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
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
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
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
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