Mathematics in Finance

Binomial model of options pricing.

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 - OptionsExample 1:

Long Call on stock S

with strike K=32,

maturity T,

price P=10.

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

strike

underlying maturity

volatility

Interest rate

Option value

dividends

Derivatives - Options

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

Problem: How can options be priced?

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

Replicating portfolio

Binomial one period method

Binomial one period method

Binomial one period method

Binomial n-period method

Binomial n-period method

Binomial n-period method

Binomial n-period methodAlgorithm for binomial method

Example

150

120

96

187.5

120

76.8

234.38

96

150

61.44

0

164.38

80

26

120.09

52.92

13.59

85.37

33.46

58.91

Numerical implementation

Some versions of binomial model

Extensions of binomial model

Black-Scholes formula

Conclusions