A Diffusion Equation
The penultimate derivation of the Black-Scholes partial differential equation is rather unusual in that it uses just pure thought about the nature of Brownian motion and a couple of trivial observations. It also has a very neat punchline that makes the derivation helpful in other modelling situations.
It goes something like this.
Stock prices can be modelled as Brownian motion, the stock price plays the role of the position of the 'pollen particle' and time is time. In mathematical terms Brownian motion is just an example of a diffusion equation. So let's write down a diffusion equation for the value of an option as a function of space and time, i.e. stock price and time, that's V(S, t). What's the general linear diffusion equation? It is
Note the coefficients a, b and c. At the moment these could be anything.
Now for the two trivial observations.
First, cash in the bank must be a solution of this equation. Financial contracts don't come any simpler than this. So plug V = ert into this diffusion equation to get
Second, surely the stock price itself must also be a solution? After all, you could think of it as being a call option with zero strike. So plug V = S into the general diffusion equation. We find
Putting b and c back into the general diffusion equation we find
This is the risk-neutral Black-Scholes equation. Two of the coefficients (those of V and д V/дS) have been pinned down exactly without any modelling at all. Ok, so it doesn't tell us what the coefficient of the second derivative term is, but even that has a nice interpretation. It means at least a couple of interesting things.
First, if we do start to move outside the Black-Scholes world then chances are it will be the diffusion coefficient that we must change from its usual o2S2 to accommodate new models.
Second, if we want to fudge our option prices, to massage them into line with traded prices for example, we can only do so by fiddling with this diffusion coefficient, i.e. what we now know to be the volatility. This derivation tells us that our only valid fudge factor is the volatility.
Black—Scholes for Accountants
The final derivation of the Black-Scholes equation requires very little complicated mathematics, and doesn't even need assumptions about Gaussian returns, all we need is for the variance of returns to be finite.
The Black-Scholes analysis requires continuous hedging, which is possible in theory but impossible, and even undesirable, in practice. Hence one hedges in some discrete way. Let's assume that we hedge at equal time periods, St. And consider the value changes associated with a delta-hedged option.
• We start with zero cash
• We buy an option
• We sell some stock short
• Any cash left (positive or negative) is put into a risk-free account.
We start by borrowing some money to buy the option. This option has a delta, and so we sell delta of the underlying
Figure 7.2: How our portfolio depends on 5.
stock in order to hedge. This brings in some money. The cash from these transactions is put in the bank. At this point in time our net worth is zero.
Our portfolio has a dependence on 5 as shown in Figure 7.2.
We are only concerned with small movements in the stock over a small time period, so zoom in on the current stock position. Locally the curve is approximately a parabola, see Figure 7.3.
Now think about how our net worth will change from now to a time St later. There are three reasons for our total wealth to change over that period.
1. The option price curve changes.
3. The stock moves
Figure 7.3: The curve is approximately quadratic.
The option curve falls by the time value, the theta multiplied by the time step:
To calculate how much interest we received we need to know how much money we put in the bank. This was
from the stock sale and
from the option purchase. Therefore the interest we receive is
Finally, look at the money made from the stock move. Since gamma is positive, any stock price move is good for us. The larger the move the better.
The curve in Figure 7.3 is locally quadratic, a parabola with coefficient |r. The stock move over a time period St is proportional to three things:
• the volatility a
• the stock price S
• the square root of the time step
Multiply these three together, square the result because the curve is parabolic and multiply that by |r and you get the profit made from the stock move as
Put these three value changes together (ignoring the St term which multiplies all of them) and set the resulting expression equal to zero, to represent no arbitrage, and you get
the Black-Scholes equation.
Now there was a bit of cheating here, since the stock price move is really random. What we should have said is that
is the profit made from the stock move on average. Crucially all we need to know is that the variance of returns is
we don't even need the stock returns to be normally distributed. There is a difference between the square of the stock prices moves and its average value and this gives rise to hedging error, something that is always seen in practice. If you hedge discretely, as you must, then Black-Scholes only works on average. But as you hedge more and more frequently, going to the limit St = 0, then the total hedging error tends to zero, so justifying the Black-Scholes model.
There are other ways of deriving the Black-Scholes equation or formulae but I am only going to give the references (see Gerber & Shiu, 1994, and Hamada & Sherris, 2003). One of the reasons why I have drawn a line by not including them is summed up very nicely by a reader (who will remain anonymous for reasons which will be apparent) who submitted a couple of possible new derivations, in particular one using 'distortion risk theory.' In an email to me he wrote: 'Unfortunately distortion risk theory is completely unknown to quants ... maybe because this theory originated in insurance mathematics, but more probably because is useless, except research paper writing. I wrote master thesis on this topic, from time perspective, completely waste of time.'
References and Further Reading
Andreason, J, Jensen, B & Poulsen, R 1998 Eight valuation methods in financial mathematics: the Black-Scholes formula as an example. Math. Scientist 23 18-40
Black, F & Scholes, M 1973 The pricing of options and corporate liabilities. Journal of Political Economy 81 637-659
Cox, J & Rubinstein, M 1985 Options Markets. Prentice-Hall
Derman, E & Kani, I 1994 Riding on a smile. Risk magazine 7 (2) 32-39
Dupire, B 1994 Pricing with a smile. Risk magazine 7 (1) 18-20
Gerber, HU & Shiu, ESW 1994 Option pricing by Esscher transforms. Transactions of the Society of Actuaries 46 99-191
Hamada, M & Sherris, M 2003 Contingent claim pricing using probability distortion operators: methods from insurance risk pricing and their relationship to financial theory. Applied Mathematical Finance 10 119-47
Harrison, JM & Kreps, D 1979 Martingales and arbitrage in multi-period securities markets. Journal of Economic Theory 20 381-408
Harrison, JM & Pliska, SR 1981 Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their
Applications 11 215-260
Joshi, M 2003 The Concepts and Practice of Mathematical Finance. Cambridge University Press.
Rubinstein, M 1976 The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics 7 407-425
Rubinstein, M 1994 Implied binomial trees. Journal of Finance 69 771-818
Wilmott, P 1994 Discrete charms. Risk magazine 7 (3) 48-51 (March)
Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd
Yakovlev, DE & Zhabin, DN 2003 About discrete hedging and option pricing