My derivation follows Almgren, R. (2002). Financial derivatives and partial differential equations.

*American Mathematical Monthly*, 109: 1-12, 2002.

In my next post I aim to discuss the idealizations deployed here and how reasonable they make it to apply (*) in actual trading strategies.

A Derivation of the Black-Scholes Model

A (call) option gives the owner the right to buy some underlying asset like a stock at a fixed price K at some time T. Clearly some of the factors relevant to the fair price of the option now are the difference between the current price of the stock S and K as well as the length of time between now and time T when the option can be exercised. Suppose, for instance, that a stock is trading at 100$ and the option gives its owner the right to buy the stock at 90$. Then if the option can be exercised at that moment, the option is worth 10$. But if it is six months or a year until the option can be exercised, what is a fair price to pay for the 90$ option? It seems like a completely intractable problem that could depend on any number of factors including features specific to that asset as well as an investor's tolerance for risk. The genius of the Black-Scholes approach is to show how certain idealizing assumptions allow the option to be priced at V given only the current stock price S, a measure of the volatility of the stock price σ , the prevailing interest rate r and the length of time between now and time T when the option can be exercised. The only unknown parameter here is σ , the volatility of the stock price, but even this can be estimated by looking at the past behavior of the stock or similar stocks. Using the value V computed using this equation a trader can execute what appears to be a completely risk-free hedge. This involves either buying the option and selling the stock or selling the option and buying the stock. This position is apparently risk-free because the direction of the stock price is not part of the model, and so the trader need not take a stand on whether the stock price will go up or down.

The basic assumption underlying the derivation of (*) is that markets are efficient so that ``successive price changes may be considered as uncorrelated random variables" (Almgren, p. 1). The time-interval between now and the time T when the option can be exercised is first divided into N -many time-steps. We can then deploy a

*lognormal*model of the change in price δ S_j at time-step j :

δ S_j = a δ t + σ S ξ_j

The ξ_j are random variables whose mean is zero and whose variance is 1 (Almgren, p. 5). Our model reflects the assumption that the percentage size of the random changes in S remains the same as S fluctuates over time (Almgren, p. 8). The parameter a indicates the overall ``drift" in the price of the stock, but it drops out in the course of the derivation.

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

δ V = V_t δ t + V_s δ S + 1/2 V_{SS} δ S^2

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

δ Π = Dδ S + r C δ t

The last two equations allow us to easily 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

δ(V-Π)=(V_t - rC)δ t + (V_S - D)δ S + 1/2 V_{SS} δ S^2

The δ S 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 δ S term can be eliminated if we assume that at each time-step the investor can adjust the number of shares held so that

D=V_S

Then we get

δ(V-Π)=(V_t - rC)δ t + 1/2 V_{SS} δ S^2

The δ S^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 the equation so that, over the whole time interval Δ t ,

Δ(V-&Pi) = (V_t - rC + 1/2 σ^2 S^2 V_{SS})Δ t

Strictly speaking, we are here applying a result known as Ito's Lemma.

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 again and assume that Δ(V-Π) is identical to the result of investing V-Π in a risk-free bank account with interest rate r . That is,

Δ(V-Π) = r (V-Π)Δ t

But given that V-Π = V - DS - C and D = V_S , we can simplify the right-hand side of this equation to

(rV - rV_S S - rC)Δ t

Given our previous equation for the left-hand side, we get

(*) V_t + 1/2 σ^2 S^2 V_{SS} + rSV_S - rV = 0

after all terms are brought to the left-hand side.

## 2 comments:

Terence Tao had a very good post a little while ago deriving the Black-Scholes equation also:

http://terrytao.wordpress.com/2008/07/01/the-black-scholes-equation/

Interestingly, a lot of the comments there (after mine) are from other people advertising their posts explaining the formula.

Kenny, thanks for the link. This looks much more detailed than what I provided! Hopefully in the next post I can add some philosophical clarification on the idealizations and their applicability.

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