# (p.303) Appendix B Black-Scholes Model

# (p.303) Appendix B Black-Scholes Model

(p.303) Appendix B

Black-Scholes Model

Given that *V* is a function of both *S* and *t*, we can approximate a change in *V* for a small time-step *δt* using a series expansion known as a Taylor series

where additional higher-order terms are dropped. Given an interest rate of *r* for the assets held as cash, the corresponding change in the value of the *replicating portfolio* Π = *DS* + *C* of *D* stocks and *C* in cash is

The last two equations allow us to represent the change in the value of a *difference portfolio*, which buys the option and offers the replicating portfolio for sale. The change in value is

The *δS* term reflects the random fluctuations of the stock price and if it could not be dealt with we could not derive a useful equation for *V*. But fortunately the *δS* term can be eliminated if we assume that at each time-step the investor can adjust the number of shares held so that

Then we get

The *δS* ^{2} remains problematic for a given time-step, but we can find it for the sum of all the time-steps using our lognormal model. This permits us to simplify B.5 so that over the whole time interval ∆*t*,

Strictly speaking, we are applying a result known as Itô’s lemma. This is glossed over somewhat in Almgren (2002), but discussed more fully in Wilmott (2007, ch. 5 and 6). What is somewhat surprising is that we have found the net change in the value of the difference portfolio in a way that has dropped any reference to the random fluctuations of the stock price *S*. This allows us to deploy the efficient market hypothesis and assume that ∆(*V* − Π) is identical to the result of investing *V* − Πin a risk-free bank account with interest rate *r*. That is,

But given that *V* − Π = *V* − *DS* − *C* and *D* = *V* _{s}, we can simplify the right-hand side of this equation to

Given our previous equation for the left-hand side, we get 7.2 after all terms are brought to the left-hand side.