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Quantitative Finance Lecture 6

AD N AN K H AN & F E R H AN A AH M E D

Arithmetic Brownian Motion

Consider

To ‘solve’ this we consider the process

From extended Ito’s Lemma

Ito Isometry

A shorthand rule when taking averages

Lets find the conditional mean and variance of ABM

Mean and Variance of ABM

We have using Ito Isometry

Geometric Brownian Motion

The process is given by

To solve this SDE we consider

Using extended form of Ito we have

Black Scholes World

The value of an option depends on the price of the underlying and time

It also depends on the strike price and the time to expiry

The option price further depends on the parameters of the asset price such as drift and volatility and the risk free rate of interest

To summarize

Assumptions

The underlying follows a log normal process (GBM)

The risk free rate is know (it could be time dependent)

Volatility and drift are known constants

There are no dividends

Delta hedging is done continuously

No transaction costs

There are no arbitrage opportunities

Derivation

We assumed that the asset price follows

Construct a portfolio with a long position in the option and a short position in some quantity of the underlying

The value of this portfolio is

Derivation

Q: How does the value of the portfolio change?

Two factors: change in underlying and change in option value

We hold delta fixed during this step

Derivation

We use Ito’s lemma to find the change in the value of the portfolio

The change in the option price is

Hence

Derivation

Plugging in

Collecting like terms

Derivation

We see two type of movements, deterministic i.e. those terms with dt and random i.e. those terms with dW

Q: Is there a way to do away with the risk?

A: Yes, choose in the right way

Reducing risk is hedging, this is an example of delta-hedging

Derivation

We pick

Now the change in portfolio value is riskless and is given by

Derivation

If we have a completely risk free change in we must be able to replicate it by investing the same amount in a risk free asset

Equating the two we get

Black Scholes Equation

We know what should be

This gives us the Black Scholes Equation

Black Scholes Equation

This is a linear parabolic PDE

Note that this does not contain the drift of the underlying

This is because we have exploited the perfect correlation between movements in the underlying and those in the option price.

Black Scholes Equation

The different kinds of options valued by BS are specified by the Initial (Final) and Boundary Conditions

For example for a European Call we have

We will discuss BC’s later

Variations: Dividend Paying Stock

If the underlying pays dividends the BS can be modified easily

We assume that the dividend is paid continuously

i.e. we receive in time

Going back to the change in the value of the portfolio

Variations: Dividend Paying Stock

The last terms represents the amount of dividend

Using the same delta hedging and replication argument as before we have

Variations: Currency Options

These can be handled as in the previous case

Let be the rate of interest received on the foreign currency, then

Variations: Options on Commodities

Here the cost of carry must be adjusted

To simplify matters we calculate the cost of carrying a commodity in terms of the value of the commodity itself

Let q be the fraction that goes toward the cost of carry, then