Mathematics in Finance

Introduction to financial markets

What to do with money?

• spend it– car– gifts– holiday– ...

• invest it– savings book– bonds – shares– derivatives– real estate– ...

I Savings book

• Lending K€, getting K(1+r)€ after a year

• bank hopes to earn a higher return on K than r

• (for example by lending it)

• practically no risk

Risk free interest rate r

• can be obtained by investing with no risk

• USA: often interest which the government pays

• Europe: EURIBOR (European Interbank Offered Rate)

• positive.• discount factor

–100 today 100(1+r) in one year

–100 in one year 100/(1+r) today

II BondsAn IOU from a government or company.

In exchange for lending them money they issue a bond that promises to pay you back in the

future plus interest.

• (IOU = investor owned utilities)• Fixed-interest bonds• Floating bonds• Zero bonds

III SharesCertificate representing one unit of ownership

in a company.

• Shareholder = owner• Particular part of nominal capital• Traded on stock exchange • No fixed payments

Earnings per share: EPS = +0

01

PPP

0

1

PDiv

IV DerivativesA derivated financing tool. Its value is derivated

from an underlying.

• Underlyings: shares, bonds, weather, pork bellies, football scores, ...

• Different derivatives:1. Forwards2. Futures3. Options

IV Derivatives - ForwardsAgreement to buy or sell an asset at a certain future

time for a certain price. Not normally traded on exchange.

• Over the counter (OTC)• Value at begin: Zero• Agree to buy long position• Agree to sell short position

IV Derivatives - FuturesAgreement to buy or sell an asset at a certain time

in future for a certain price. Normally traded on exchange.

• Standardized features• Agree to buy long position• Agree to sell short position• Exchanges: CBOT, CME, ...

IV Derivatives - OptionsGive the holder the right to buy or sell

the underlying at a certain date for a certain price. (European options)

• Right to buy call option• Right to sell put option• Payoff function• Cash settlement• Exchanges: AMEX, CBOT, Eurex, LIFFE, EOE, ...

IV Derivatives - OptionsDenotations:• Strike you can buy or sell for that price• Maturity date when the option expires• Buy option long position (holder)• Sell option short position (writer)

Exercising ...... only at maturity possible European... at any date up to maturity possible American

IV Derivatives - OptionsExample 1:

Long Call on stock S

with strike K=32,

maturity T,

price P=2.

Payoff function:

f(S) = max(0,S(T) – K)-4

-2

0

2

4

6

8

10

12

14

16

1 5 9 13

17

21

25

29

33

37

41

45

underlying at T

IV Derivatives - OptionsExample 2 (how to use options):

1.1.: 100 shares of S, each 80 €30.6: must pay 7500€ (by selling the shares)

Problem: price of shares could fall under 75€Solution: buy 100 puts with strike 77 each option costs 2

Result: S(T) > 77 you have > 7700€ -200€ S(T) < 77 you have = 7700€ -200€

IV Derivatives - OptionsExample 3 (how to use options):

Situation: You think the prices of S will raise & want to profit from that. One share costs 100€. You have 10000€.

Solution 1: you buy 100 shares.Solution 2: you buy calls (10€) with strike 100.

Result if the prices raise to 120:Case 1: your profit 100*20€ = 2000€Case 2: your profit 1000*20€-1000*10€ = 10000€

IV Derivatives - OptionsExample 4 (how to use options):

Call with strike 105 costs 2€ each

Put with strike 110 costs 2€ each (same maturity)

Action: Buy 100 calls and 100 puts.

Result at T: Costs 200*2€ = 400€Income (110€-105€)*100 = 500€

Riskless profit (arbitrage)

IV Derivatives - Options

Other options:• Spreads f(S)=max(0,K-S)+max(0,S-K) • Strangles f(S)=max(0,K-S)+max(0,S-L)• Pathdependant options:

– Floating rate optionsF(S) = max(0,S(T)-mean(S))

– ...• Options on options• ...

strike

underlying maturity

volatility

Interest rate

Option value

dividends

II Derivatives - Options

Call Putstrike up down upunderlying up up downmaturity approaching down downvolatiliy up up upinterest rates up ~ ~dividends are paid down up

Summary

Assets:• Savings book (risk free)• Bonds• Shares• Derivatives Futures

ForwardsOptions

Problem: How can options be priced?

– Modelling– Black-Scholes– Solving partial differential equations– Monte-Carlo simulation– ...