Derivative Financial Products

Donald C. WilliamsDoctoral Candidate

Department of Computational and Applied Mathematics, Rice University

Thesis AdvisorsDr. Richard A. Tapia, Department of Computational and Applied Mathematics

Dr. Jeff Fleming, Jesse H. Jones Graduate School of Management

10 September 2003

Computational Finance Seminar

Outline

• Motivation– Derivative Securities Markets

– Nature of Derivative Financial Products

• Modeling– Relevant Parameters

– Option Valuation Problem

• Mathematical Machinery• Concluding Remarks

Derivative Securities Markets

The valuation of various derivative securities is an area of extreme importance in modern finance theory and practice:

• Market Growth. Gross market value of outstanding over-the-counter derivative contracts stood in excess of $US 128 trillion (BIS, 2002)

• Product Innovations. New derivative products are becoming more complex to fit desired exposure of clients.

• Quantitative Evolution. Valuation and hedging techniques must evolve to effectively manage financial risk.

• Decision Science. Corporate decision strategy.

Nature of Derivatives

A derivative (or derivative security) is a financial instrument whose value depends on the value of other, more basic underlying assets.

What is a derivative?

Nature of Derivatives

• Equity (e.g., common stock)

• Agricultural (e.g., corn, soybeans)

• Energy (e.g., oil, gas, electricity)

• Bandwidth (e.g., communication)

100 sharesIBM

class ACommon

Basic Underlying Assets

General Derivative Contracts

• Forward Contracts• Futures Contracts• Swaps• Options

An option is a particular type of derivative security that gives the owner the right (without the obligation) to trade the underlying asset for a specific price (the strike or exercise price) at some future date.

Market Structure

Financial Institution

A

Exchange

General Financial Market

Over-The-Counter (OTC)

Financial Institution

B

Financial Institution

C

Individual A

Individual B

Individual B

Option Contract Specification

An American-style option is a financial contract that provides the holder with the right, without obligation, to buy or sell an underlying asset, S, for a strike price K, at any exercise time where T denotes the contract maturity date.

T,0

Basic Financial Contracts:

An European-style option is similarly defined with exercise restricted to the maturity date, T .

Option Types

A call option gives the holder the right to buy the underlying asset.

A put option gives the holder the right to sell the underlying asset.

Two Basic Option Types:

Payoff: Fundamental Constructs

Payoff Functions

ST STKK

ST STKK

Call Option)0,max(),( KStS

)0,max(),( SKtS Put Option

Long position Short position

Ex. : Put Option - Hedge

Example. ABC is an oil company that will produce a 1,000 barrels of oil this

year and sell them in December. The expected selling price is $20/bbl. Assume

ABC can buy a put option contract (hedge) on a thousand barrels of oil for

$500, with strike, X=$20/bbl and December expiration.

Unhedged-2,500-5001,5003,5005,500

Hedged1,0001,0001,0003,0005,000

Cash flow variations with the price of oil:

Oil Price16.0018.0020.0022.0024.00

Revenue16,00018,00020,00022,00024,000

Fixed Cost18,50018,50018,50018,50018,500

Put Payoff3,5001,500-500-500-500

Profits

Ex.: Call Option - Speculation

Profit from buying an IBM European call option: option price = $5, strike price = $100, option life = 2 months

30

20

10

0-5

70 80 90 100

110 120 130

Profit ($)

Long Call on IBM:

)0,max(),( KSTS TT

Maturity @

PriceAsset

TS

Modeling

How do we mathematically model the value of option contracts?

Basic Question:

Financial specificationsand intuition

Highly sophisticatedquantitative models

Modeling

Option valuation models establish a functional relationship between the traded option contract, the underlying asset, and various market variables (e.g., asset price volatility).

Modeling

Idea: Express the value of the option as a function of the underlying asset price and various market parameters, e.g.,

drtSTKSftSU ,,,,,,),(

• S and t are asset price and time

• volatility of underlying asset price

• K and T are contract specific parameters

• r is the interest rate associated with underlying currency

• d is the expected dividend during the life of the option

Modern Option Valuation

• Uses continuous-time methodologies– (1900) Louis Bachelier (one of the 1st analytical treatments)– (1973) Fisher Black and Myron Scholes– (1973) Robert Merton

>>>> Black-Scholes-Merton PDE <<<< >>> 1998 Nobel Prize in Economic <<<• Employs mathematical machinery that derives from

– Stochastic calculus– Probability– Differential equations, and– Other related areas.

Modeling Building Framework

• Define State Variables. Specify a set of state variables (e.g., asset price, volatility) that are assumed to effect the value of the option contract.

• Define Underlying Asset Price Process. Make assumptions regarding the evolution of the state variables.

• Enforce No-Arbitrage. Mathematically, the economic argument of no-arbitrage leads to a deterministic partial differential equation (PDE) that can be solved to determine the value of the option.

Asset Price Evolution

At the heart of any option pricing model is the assumed mathematical representation for underlying asset price evolution.

By postulating a plausible stochastic differential equation (SDE) for the underlyinging price process, a suitable mathematical model for asset price evolution can be established.

Asset Price: Single Factor Model

This fundamental model decomposes asset price returns into two components, deterministic and stochastic, written in terms of the following SDE

This geometric Brownian motion is the reference model from which the Black-Scholes-Merton approach is based.

tt

t dWdtS

dS

Asset Price Trajectory

Assume: SDE models asset price dynamics

tttt dWSdtSdS

x

t

t

x

t

t

xt dWSdxSSS 00

0

00with SSt

European Option Valuation Problem

Black-Scholes-Merton Equation

0)2

( 22

rUrSUUSU SSSt

where appropriate initial and boundary conditions are specified.

Concluding Remarks

• Black-Scholes Assumption– The market is frictionless

– There are no arbitrage opportunities

– Asset price follows a geometric Brownian motion

– Interest rate and volatility are constant

– The option is European

• Circumventing the limitations inherent in the aforementioned assumption is a large part of option pricing theory.

The value, U(S,t), of an American option must satisfy the followingpartial differential complementarity problem (PDCP):

rateinterest freerisk denotes

payoffoption specifies ),(

)2

()],(L[ 22

r

tS

rUrSUUSUtSU SSSt

0)]),())(L[,(),((

0),(),((

0)],(L[

tSUtStSV

tStSV

tSU

where,

American Option Valuation Problem

Results from Stochastic Volatility Model