Tuesday, October 6, 2009

Mathematics, Financial Economics and Failure

In a recent post I noted Krugman's point about economics being seduced by attractive mathematics. Since then there have been many debates out there in the blogosphere about the failures of financial economics, but little discussion of the details of any particular case. I want to start that here with a summary of how the most famous model in financial economics is derived. This is the Black-Scholes model, given as (*) below. It expresses the correct price V for an option as a function of the current price of the underlying stock S and the time t.

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


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.


Kenny said...

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


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

Chris Pincock said...

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.