What is Jensen's Inequality and What is its Role in Finance?

Short answer

Jensen's Inequality states^{[1]} that if f() is a convex function and x is a random variable then

E [f(x)] > f (E[x]).

This justifies why non-linear instruments, options, have inherent value.

Example

You roll a die, square the number of spots you get, you win that many dollars. For this exercise f(x)is x2 a convex function. So E [f(x)] is 1 + 4 + 9 + 16 + 25 + 36 = 91 divided by 6, so 15 1/6. But E[x]is 3 1/2so f (E[x]) is 12 1/4.

Long answer

A function f()is convex on an interval if for every x and y in that interval

f(Xx + (1 - X)y) < Xf(x) + (1 - X)f(y)

for any 0 < X < 1. Graphically this means that the line joining the points (x, f(x)) and (y, f(y)) is nowhere lower than the curve. (Concave is the opposite, simply — f is convex.)

Jensen's Inequality and convexity can be used to explain the relationship between randomness in stock prices and the value inherent in options, the latter typically having some convexity.

Suppose that a stock price S is random and we want to consider the value of an option with payoff P(S). We could calculate the expected stock price at expiration as E[St ], and then the payoff at that expected price P(E[Sj]). Alternatively, we could look at the various option payoffs and then calculate the expected payoff as E[P(St)]. The latter makes more sense, and is indeed the correct way to value options, provided the expectation is with respect to the risk-neutral stock price. If the payoff is convex then

We can get an idea of how much greater the left-hand side is than the right-hand side by using a Taylor series approximation around the mean of S. Write

where S = E[S], so E[e] = 0. Then

Therefore the left-hand side is greater than the right by approximately

This shows the importance of two concepts:

• f"(E[S]): The convexity of an option. As a rule this adds value to an option. It also means that any intuition we may get from linear contracts (forwards and futures) might not be helpful with non-linear instruments such as options.

• E [e2]: Randomness in the underlying, and its variance. Modelling randomness is the key to modelling options.

The lesson to learn from this is that whenever a contract has convexity in a variable or parameter, and that variable or parameter is random, then allowance must be made for this in the pricing. To do this correctly requires a knowledge of the amount of convexity and the amount of randomness.

References and Further Reading

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

What is Ito's Lemma?

Short answer

Ito's lemma is a theorem in stochastic calculus. It tells you that if you have a random walk, in y, say, and a function of that randomly walking variable, call it f (y, t), then you can easily write an expression for the random walk in f . A function of a random variable is itself random in general.

Example

The obvious example concerns the random walk

commonly used to model an equity price or exchange rate, S. What is the stochastic differential equation for the logarithm of S,ln S?

The answer is

Long answer

Let's begin by stating the theorem. Given a random variable y satisfying the stochastic differential equation

where dX is a Wiener process, and a function f (y, t)that is differentiable with respect to t and twice differentiable with respect to y, then f satisfies the following stochastic differential equation

Ito's lemma is to stochastic variables what Taylor series is to deterministic. You can think of it as a way of expanding functions in a series in dt, just like Taylor series. If it helps to think of it this way then you must remember the simple rules of thumb as follows:

1. Whenever you get dX2 in a Taylor series expansion of a stochastic variable you must replace it with dt.

2. Terms that are O(dt3/2) or smaller must be ignored. This means that dt2, dX3, dt dX, etc. are too small to keep.

It is difficult to overstate the importance of Ito's lemma in quantitative finance. It is used in many of the derivations of the Black-Scholes option-pricing model and the equivalent models in the fixed-income and credit worlds. If we have a random walk model for a stock price S and an option on that stock, with value V(S, t), then Ito's lemma tells us how the option price changes with changes in the stock price. From this follows the idea of hedging, by matching random fluctuations in S with those in V. This is important both in the theory of derivatives pricing and in the practical management of market risk.

Even if you don't know how to prove Ito's lemma you must be able to quote it and use the result.

Sometimes we have a function of more than one stochastic quantity. Suppose that we have a function f(yi,y2, yn, t)of n stochastic quantities and time such that

where the n Wiener processes dXi have correlations pij, then

We can understand this (if not entirely legitimately derive it) via Taylor series by using the rules of thumb

Another extension that is often useful in finance is to incorporate jumps in the independent variable. These are usually modelled by a Poisson process. This is dq such that dq = 1 with probability k dt and is 0 with probability 1 — k dt. Returning to the single independent variable case for simplicity, suppose y satisfies

where dq is a Poisson process and J is the size of the jump or discontinuity in y (when dq = 1) then

And this is Ito in the presence of jumps.

References and Further Reading

Joshi, M 2003 The Concepts and Practice of Mathematical Finance. CUP

Neftci, S 1996 An Introduction to the Mathematics of Financial Derivatives. Academic Press

Wilmott, P 2007 Paul Wilmott Introduces Quantitative Finance, second edition. John Wiley & Sons Ltd

[1] This is the probabilistic interpretation of the inequality.

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