I've been asked this question quite a couple of times lately. Since I'm convinced that 1. the Black-Scholes-Merton model will always be the starting point for anybody interested in option pricing and that 2. some people wrongly blame the model and not the people who (allegedly) sheepishly apply it without understanding its limitations/strict assumptions/shortcomings, I gladly answer it.
Ok, so most people who hear that the price of a call option should be the same for a stock with an expected return of 5% and a stock with an expected return of 20% (given that both have the same volatility) are puzzled. After all, the call option on the second stock is more likely to be exercised/end in the money! The nicest explanation--which does not depend on the overall market structure (e.g. the beta of the stocks)--goes as follows:
Assume you go long a call (C) and short Δ units of the underlying stock (S). Π, the value of your portfolio, is given as:
Delta (Δ) is the ratio of the change in the option value to the change in the stock price, when the change in the stock price is (infinitesimally) small. After this small change in the stock price occurs we have:
To a first approximation, the change in the call price, dC, is ΔdS, i.e. the random movement in the stock is cancelled by the movement of the call position in the opposite direction. As Black and Scholes note (p. 642):
You can clearly see that the ΔdS terms cancel each other. What's left is a term involving the time decay of the option (∂C/∂t = Θ) and a term giving the profit from the stock move, i.e. the volatility weighted Γ (∂2C/∂2S = Γ):
With the given position/portfolio, you make money when the stock moves up or down (long gamma) but loose money when the stock doesn't move (short theta). Maybe at this point statements like: "I'm long vol up the ying-yang and bleeding theta" start making sense to you!
Since the return on the hedged position has become certain, i.e. the change has become riskless, we should earn the risk-free rate:
And by putting all those pieces together, you end up with the famous Black-Scholes-Merton equation:
related items:
The Put-Call Parity, Mahalanobis
Paul Wilmott on Quantitative Finance, Paul Wilmott
Options, Futures, and other Derivatives, John Hull
Ok, so most people who hear that the price of a call option should be the same for a stock with an expected return of 5% and a stock with an expected return of 20% (given that both have the same volatility) are puzzled. After all, the call option on the second stock is more likely to be exercised/end in the money! The nicest explanation--which does not depend on the overall market structure (e.g. the beta of the stocks)--goes as follows:
Assume you go long a call (C) and short Δ units of the underlying stock (S). Π, the value of your portfolio, is given as:
Delta (Δ) is the ratio of the change in the option value to the change in the stock price, when the change in the stock price is (infinitesimally) small. After this small change in the stock price occurs we have:To a first approximation, the change in the call price, dC, is ΔdS, i.e. the random movement in the stock is cancelled by the movement of the call position in the opposite direction. As Black and Scholes note (p. 642):
As the variables S and t change, the number of [stocks] to be sold short to create a hedged position with one [call] changes. If the hedge is maintained continuously, then the approximations mentioned above become exact, and the return on the hedged position is completely independent of the change in the value of the stock. In fact, the return on the hedged position becomes certain.3That's the answer! More precisely: Assuming that the stock price follows a geometric Brownian motion, the Itô-Döblin formula (stochastic Taylor series expansion) gives us:
3 This was pointed out to us by Robert Merton.
You can clearly see that the ΔdS terms cancel each other. What's left is a term involving the time decay of the option (∂C/∂t = Θ) and a term giving the profit from the stock move, i.e. the volatility weighted Γ (∂2C/∂2S = Γ):With the given position/portfolio, you make money when the stock moves up or down (long gamma) but loose money when the stock doesn't move (short theta). Maybe at this point statements like: "I'm long vol up the ying-yang and bleeding theta" start making sense to you!Since the return on the hedged position has become certain, i.e. the change has become riskless, we should earn the risk-free rate:
And by putting all those pieces together, you end up with the famous Black-Scholes-Merton equation:related items:
The Put-Call Parity, Mahalanobis
Paul Wilmott on Quantitative Finance, Paul Wilmott
Options, Futures, and other Derivatives, John Hull
Mahalanobis - am 2009-01-12 22:59 - Rubrik: Finance
