The easy route to risk-neutral measure pricing

The principle of pricing in the risk-neutral measure is the foundation of quantitative analysis.

I have already written a post which gives an intuitive description of the concept of a risk premium and which discusses some aspects of the risk-neutral approach (see here).

In this post I want to look again at risk-neutral pricing. It recently occurred to me that we get a much simpler, step-by-step explanation if we take the Fundamental Theorem of Arbitrage-Free Asset Pricing as our starting point.

Moreover, we can even incorporate an intelligent discussion of the real-world measure.

(In my opinion it is a pity that the literature does not really investigate the relationship between the risk-neutral measure and the real-world measure; when you look more carefully at the two measures together you begin to get a much better feel for how the real world of trading functions.)

Throughout most of the literature, the Fundamental Theorem of Arbitrage-Free Asset Pricing tends to be deduced from a hedging-analysis of a collection of assets. That’s all very fine, but it means you spend a lot of time thinking about that result itself, and not much on its consequences.

But if we take the FTAFAP as our starting point, I think we get a nice, compact and clearer sequence through all the steps we need.

Comments welcome, of course!

The Fundamental Theorem of Aribitrage-Free Asset Pricing states:

A model of investment assets is arbitrage free if, and only if, there is an equivalent measure under which all the discounted asset prices are martingales.

The proof of the theorem in finite dimensions is do-able. In infinite dimensions the proof is sufficiently complex that you won’t really want to read it (the complexity mostly derives from the fact that the things that you would like to be obvious are not actually that easy to define precisely and robustly for infinite-dimensional processes).

We do not even want to prove the theorem, but instead just want to use it to get a good feel for the basic intuition that:

Arbitrage-free models are basically just a a bunch of martingales.

A simple example of there being no martingale measure, and the corresponding arbitrage

Suppose that W_t is a brownian motion under a certain measure \mathbb{P}.

Let us create two assets, X_1(t) and X_2(t), defined as

  • X_1(t)=W_t
  • X_2(t)=W_t+t

(We suppose that interest rates are 0.)

Obviously there is no measure under which both of these assets can be a martingale.

Therefore there must be an arbitrage, according to the Fundamental Theorem.

And there is: at time 0 we go long X_1 and go short X_2; it is a zero-cost strategy, a costless portfolio.

At any future time t our portfolio is worth W_t+t-W_t=t, so we have created guaranteed profits from a costless portfolio. Hello, that is an arbitrage.

The important point of this example is that it makes you realize:

Your model can only have as many assets as there are martingales.

So if you want to build a model of a 5-asset portfolio you are probably going to need 5 sources of randomness in your model, and they will probably be 5 brownian motions.

How many martingales can you think of?

Without thinking about it too much, I really only know two fundamental types of martingale:

  1. plain-ol’ brownian motion W_t,
  2. snazzy geometric brownian motion e^{W_t -t/2}.

Drifts are allowed — but only one type

According to the FT, the most simple arbitrage-free model of assets is made out of a collection of brownian motions.

Separating the concepts

We have the martingale principle, and we have the idea of risk-elimination via hedging, which allows us to use the martingale approach.

It is nice to see the two parts separated. The classic Black-Scholes derivation mixes the two up together.

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