Category Archives: Finance

Disavowing Lebesgue

No, you cannot disavow Lebesgue. Forget about it.

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:

Theorem
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? What a laughable idea.

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

Of course, my book Pricing and Hedging of Derivative Securities uses measure theory and Lebesgue integration throughout. That is just part of the craft of doing mathematics properly.

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

Hedging the Prices of Risk

If the intercept and slope of the instantaneous capital market line (ICML) is deterministic, then investors will optimally choose a portfolio strategy which is a combination of riskless borrowing / lending with the logarithmic (growth optimal) portfolio. This is true even if the means, variances and covariances of securities prices are time varying and stochastic in all kinds of wild ways. All that matters is that the ICML is deterministic. This was shown in Nielsen and Vassalou, The Instantaneous Capital Market Line, Economic Theory 28, No. 3, August 2006, 651-664.

But what if the ICML moves around in a stochasitc manner? Must investors then take all the stochastic dynamics of means, variances, and covariances into account in forming their optimal portfolios?

I once thought that all that mattered would be changes in the ICML. In other words, you hold the logarthmic portfolio, the riskless asset, and a couple of hedge portfolios that hedge against changes in the slope and position of the ICML. That turns out not to be true.

Probably what is going on is this. Investors hold the logarthmic portfolio and the riskless asset. The return characteristics of these positions are stochastic, and so they also hold a couple of hedge portfolios to hedge against that. But then they realize that the return characteristics of the hedge portfolios themselves are stochastic, so they need to hold additional hedge portfolios. And so on.

What I do know to be true is that all that matters is changes in the interest rate and the prices of risk. You hold the riskless asset, the logarithmic portfolio, and a bunch of hedge portfolios that hedge against changes in the interest rate and the prices of risk.

The problem is that I have not followed this literature for a number of years, so I do not know what people have come up with recently along these lines. It will take some time to find out.

Here is a reasonably precise formulation of the result. Assume:

  • Markets are dynamically complete
  • There is a vector of state variables which is driven by the same sources of uncertainty as the securities prices
  • Because of complete markets, there exist hedge portfolios that are perfectly correlated with the state variables.
  • The logarithmic portfolio can be replicated by trading in the hedge portfolios.

Under these assumptions, if the interest rate and the prices of risk are functions of the state variables, then investors need only trade the hedge portfolios and the money market account.

That is, investors trade the money market account, the logarithmic portfolio, and hedge portfolios that hedge against changes in the interest rate and the prices of risk.

Work in progress.

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.