Chapter 11: The Black-Scholes Model

Setting up the continuous time model

we wish to look at the return on investments... how should this be modeled?

For the Black-Scholes model we model the return on investments...

contingent claim
(continuous time)
ECC

Portfolio
(continous time)

Value of a portfolio
(continuous time)

A portfolio is said to be replicating if....
and self-financing if....

In out B-S model, self-financing means:

Stochastic exponential and SDE it satisfies

What is the probability measure that we define in order for the discounted stock price to be a martingale?

The probability measure for which the discounted stock price is a martingale

What type of stochastic process does the discounted stock price follow?

Claim martingale

Apply the martingale representation theorem to the claim martingale

Black-Scholes Portfolio

Value of the Black-Scholes portfolio

arbitrage price of an option under the B-S model

Derive the B-S pricing formula for a general ECC

Black-Scholes pricing formula for a general ECC, X

Black-Scholes pricing formula for a European Call option

Black-Scholes PDE

Delta hedging

Foreign Exchange set up

What does the market look like from the point of view of a dollar investor?

What is the arbitrage price (in dollars) of a contingent claim X?

What does the market look like from the point of view of a sterling investor?

What is the arbitrage price in sterling of a contingent claim X?

So that arbitrage opportunities are not created there must be certain relationships between U(t) and V(t), and the respective martingale measures, what are these?

Continuous dividend payments, continuously reinvested into the stock

How do you price Z(t)?

dividends paid at regular intervals, payments immediately reinvested into the stock