Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

[DP] Risk-neutral probabilities, state prices and the stochastic discount factor

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Today we began our second year doctoral (FPM) course on derivatives pricing, which is essentially meant to be an introduction to multi-period securities markets in continuous time. We are following Stanley Pliska’s Mathematical Finance text for much of the initial theory in discrete time, and then plan to move to Steven Shreve’s Stochastic Calculus for Finance – Vol. 2 for continuous time. The course outline is here.

The discussion today was just a recap of the basic asset pricing theory ala Cochrane and pointing out why derivatives pricing is done the way it is.

The Cochrane story is that price p_t of any risky payoff x_{t + 1} is given by the following expectation:

p_t = \mathbb{E}[m_{t + 1} x_{t + 1}]

where

m_{t + 1} = \beta \dfrac{U'(C_{t + 1})}{U'(C_t)}

is the stochastic discount factor.

Since this post is meant to point out specificity about derivatives pricing, suppressing the time dependence, we first interpret the pricing equation as:

\begin{aligned} p &= \mathbb{E}[m_{t + 1} x_{t + 1}] \\ &= \sum \pi(s) m(s) x(s) \end{aligned}

where \pi(s) represents the probability of state s.

If there exist state contingent securities (also called state prices), z(s), which give a payoff of 1 when that state s occurs and nothing otherwise (think of them as unit vectors i, j, k etc. in physics), then one can also use z(s) to write the same price as:

p = \sum z(s) x(s)

which gives our first equivalence between the stochastic discount factor m(s) and the state price z(s) as:

\boxed{z(s) = \pi(s) m(s)}

That is, the state price z(s) embeds both information – that of probability of different states \pi(s) and the associated discount factor m(s). If one were to observe such state price in the market, or could show their existence, one can simply proceed with pricing in their terms. So even if one does not know the underlying stochastic discount factor, it does not matter as long as one could observe – or potentially observe – z(s).

Another way to reinterpret the fundamental equation of asset pricing is to write \tilde{\pi}(s) =\pi(s) m(s) (1 + r) where r is the risk-free one-period rate and then replace \pi(s) with \tilde{\pi}(s):

\begin{aligned} p &=\displaystyle \sum \pi(s) m(s) x(s) \\&= \sum \dfrac{\tilde{\pi}(s) x(s)}{1 + r} \\&= \dfrac{\mathbb{\tilde{\mathbb{E}}}[x(s)]}{1 + r} \end{aligned}

That is, \tilde{\pi}(s) may now be interpreted as those (suitably revised) probabilities under which price is simply expected payoff discounted using the risk free rate. That is, there is no more any correction required for risk aversion.

With this reinterpretation, we can now also connect \tilde{\pi}(s) with the state price z(s) as:

\boxed{\tilde{\pi}(s) = z(s)(1 + r)}

which suggests that one can now work with even these ‘synthetic’ (‘risk-neutral’) probabilities \tilde{\pi}(s) instead of the true ones and still not worry about the underlying stochastic discount factor m(s), because the state prices are intimately connected to these \tilde{\pi}(s).

Finally, noticing that \mathbb{E}[m] = \dfrac{1}{(1 + r)} now allows us to write the following:

\begin{aligned}\tilde{\pi}(s) &= \dfrac{z(s)}{\mathbb{E}[m]} \\&= \dfrac{m(s)}{\mathbb{E}[m]}\pi(s) \end{aligned}

That is, the risk-neutral probabilities are nothing but suitably rescaled original probabilities, making our earlier intuition formal.

If one can talk about state-contingent prices or the risk-neutral probabilities, that’s all that one needs sometime. From the point of view of pricing it is also extremely convenient, as now the whole machinery of probability theory becomes available.

Although as such there is nothing new here (and Cochrane builds all this up in his book), this intimate connection between the stochastic discount factor, state prices and risk-neutral probabilities suggests why derivatives pricing need not start from the fundamental stochastic discount factor.

This is, of course, only the larger picture. For all of this to work, at the time of actual modeling one needs a bit more structure. So much of the detail in derivatives pricing is about putting such structures in place, so that one does not end up with situations like ‘free money’ lying on the table (arbitrage possibilities).

The theory itself can be done at multiple levels:

  • Single-period, discrete time, finite states of the world: t = \{0, 1\}, s = \{1, 2, 3, \cdots, K\}; K < \infty
  • Multi-period, discrete time, finite states of the world: t = \{0, 1, 2, \cdots, T\}, s = \{1, 2, 3, \cdots, K\}; K < \infty, T < \infty
  • Continuous time, infinite states of the world: t \in \mathbb{R} (meaning time is not discrete but continuous), s \in \mathbb{R} (meaning that random variables S_i(t) are continuous; the number of random variables are still finite though, i.e. i = \{1, 2, \cdots, N\} and N \in \mathbb{N}).

And one can do all of the above in the context of complete and incomplete markets.

We’ll go through all the above three versions – starting first with single-period discrete time and ending with multi-period continuous time. But much of what we would do in this course would be done in the context of complete markets. We would spend some time on what happens when markets are incomplete, but it is unlikely we would have the time to get into all the gory details.

Do state prices exist?

While state prices/risk-neutral probabilities do not always exist, in some markets binary options and butterfly spreads are quite liquid, so at least in-part they can be directly observed. In general, the Breeden-Litzenberger (1978) results ensures that given enough traded derivatives one can infer state-prices/risk-neutral probabilities from market prices, so all of derivatives pricing can be thought of in this ‘relative way’ (as against the ‘absolute way’ using the stochastic discount factor approach). The fact that derivative instruments are typically defined in terms of some underlying (derivatives are often referred to as contingent claims), this intuition works economically too.

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Written by Vineet

September 9, 2014 at 10:21 pm

Posted in Teaching: DP

Tagged with , ,

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