Derivation of BS equation

In the BS world, the evolution of the stock price S is given by:   Bs 1  for MuOmega > 0. 

Assumptions: 

- stock does not pay any dividends

- no transaction costs

- continuously compounding IR is r > 0 (constant). 

 

The later assumption implies the evolution of the R-F asset Bt is given by: Rf asset

It means that the R-F bank account grows at the continuously compounding rate r and hence: Bt = ert --> Rf asset.

 

We are interested in pricing an option which is a function of the stock price at time T > 0, ST. Ex: Call. 

- the form of the payoff is not so important

- the fact that it is a function of the stock price at time T, and only time T, is important. 

 

Under this condition, we can show that call price is a function of current time t and current stock price St only. 

 

To price a derivative in BS world, we must do so under a measure which does not allow arbitrage --> Risk-neutral measure. 

One can show that under this measure, the drift term of the stock price changes so that: Bs rn

 

In R-N world, C(t,St)/Bt is a martingale and hence if we calculate its differential, we know it must have zero drift. 

A simple explanation of its meaning is that we expect it to have zero growth. 

Our option price is expected to grow at same rate as bank account and growth of each cancels out in the given process. 

This is what it means to be a martingale. We do not expect change over time so we have zero expected growth. 

This translates to the discounted price having a zero drift term. 

 

Applying Ito's lemma to C(t,St) gives: Bs ito lemma

 

Under the R-N dynamics of St and recalling that: 

- (dWt)2 = dt

- dWtdt = dt2 = 0

 

Bs3

 

Using the Ito product rule:    Bs ito product

 

Since C(t,St)/Bt is a martingale and hence must have zero drift: Bs equation

This is the BS equation. 

It is a PDE describing the evolution of the option price as a function of the current stock price and the current time. 

The equation does not change if we vary the payoff function of the derivative. 

However, the associated boundary conditions, which are required to solve the equation do vary!

 

The above implies that 2 stocks with the same volatility but different drifts will have the same option prices. 

The pricing of any derivative must be done in the R-N measure in order to avoir arbitrage. 

Under this measure, we have seen that the drift changed and was independent of the drift of the stock. 

Financially, this reflects the fact that the hedging strategy ensures that the underlying drift of the stock is balanced against the drift of the option. The drifts are balanced since drift reflect the risk premium demanded by investors to account for uncertainty and that uncertainty has been hedged away. 

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