vega neutral both Option 1 and 2 should be used:

and

where w1 is quantity of Option 1;

w2 is quantity of Option 2.

The delta of the portfolio after addition of Option 1 and 2 changes to:

Therefore 3 240 units of the underlying need to be sold to maintain delta neutrality.

−5000 +0.5𝑤1 + 0.8𝑤2 = 0

−8000 +2𝑤1 + 1.2𝑤2 = 0

𝑤1 = 400 𝑎𝑛𝑑 𝑤2 = 6000

= 3240400 × 0.6 + 6000 × 0.5


Options, when first sold, are usually close to at the money…

so have relatively high Gammas and Vegas….

with time the price of underlying usually changes enough to make option deep in the money or out of money…

thus both Gammas and Vegas are very small…

consequently focus on delta while hedging


Hedging, delta, Value of the Option…and Black-Scholes

What can we do?

1. Do nothing: have naked position

Consequences:

If ST < K option will not be exercised, we made £300 000

If ST > K option will be exercised, our cost is 100 000 (ST – K)

e.g. ST = £60, the cost is 10 * 100 000 = 1000 000, which is much greater than £300 000

2. Buy 100 000 shares as soon as we sold the option: covered position

Consequences:

If ST < K option will not be exercised, we have lost (S0 – ST)* 100 000 on position in stock

if ST = £40, we lost £900 000 on stock

If ST > K option will be exercised, we gain (K-S0)*100 000

3. Stop- loss strategy:

Buy stock if price raise above K, sell if it falls below K

Can be costly

4. Perfect Hedge would make the cost of option be equal to Black Scholes price…

Dynamic Delta Hedge

Only with dynamic delta hedge we have profit of £60 000

Example: We sold for £300 000 European call option on 100 000 shares of

a non-dividend paying stock. We also know that:

S0 = 49, K = 50, r = 0.05, σ = 0.2, τ = 0.3846

The Black- Scholes price of the option is £ 240 000.

Have we just made £60 0000 profit?

Not necessarily…

…there are risks



Simulation of delta hedging.

Option closes in the money.

Cost of hedging £263 300.

Option is exercised

We get 50*100 000 for the

shares we have.

2557.8 + 2.5 – 308 = 2252.3



Simulation of delta hedging.

Option closes out of the money.

Cost of hedging £256 600.

The difference in the cost

of hedging the position in

option and Black-Sholes

price come from the

frequency of hedge

rebalancing.

Still even weekly

rebalancing locks us in

profit…


European Options

Pay-off Diagrams,


Options

Option Valuation

Binomial Model

Black-Scholes Model

Put-Call Parity

American Options

Dividends

Delta Hedging

EXERCISE

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

6.Value at Risk

7.Credit Risk

Lecture 3:

6. Value at Risk

VaR

Calculating VaR:

Historical Simulation;

Model Building Approach

VaR of Option Portfolio

Value at Risk

p = (100 - X)% = 3%

Value at Risk (VaR)

Each of the Greeks (delta, gamma and vega) were describing different aspect of risk of a portfolio.

Thus, VaR is the loss level (V) over N days that has a probability of only (100 - X)% of being exceeded.

VaR is a function of:

(1) time horizon (N days) and

(2) confidence interval (X%).

VaR is the loss corresponding to the (100 - X)th percentile of the distribution of the gain in the value of the portfolio over the

next N days.

Example: If N = 5 and X = 97 what is the VaR?

Value at Risk (VaR) is a measure that attempts to summarize the total risk of a portfolio and evaluates ‘how bad things can get’.

VaR (V) can be best described by following statement:

‘We are X percent sure there will not be a loss of more than V in the next N days’.

gain over N days

VaR is a 3rd percentile of

the distribution of gain in

the value of the portfolio in

the next 5 days.

Value at Risk VaR: Introduction

Time Horizon

In practice N is usually set to 1 and the usual assumption is that:

This formula is true if changes in the value of portfolio have iind (with mean 0), otherwise it is just an approximation.

Example 1: 10-day 99% VaR can be calculated as:

10 = 3.162 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 1 − 𝑑𝑎𝑦 99% 𝑉𝑎𝑅

Example 2:

What is the relationship between volatility per year σyear (used in option pricing) and volatility per day σday (used at VaR).

Assuming 252 trading days it is:

𝜎𝑦𝑒𝑎𝑟 = 𝜎𝑑𝑎𝑦 252

Daily volatility is about 6% of annual volatility.

(𝑁 − 𝑑𝑎𝑦 𝑉𝑎𝑅) = (1 − 𝑑𝑎𝑦𝑉𝑎𝑅) × 𝑁

iind: independent identical normal distribution.

Value at Risk VaR: Introduction

Calculating VaR: Historical Simulation

The distribution of daily loss in the value of the portfolio depends on the value of x market variables (v).

Define vi as the value of a market variable on Day i and suppose that today is Day n.

The ith scenario assumes that the value of the market variable tomorrow will be:

Example 1: Calculate 1-day VaR using 99% confidence level on 501 days of data.

First identify the factors affecting the Value of the portfolio.

Next calculate the daily changes in those factors.

You should have 500 such changes.

Create scenarios of the value of the portfolio based on those changes.

Since there are 500 scenarios, the 99th percentile of the distribution is the 5th highest loss.

𝑣𝑎𝑙𝑢𝑒 𝑢𝑛𝑑𝑒𝑟 𝑖𝑡ℎ 𝑠𝑐𝑒𝑛𝑎𝑡𝑖𝑜 = 𝑣𝑛

𝑣𝑖

𝑣𝑖−1

Value at Risk Calculating VaR: Historical Simulation

Example 2: Today the value of the portfolio is 10 000USD.

There are 4 assets in the portfolio, and their value is: (1) DJIA: 4 000USD, (2) FTSE 100: 3 000USD, (3)

CAC 40: 1 000USD and Nikkei 225: 2 000USD.

What is the one-day 99% VaR?

Scenario 1: Value of DJIA

11022.06 ×11173.59

11219.38= 10977.08

value today1st possible growth rate


Scenario 1: Value of the portfolio

Thus the portfolio has a gain of 14 USD under Scenario 1.

4000 ×10977.08

11022.06+ 3000 ×

5180.40

5197+ 1000 ×

4229.64

4226.81+ 2000 ×

12224.10

12006.53= 10014


The one-day 99% VaR can be estimated as the 5th worst loss.

This is 253,385.


Calculating VaR: Model-Building Approach

Consider portfolio worth P that consist of n assets with an amount αi being invested in each asset i (1 ≤ i ≤ n).

Define Δxi as the return on asset i in one day.

The change in the value of the investment in asset i in one day is αi Δxi and the change in the value of the portfolio in one

day is:

If Δxi are multivariate normal, then ΔP is normally distributed.

Assume that the expected value of Δxi change is zero (in reality it is different than zero, but small in comparison to the

standard deviation), the mean of is zero.

Therefore to calculate VaR one need to calculate the standard deviation of ΔP (σP ):

where σi is the daily volatility of the ith asset, and ρij is the correlation coefficient between returns on asset i and asset j.

𝜎𝑝2 =

𝑖=1

𝑛

𝑗=1

𝑛

ρ𝑖𝑗α𝑖α𝑗σ𝑖σ𝑗

∆𝑃 =

𝑖=1

𝑛

𝛼𝑖∆𝑥𝑖

Value at Risk Calculating VaR: Model Building Approach

Example:

Consider a portfolio consisting of 10 000GBP of shares A, and 5 000GBP of shares B.

The returns on those two shares have bivariate normal distribution with a correlation of 0.3.

The volatility of A is 2% a day and the volatility of B is 1%.

What is 10-day 99% VaR?

Share A over 1-day period has a standard deviation of 200 GBP (= 10 000* 0.02), whereas share B of 50 GBP.

The standard deviation of the portfolio is therefore:

The mean change is assumed to be zero, and the change in the value of the portfolio is normally distributed.

N(-2.33) = 0.01 means that there is 1% probability that normally distributed variable will decrease in value by more than

2.33 standard deviations.

Therefore 1-day VaR is:

Whereas 10- day VaR is:

220 × 2.33 = 512.6

10 × 512.6 = 1620

𝜎𝑃 = 2002 + 502 + 2 × 0.3 × 200 × 50 = 220

Value at Risk Calculating VaR: Model Building Approach


Consider portfolio of options of single stock. Delta of the portfolio is:

Define Δx as the percentage change in the stock price in 1 day:

We know that the approximate relationship between ΔP and Δx is:

If there are several underlying, then approximate relationship between ΔP and Δxi is similar:

Define 𝛼𝑖 = 𝑆𝑖𝛿𝑖 we have:

𝛿 =Δ𝑃

Δ𝑆

∆𝑃 =

𝑖=1

𝑛

𝛼𝑖∆𝑥𝑖

∆𝑃 =

𝑖=1

𝑛

𝑆𝑖𝛿𝑖∆𝑥𝑖

∆𝑥 =Δ𝑆

𝑆

∆𝑃 = 𝑆δΔ𝑥

Value at Risk VaR of Option Portfolio

Example:

Portfolio consists of options on stock X and Y.

The option on stock X has delta of 1 and on stock Y of 20.

X trades for 120 and Y for 30.

What is 5 day 95% VaR?

It is approximately true that:

The portfolio is assumed to be equivalent to an investment of 120 in X and of 600 in Y.

Assuming that the daily volatility of X is 2% and of Y is 1% (and assuming the correlation is 0.3), the standard deviation of

is:

Since N(-1.65) = Φ(-1.65) = 0.05, thus 5-day 95% VaR is:

∆𝑃 = 120 × 1 × ∆𝑥1 + 30 × 20 × ∆𝑥2

= 120∆𝑥1 + 600∆𝑥2

1.65 × 5 × 7.099 = 26.19

120 × 0.02 2 + 600 × 0.01 2 + 2 × 120 × 0.02 × 600 × 0.01 × 0.3 = 7.099

Value at Risk VaR of Option Portfolio

positive gamma

When gamma is positive probability distribution of the value of the portfolio is positively skewed.

long call

Tend to have less heavy left tail than the

normal distribution.

If the distribution is assumed to be normal,

then the VaR is overestimated

When options are included in portfolio linear model is approximation.

Gamma of the portfolio should be taken into account as well.

Value at Risk VaR of Option Portfolio A

negative gamma

When gamma is positive probability distribution of the value of the portfolio is positively skewed.

When gamma is negative, it is negative skewed.

short call

Tend to have heavier left tail than the

normal distribution

If the distribution is assumed to be normal,

then the VaR is underestimated.

When options are included in portfolio linear model is approximation.

Gamma of the portfolio should be taken into account as well.


Thus both delta and gamma should be used to calculate the change in the value of portfolio.

In portfolio depended on single asset:

Using earlier formulas:

In portfolios with n underlying market variables, with each individual instrument depended on one market variable (7.18)

becomes:

∆𝑃 = 𝑆δΔ𝑥 +1

2𝑆2𝛾 ∆𝑥 2

∆𝑃 =

𝑖=1

𝑛

𝑆𝑖𝛿𝑖∆𝑥𝑖 +

𝑖=1

𝑛1

2𝑆𝑖

2𝛾𝑖 ∆𝑥𝑖2

∆𝑃 = δΔ𝑆 +1

2𝛾 ∆𝑆 2


VaR

Calculating VaR:

Historical Simulation;

Model Building Approach


Value at Risk

EXERCISE

7. Credit Risk

Credit Risk: Introduction

Credit Risk Models

Structural Models

Merton Model

Reduced- form Models

Two- and Multiple-state Models

Credit Risk in Derivatives Transaction

Credit Risk Mitigation

Credit Risk

The outcome of a default may be that the contracted payment stream is:

rescheduled

cancelled by the payment of an amount that is less than the default-free value of the original contract

continue but at reduced rate

totally wiped out.

failure to pay either capital or a coupon

loss event

bankruptcy

rating downgrade

Credit risk is a risk that borrowers and/or counterparties in derivative transaction will default, i.e. will fail to make a

payment they have promised.

Default-free bond is one where the stream of payments due from the bond will definitely be paid in full and on time.

Credit risk event is an event that will trigger the default of a bond and includes the following:

Credit Risk Introduction

It could be also defined as the market value of the bond a few days after a default.

It is expressed as a percentage of face value.

Source: Moody’s

Credit Risk Introduction: Recovery Rate

Recovery rate (R or δ ) is the fraction of the defaulted amount that can be recovered through bankruptcy proceedings or

some other form of settlement in case of default.

Only bonds of Baa or above rating are investment grade.

Probability of default of investment grade bonds tend to be an increasing function of time, whereas for bonds of

poor rating it is often a decreasing function of time.

Source: Moody’s

Credit Risk Introduction: Rating Agencies

Rating agencies (Moody’s, S&P) are providing ratings describing creditworthiness of corporate bonds.

0.025

1.793

It is calculated as a ratio of unconditional probability that the bond defaults in period t to the cumulative probability that

the bond will survive until period t (or until the end of period t-1) , i.e. (V(t)):

Example. Based on table above what is the 1-year hazard rate for B grade bond for 3rd year

(i.e. what is the probability that the bond defaults in year 3 providing it didn’t default in year 1 and year 2)?

Unconditional probability of default in year 3:

V(3) – V(4) =

= 100 – 6.460 – (100 – 9.424)

= 9.424 – 6.460

= 2.9640 (%)

The cumulative probability that the bond will survive until end of year 2:

V(3) =

= 100 – 6.460

= 93.54 (%)

The hazard rate at year 3:

λ (3) =

= 2.964 / 93.54

= 0.0317

𝜆 𝑡 =𝑉 𝑡 − 𝑉(𝑡 + 1)

𝑉(𝑡)

Credit Risk Introduction: Hazard Rate

Hazard rate λ(t) (or default intensity) is the conditional probability that a bond will default in time period t, providing it didn’t

default until period t (i.e. in previous periods).

Consider short period of time Δt.

The hazard rate λ(t) is defined so that λ(t)Δt is the probability of default between t and t+Δt conditional on no earlier default:

Taking limits:

From which

If Q(t) is the probability of default by time t, so that Q(t) = 1-V(t), then:

where 𝜆(𝑡) is the average hazard rate between time 0 and t.

𝜆 𝑡 ∆𝑡 =𝑉 𝑡 − 𝑉(𝑡 + 𝛥𝑡)

𝑉(𝑡)

−𝜆 𝑡 𝑉 𝑡 ∆𝑡 = 𝑉 𝑡 + 𝛥𝑡 − 𝑉(𝑡)

𝑑𝑉(𝑡)

𝑑𝑡= −𝜆 𝑡 𝑉(𝑡)

𝑉 𝑡 = 𝑒− 0𝑡𝜆 𝜏 𝑑𝜏

𝑄 𝑡 = 1 − 𝑒− 0𝑡𝜆 𝜏 𝑑𝜏

= 1 − 𝑒− 𝜆 𝑡 𝑡


Assume that the only reason corporate bonds sell for less than similar risk-free bonds is the possibility of default.

Then, the probability of default of a company can be estimated from prices of bonds it has issued:

where 𝜆 is the average hazard rate per year,

s is the spread of the corporate bond yield over risk-free rate and

R is expected recovery rate.

Example: A bond yields 300basis points more (or 3% more) than similar risk-free bond and the expected recovery rate is 50%.

Thus the holder of the bond is expecting to lose 3% in case of default.

The default probabilities implied from bond yields are risk-neutral probabilities.

The default probabilities implied from historical data are real-world default probabilities (physical probabilities).

There is a significant difference between those two probability types.

𝜆 =0.03

1 − 0.5= 0.06

𝜆 =𝑠

1 − 𝑅


Credit Risk Models

Structural models Reduced-form models

Intensity-based models

They are type of continuous-time

reduced form model that model

‘jumps’ between different states

(e.g. credit ratings).

The ‘jumps’ are modelled using

transition intensities.

These models are explicit

models of a corporate

entity issuing both debt

and equity.

They aim to link default

events explicitly to the

fortunes of the issuer.

They derive the default

probability from the

variation in the value of

firm assets.

Example: Merton model.

These models are statistical models

which use market statistics (such as

credit ratings) rather than specific

data relating to the issuer, and give

statistical models for their movement.

No assumptions are made concerning

why a default occurs. Rather, the

dynamics of default are exogenously

given by the default rate (or

intensity).

Prices of credit sensitive securities

can be calculated as if they were

default free using the risk free rate

adjusted by the level of intensity.

Credit Risk Credit Risk Models

Structural models: Merton model

Merton model assumes that the company’s equity is an option on the asset of the company.

Moreover it relates price of a bond (subject to default) to the price of the option.

Consider a company that issues both equity and debt (and doesn’t pay dividends nor coupon).

The total value of company asset at time t is V(t).

This value varies in time as a result of actions taken by the company.

Part of company value is zero-coupon debt with a promised repayment amount of D at future time T.

At time T the reminder of the value of company is distributed to shareholders.

Providing the company has enough funds to pay the debt, the shareholders will receive V(T) - D.

If V(T) < D the company will default.

In this situation the bondholders will receive V(T) instead of D and shareholders will receive nothing.

Therefore the bondholders receive:

whereas the shareholders receive:

B(T) = min[V(T), D]

E(T) = max[V(T)-D,0] at time T.

Credit Risk Credit Risk Models: Merton Model

From shareholders point of view the equity is equivalent to having a European call option on the assets of the company with

maturity T and strike price equal to the value of the debt:

where:

𝛷 𝑑2 represents the risk-neutral probability that the option will be exercised, which means that the debt will be repaid.

Therefore 1- 𝛷 𝑑2 = 𝛷 −𝑑2 represents the risk-neutral probability of default.

The current value of the company’s risky corporate bond can be found as:

B(t) = V(t)- E(t)

Total value of equity is equal to the market capitalization, thus E(t) is known.

What we don’t know is the company value V(t) and its volatility σV.

However under certain assumptions:

where σE is the volatility of the equity value that can be observed from market data.

This are 2 equations with 2 unknown variables V(t) and σV which can be solved numerically.

𝐸 𝑡 = 𝑉 𝑡 𝛷 𝑑1 − 𝐷𝑒−𝑟 𝑇−𝑡 𝛷 𝑑2

𝑑1 =𝑙𝑛

𝑉(𝑡)𝐷 + 𝑟 + 1/2𝜎𝑉

2 𝑇 − 𝑡

𝜎𝑉 𝑇 − 𝑡𝑑2 = 𝑑1 − 𝜎𝑉 𝑇 − 𝑡

𝜎𝐸𝐸 𝑡 = 𝜎𝑉𝑉 𝑡𝑑𝐸(𝑡)

𝑑𝑉(𝑡)= 𝜎𝑉𝑉 𝑡 𝛷 𝑑1


Example Company X has just issued 3-year zero-coupon bonds with a nominal value of £2.5 million.

The total value of Company X now stands at £3 million.

A constant (continuously-compounding) risk-free rate of return is 2% pa.

(a) Use the Merton model to calculate the theoretical price of £1000nominal of Company’s X bonds (assuming 25% of the

annual volatility of the value of the company’s assets).

(b) Estimate the risk-neutral probability of default on Company’s X bonds.

(a) Under Merton model shareholders in the company receive a payoff after 3 years equivalent to that from a call option

with strike price equal to the amount to be repaid to the bondholders.

Use BS formula for call option to assess the current value of a shareholding:

where

and

E(0) = 3 Φ(d1) - 2.5*exp (-0.02*3) Φ(d2)

d1 = ({ln(3/2.5)+(0.02+(0.25^2)/2)*3})/(0.25*3^0.5)

= 0.7761

d2 = d1-0.25*3^0.5

= 0.34311131


Example Company X has just issued 3-year zero-coupon bonds with a nominal value of £2.5 million.

The total value of Company X now stands at £3 million.

A constant (continuously-compounding) risk-free rate of return is 2% pa.

(a) Use the Merton model to calculate the theoretical price of £1000nominal of Company’s X bonds (assuming 25% of the

annual volatility of the value of the company’s assets).

(b) Estimate the risk-neutral probability of default on Company’s X bonds.

Thus the value of the bond at t0 is

So the theoretical price of 1000GBP nominal of these bonds is:

(b) Under Merton model the probability of default is:

E(0) = 3 Φ(0.7761) - 2.5*exp (-0.02*3) Φ(0.34311131)

= 0.85

B(0) = 3 – 0.85

= 2.15

(2.15/2.5)*1000 = 860

1-Φ(d2) = 0.36576= 1 – 0.63424


Reduced form models: Two-state credit rating models

Consider two-state intensity-based model, which is the simplest continuous-time reduced-form model

A simple model can be set up in continuous time with two states:

N = not previously defaulted

D = defaulted

Under this models it is assumed that the default-free interest rate term structure is deterministic with r(t) = r for all t.

If λ(t) represents the transition intensity from N to D at time t under the real-world measure P, the model can be represented

as:

This is two – stage continuous time Markov model.

If X(t) is the state at time t, then λ(t) can be interpreted in terms of the transition probabilities over the infinite small time

period [t, t+dt]:

as dt → 0,

and

as dt → 0.

where odt is smaller order quantity that disappears as the length of the time interval dt tends to zero.

λ(t)No default, N Default, D

P[X(t+dt) = N|X(t) = N] = 1 - λ(t)dt + odt

P[X(t+dt) = D|X(t) = N] = λ(t)dt + odt

Credit Risk Credit Risk Models: Reduced Form Models: Two state models A

Let B(t, T) be the price at time t of £1 of a zero-coupon bond that matures at time T, and λ𝑠 be deterministic, then:

where 1 − 𝑒𝑥𝑝 − 𝑡𝑇 λ𝑠𝑑𝑠 is the risk-neutral probability of default for the bond (Q(t)).

Example: A company has zero-coupon bonds that redeems in 1 year and the current price of £100 nominal is £94.50.

The continuously compounded risk-free rate is 1.5% for the next year.

Determine the implied risk-neutral probability of default for bond.

Assume that two-state model for credit rating holds, and a recovery rate of bond A is 50%.

Using the formula:

B(t,T) = exp(−r(T−t))*[1 − (1 − δ)(1 − exp(- 01𝜆𝑆 𝑑𝑠 ))]

where B is the bond price, λ is the risk-neutral default rate, (1-δ) is the recovery rate, and r is the risk-free rate:

so that

𝐵 𝑡, 𝑇 = 𝑒𝑥𝑝 −𝑟 𝑇 − 𝑡 1 − 1 − 𝛿 1 − 𝑒𝑥𝑝 − 𝑡

𝑇 λ𝑠𝑑𝑠

0.945 = exp(−0.015)*[1 − 0.5(1 − exp(− 01𝜆𝑆 𝑑𝑠))]

Q(bond defaults) = (1 – exp(− 01𝜆𝑆 𝑑𝑠)) = 0.081436

Credit Risk Credit Risk Models: Reduced Form Models: Two state models A

Reduced form models: Multiple-state credit rating models

Jarrow-Lando-Turnbull (JLT) model allow for more than 2 ratings (vs. default & non default previously).

It allows for n-1 credit ratings plus default.

Transfer is possible between all stated (but default state n).

Credit Risk Credit Risk Models: Reduced Form Models: Multiple state models A

Credit risk in derivative transactions

Credit exposure on derivative transactions is more complicated than on loan (bonds).

In case of (a) there is no credit risk related to the derivatives.

If the counterparty goes bankrupt there will be no loss.

The derivative is an asset to the counterparty, so in it is likely it will be retained, closed out or sold to third party.

In case of (b) there is always credit risk related to the derivative.

If the counterparty goes bankrupt the loss is likely to be experienced.

The derivative is counterparty’s liability, so in case of default of counterparty the holder will need to make a claim against

counterparty assets.

Derivatives in (c) may or may not have credit risk.

If the counterparty defaults when the value of derivative is positive then claim will be made towards the assets of the

counterparty and loss is likely to be experience.

If it defaults when derivative has negative value no loss is to be incurred.

One outstanding derivative contract can mean:

(a) always a liability (e.g. short option contract)

(b) always an asset (e.g. long option contract)

(c) either and asset or liability (e.g. forward contract)

Credit Risk Credit Risk in Derivative Transactions

Consider derivative that lasts until T and has a value of f0 assuming no default.

Default can take place at t0, t1, …tn (where tn = T),

The value of derivative at ti is fi.

qi is the risk neutral probability of default at ti (independent of the value of the derivative)

R is expected recovery rate (independent of the value of the derivative).

The exposure (max (fi, 0)) is the potential loss.

The risk neutral expected loss from default at time ti is:

where 𝐸 is expected value in a risk-neutral world.

In case of (a), fi is negative so above is equal to 0, therefore no adjustment of the value of derivative for credit risk is

needed.

In case of (b) above becomes qi(1-R)fi.

𝑞𝑖 1 − 𝑅 𝐸 max 𝑓𝑖 , 0

Credit Risk Credit Risk in Derivative Transactions

Credit risk mitigation

In many cases the analysis above overstates the credit risk in derivative transactions as there are number of clauses (in the

contract) that can mitigate the risk. For example:

Netting

If a counterparty defaults on one transaction, it must default on all outstanding transactions with that counterparty.

Collateralization

Transactions are valued periodically and collateral needs to be provided that equals the difference between the value of

transactions and certain threshold.

Downgrade Triggers

If the credit rating of the counterparty falls below certain level it is possible to close out the derivative transaction (at its

market value).

Credit Risk Credit Risk Mitigations

Credit Risk: Introduction

Credit Risk Models

Structural Models

Merton Model

Reduced- form Models

Two- and Multiple-state Models

Credit Risk in Derivatives Transaction

Credit Risk Mitigation

Credit Risk

EXERCISE

Question 1:

What is 1-year hazard rate for C grade bonds for 4th year if average cumulative default

rate for year 3 is 22.753 and for year 4 is 26.05?

Question 2:

To fund an expansion in its operations Company X has just issued 5-year zero coupon

bonds with a total face value of 10 million, taking its total asset value to 15 million.

(a) Explain how the value of the bonds can be expressed in terms of a European put

option.

(a) Hence calculate the fair price of a holding of company bonds with face value of 100

using Black-Scholes model, given that the price of a 5-year zero-coupon

government bond is 77.88. Assume that the annualized volatility of the company’s

asset’s over 5-year period is 25%.

(a) Explain what is meant by credit spread and calculate its value for the company

bonds.

Question 3:

Company Y has just issued 4-year zero-coupon bonds with a nominal value of £4

million. The total value of Company Y now stands at £5.5 million. A constant

(continuously-compounding) risk-free rate of return is 3% pa.

(a) Use the Merton model to calculate the theoretical price of £100 nominal of

Company’s Y bonds (assuming 35% of the annual volatility of the value of the

company’s assets).

(b) Estimate the risk-neutral probability of default on Company’s Y bonds.

9. Real Options

Test

Lecture 4:

9. Real Options

A

Real Options: Introduction

Market Price of Risk

Types of Real Options

Valuation of Real Options: Examples

Option Pricing in Equity Valuation

Real Options

FINANCIAL Options

OPTIONS

REAL Options

Real Options Introduction

The right, but not the obligation to undertake certain business activities, such as deferring, abandoning, expanding capital investment.

Represents decision or choices to be made during life of investment project

Deals with capital budgeting or resource allocation decision

Underlying can be illiquid, hard to trade or traded on inefficient market

The right, but not the obligation, to buy or sell underlying asset.

Underlying traded on liquid market

Real Options

NPV

Risk-free rate

Risk-neutral probabilities

Requires market price of risk (λ) for stochastic variables

Real Options

Valuation of potential capital investment project

NPV of a project = PV of expected future incremental cash flows

CF: Real life probabilities Risk-adjusted discount rate (from e.g. CAMP)

NPV > 0: undertake the project It creates value to the shareholders

Projects usually have options build within them Different risk than the project Different discount rate needed

𝑁𝑃𝑉 = −𝐶𝐹0 +𝐶𝐹1

1 + 𝑟+ ⋯+

𝐶𝐹𝑛1 + 𝑟 𝑛

λ =𝜇 − 𝑟𝑓

𝜎

𝑝 =𝑟𝑓 − 𝐷

𝑈 − 𝐷

Underestimates the value of the

project with embedded option

Premium should be

paid on project

with embedded option (vs

NPV)

Introduction

A bad investment…

today

+100

-120

100*0.5 + (-120)*0.5

= -10


A bad investment… becomes a good one

today

+100

-120

-20

+20

+80

-70

30

15

Learn at the 1st

stage

The keys to real option value come from

learning and adaptive behaviour.

NPV: expected cashflows from today’s point without considering

other pathways given what happens in the first year, etc.


NPV

Real Options

Can the valuation be the same?

If you modified decision tree analysis to:

Estimate risk-neutral probabilities to estimate an expected value, Adjust the expected value for the market risk in the investment and Use the riskfree rate to discount cashflows in each branch

… it could yield the same values as option pricing models


Suppose that real asset depends on several variables θi (i=1,2,..), where:

λi is the market price of risk of θi

Risk-neutral valuation: any asset dependent on θi can be valued by:

Reducing the expected growth rate of each θi from mi to mi - λisi.

Discounting cash-flows at the risk free rate

𝑑𝜃𝑖

𝜃𝑖= 𝑚𝑖𝑑𝑡 + 𝑠𝑖𝑑𝑊

𝑑1 =𝑙𝑛

𝐸(𝑉)𝐾

+𝜔2

2𝜏

𝜔 𝜏

𝐸 𝑚𝑎𝑥 𝑉 − 𝐾, 0 = 𝐸 𝑉 𝑁 𝑑1 − 𝐾𝑁 𝑑2

Assume V has lognormal distribution

𝑑2 =𝑙𝑛

𝐸(𝑉)𝐾

−𝜔2

2𝜏

𝜔 𝜏

where ω is the volatility of V

Real Options Market Price of Risk

Example: Current cost of renting 1m2 is £30. Cost is quoted as amount per 1m2 per year in 5-year

rental agreement. The expected growth rate in the cost is 12% pa, volatility 20% pa, market price of

risk 0.3. What is the value of an opportunity to pay £1m now for option to rent 100 000m2 at £35

for 5 years in 2 years time? Assume 5% risk-free rate pa.

Let V be cost per 1m2 in 2 years time.

The pay off from the option is: 100 000 × 𝐴 × 𝑚𝑎𝑥(𝑉 − 35, 0) call

The expected pay off in risk neutral world:

A: annuity factor4.5355

Expectations in risk neutral world

100 000 × 4.5355 × 𝐸 𝑚𝑎𝑥(𝑉 − 35, 0)

= 453 550 × 𝐸 𝑚𝑎𝑥(𝑉 − 35, 0)

= 453 550 × 𝐸 𝑉 𝑁 𝑑1 − 35𝑁 𝑑2

𝑑1 =

𝑙𝑛 𝐸(𝑉)35

+0.22

2 2

0.2 2

𝐸 𝑉 = 30 exp((𝑚𝑖 − 𝜆𝑖𝑠𝑖) ∗ 2)

= 30 exp((0.12 − (0.3 ∗ 0.2)) ∗ 2)

= 30 exp(0.06 ∗ 2)

= 33.83

= 1 501 500

Value of the option:

1 501 500 ∗ exp(−0.05 ∗ 2) = 1 358 600

The opportunity is worth: 1 358 600 − 1 000 000 = 358 600


When historical data are available market price of risk can be estimated using:

λ: market price of risk

ρ: instantaneous correlation between the percentage change in the variable and returns on stock

market index

σm: volatility of return on stock market index

μm: expected return on stock market index

rf: short term risk-free rate

λ =𝜌

𝜎𝑚𝜇𝑚 − 𝑟𝑓

Example:

Percentage changes in company’s sale have a correlation of 0.3 with returns on FTSE100 index. The

volatility of FTSE100 returns is 20%pa, the expected excess returns of FTSE100 over risk-free rate

is 5%.

The market price of risk is:

λ =0.3

0.2∗ 0.05 = 0.075


Types of Real Option

Most investment projects involve options. Those options can add substantial value to the project.

Examples of options embedded in the project:

Option to Abandon

Option to sell or close down a project.

It is an American put option on the project’s value with the strike price being the

liquidation (or resale) value less closing- down costs.

It mitigates impact of poor investment performance.

Option to Expand

Option to make further investments if conditions are favourable.

It is an American call option on the value of additional capacity. The strike price is the cost

of creating this additional capacity discounted to the time of option exercise.

The strike price depends on initial investment.

Option to Wait/ Delay

This is an American call option on the value of the project.

R&D

Oil exploration

Patent

Project can include more than one option.

Real Options Types of Real Options

Black-Scholes

OPTION Valuation Models

Binomial Model

Can price American Option

Majority of real options are exercised before maturity (early exercised)

Often underlying assets are discontinuous

Binomial tree with outcomes at each node looks like a decision tree from capital budgeting.

For European option without dividend

Can be adjusted for dividend

What about American option?

American Call Option will never be exercised prior maturity…

Still getting the inputs to a binomial model can be difficult…

Real Options Option Valuation

Valuing product patent as an option

A patent provides the company with the right to develop and market the product.

The product will be developed and marketed only if the present value of the expected cash flows

from the product sales (V) exceed the cost of development (I).

If this does not occur, the patent will not be used and non further cost will be incurred.

The payoffs from owning a product patent can be written as:

Max ( 0, V – I )

.

VI

Pay-off

Real Options Option Valuation: Examples

Input Estimation Process

St: Value of the UnderlyingAsset

PV of Cash Flows from taking the project now

σ2: Variance in value of underlying

Variance in CF of similar asset or firm Variance in PV from capital budgeting

simulation

K: Exercise Price on Option Cost of making investment in the project

τ: Expiration of the Option Life of the patent

δ: Dividend Yield Cost of delay Each year of delay means less years of CF

𝑐𝑡 = 𝑆𝑡exp(−𝛿𝜏)Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2𝑑1 =

𝑙𝑛𝑆𝑡𝐾 + 𝑟 − 𝛿 +

𝜎2

2 𝜏

𝜎 𝜏

𝑑2 = 𝑑1 − 𝜎 𝜏


Example:

Company X, a bio-technology company, has a patent on ABC, a drug to treat multiple sclerosis, for the

next 17 years. X plans to produce and sell the drug by itself.

The key inputs on the drug are as follows:

St: PV of Cash Flows from Introducing the Drug Now = £3.422 billion

K: PV of Cost of Developing Drug for Commercial Use = £2.875 billion

τ: Patent Life = 17 years

r: Riskless Rate = 6.7% (17-year T.Bond rate)

σ2: Variance in Expected Present Values = 0.224 (Industry average firm variance for bio-tech

companies)

δ: Expected Cost of Delay = 1/17 = 5.89%

Implementing BS model:

d1 = 1.1362 N(d1) = 0.8720

d2 = -0.8512 N(d2) = 0.2076

Call = 3,422*exp(-0.0589*17)*(0.8720) - 2,875*(exp(-0.067*17)*(0.2076)

= £907 million


NPV of this project:

= 3422 – 2875

= £547 million


Valuing natural resources

The underlying asset is the resource and the value of the asset is based upon two variables: (1) the

quantity and (2) the price of the resource.

Usually there is a cost associated with developing the resource, and the difference between the

value of the asset extracted and the cost of the development is the profit to the owner of the

resource.

Define the cost of development as X, and the estimated value of the resource as V.

The payoffs from a natural resource option can be written as:

Max ( 0, V – X )

VX

Pay-off


Input Estimation Process

St: Value of Available Reserves of the Resource

Expert estimates (Geologist) PV of cash flows from the recourse

σ2: Variance in value of underlying

Based on variability of the price of the resource and variability of available reserves

K: Cost of Developing Reserve Past costs and the specifics of the investment

τ: Time to Expiration Relinquishment Period Time to exhaust inventory

δ: Net Production Revenue

(Dividend Yield) Net production revenue every year as per cent of

market value

Development lag Calculate PV of reserve based upon the lag

𝑐𝑡 = 𝑆𝑡exp(−𝛿𝜏)Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2𝑑1 =

𝑙𝑛𝑆𝑡𝐾 + 𝑟 − 𝛿 +

𝜎2

2 𝜏

𝜎 𝜏

𝑑2 = 𝑑1 − 𝜎 𝜏

Uncertainty about:Price, Quantity, Costs


Example:

Consider oil reserve of 50 million barrels, with PV of the development cost $12 per barrel and the

development lag 2 years. Company X has the right to exploit this reserve for the next 20 years. The

marginal value per barrel of oil is $12. Once developed, the net production revenue each year will be

5% of the value of the reserves.

The key inputs on the drug are as follows:

St: Value of developed reserve discounted back the length of development lag at the

dividend yield = $12*50/(1.05)^2= $ 544.22 million

K: PV of development Costs= $12 * 50 = $ 600 million

τ: Time to expiration of the option = 20 years

r: Riskless Rate = 8%

σ2: Variance in ln(oil prices) = 0.03

δ: Dividend yield= Net production revenue/Value of reserve = 5%

Implementing BS model:

d1 = 1.0359 N(d1) = 0.8498

d2 = 0.2613 N(d2) = 0.6030

Call = 544 .22 exp(-0.05*20) (0.8498) -600 (exp(-0.08*20) (0.6030))

= $97.08 million


NPV of this project:

= 544.22 – 600

= - $55.78 million



Equity in a troubled company

Company with high leverage, negative earnings and a significant chance of

bankruptcy

Equity can be viewed as a call option (option to liquidate the company).

Natural resource companies

The undeveloped reserves can be viewed as options on the natural resource.

Start-ups or high growth companies

Companies which derive the bulk of their value from the rights to a product or a

service (eg. a patent)

In late 90s dot.com companies were valued as options to enter e-commerce market

Huge premiums

One could invest in Nokia or GE to enter the same market (lack of exclusivity)

Real Options Option Pricing in Equity Valuation

Real Options: Introduction

Market Price of Risk

Types of Real Options

Valuation of Real Options: Examples


Real Options

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