Financial Markets

1
Up and out European call

• European call

Knock out condition

• Value of the option at time 0

1. Find the underlying pdf using reflection
principle
and the Girsanov theorem
2. Compute the expectation
- Complete the square in the exponent
- Recognize a cumulative normal distribution
2
Stopping time
3
Black-Scholes equation

• Itos formula for

• Integrate from 0 to

4

0
5
Asian options

• The payoff depends on the average price of stock

e.g.,
- Value of the option at time 0
- Auxiliary process
6
Markov and martingale properties
- Markov property
- Martingale property
7
Itos formula

• Itos formula for

8
Feynman-Kac formula

0
(Martingale property)
9

• Value of the option at time t

10

• Multi-dimensional market model

• d-dimensional Girsanov theorem

• An outside barrier option

11
d-dimensional Girsanovs theorem
12
Multi-dimensional market model

• m stocks

• Discount factor

• Itos rule

13
(No Transcript)
14

0
15
An outside barrier option
- Barrier process
- Stock

0
- Payoff of the option
16
Market price of risk equations

r

r
Unique solution
17
Markov Martingale properties

• Markov property

• Martingale property

18
Itos formula

• Itos formula for

19
Black-Scholes equation
20
Conditional Expectation Martingale theory Markov
processes Ito integral and Itos
formula Black-Scholes formula Girsanov theorem
Pricing and Hedging on - European option -
American option - Exotic option
21
Stochastic Simulation in Finance
The End
Yuji Yamada
Control and Dynamical Systems California
Institute of Technology
First Term Fall 2001