Category Archives: Finance

Disavowing Lebesgue

No, you cannot disavow Lebesgue.

Lebesgue integration (and measure theory) are indispensable in stochastic calculus, especially for finance applications.

Take for example the martingale representation theorem, which is central to the theory of dynamic hedging and replication. This version is essentially Rogers and Williams (1987), Theorem 36.5:

The martingale representation theorem
Let \(W\) be a standard Brownian motion of dimension \(K\). If \(X\) is a martingale with respect to the augmented filtration generated by \(W\), then there exists a process \(b \in \mathcal{L}^{2}\) such that
$$X(t) -X(0) = \int_{0}^{t} b \, dW$$ (in the sense that the two processes are indistinguishable).

But what does it mean to say that the process \(b\) is in \(\mathcal{L}^{2}\)? It means that \(b\) is measurable and adapted and pathwise square integrable on bounded intervals. In other words,
$$\int_{0}^{t} \|b\|^{2} \, ds < \infty $$ with probability one, for all \(t \in [0,\infty)\). Or, to write it out in detail, if the underlying probability space is \((\Omega,\mathcal{F},P)\), then the requirement is that
$$\int_{0}^{t} \|b(\omega,s)\|^{2} \, ds < \infty $$ for \(P\)-almost all \(\omega \in \Omega\), for all \(t \in [0,\infty)\).

For each \(\omega\), the time integral is a Lebesgue integral. The martingale representation theorem does not in any way guarantee that the function \(\|b\|^{2}\) is pathwise Riemann integrable. How could it?

So the statement of the martingale representation theorem requires Lebesgue integration and measure theory.

N(d1) and N(d2)

No, N(d1) is not the probability of exercise.

In one of my classes I derived the formulas for the values of standard options and various digital options in the Black-Scholes models, the point being to illustrate various concepts – the state price process, risk-adjusted probabilities, and the use of different numeraires.

We got into an argument about the meaning of N(d1) and N(d2).

I published a paper about this a number of years ago in Revue Finance, the journal of the French finance association: “Understanding N(d1) and N(d2): Risk-Adjusted Probabilities in the Black-Scholes Model,” Revue Finance (Journal of the French Finance Association) 14 (1993), 95-106. [Abstract][Abstract on the journal’s website][Paper (pdf)]

The paper explains N(d1) and N(d2) and relates them to the single-period and multi-period binomial models.

After so many years, people are still puzzling over this.

What is N(d1)? Well, N(d2) is the risk-adjusted probability of exercise. N(d1) is something else: It is the factor by which the present value of contingent receipt of the stock (contingent upon exercise) falls short of the current stock price.

Instantaneous Arbitrage and the CAPM

This working paper from 2004–2006 studies the concept of instantaneous arbitrage in continuous time and its relation to the instantaneous CAPM. Absence of instantaneous arbitrage is equivalent to the existence of a trading strategy which satisfies the CAPM beta pricing relation in place of the market. Thus the difference between the arbitrage argument and the CAPM argument in Black and Scholes (1973) is this: the arbitrage argument assumes that there exists some portfolio satisfying the CAPM equation, whereas the CAPM argument assumes, in addition, that this portfolio is the market portfolio.

Instantaneous Arbitrage and the CAPM

Partial Derivatives of Option Prices

Someone recently asked me how to calculate the partial derivative of the Black–Scholes option price with respect to the strike. Indeed, that kind of calculation easily leads to a complete mess. You end up with page after page of hopelessly complicated expressions. You need to simplify, but how?

Fortunately, there is a trick. Express the option price as a function of d1, and write d2 as d1 minus sigma times the square root of remaining time to maturity. It turns out that the derivative of the option price with respect to d1 is zero. So when you calculate the partial derivative, for example with respect to the strike, by the chain rule, the effects that come from the dependence of d1 on the strike will disappear.

Sure enough, this trick is explained on pages 216–217 of my 1999 book, Pricing and Hedging of Derivative Securities

Low-brow, but useful.