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Jensen Inequality in Finance: All You Need to Know

By The Whispering Void #Convex function, #Jensen Inequality, #risk neutral

What is Jensen Inequality?

Jensen Inequality states that if f(.) is a convex function and x is a random variable, then

E[f(x)] \geq f\ [E(x)]

This explains why non-linear instruments and options have inherent value.

Jensen Inequality

A Short Example

When you roll a die and square the number of spots, you will win the specified amount of money. In this exercise, f(x) represents x^{2}, a convex function. E[f(x)] = 1 + 2 + 9 + 16 + 25 + 36 = 91 divided by 6, resulting in 15 1/6. E[x] = 3 1/2, hence f(E[x]) = 12 1/4.

A Deep Dive into Jensen Inequality

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

f (\lambda x + (1 - \lambda)y)\geq \lambda f (x) + (1 - \lambda )f (y)

For every 0 \leq \lambda \leq 1. This means that the line connecting points (x, f (x)) and (y, f (y)) is not lower than the curve. (Concave is the reverse of convex, therefore -f is convex.)

Jensen’s inequality and convexity help explain how unpredictability in stock prices affects the value of options, which are often convex.

Assuming a random stock price S, we can calculate the value of an option with payment P(S). We may calculate the expected stock price at expiration (E[S_T]) and the payment (P(E[S_T]). Alternatively, we can analyse the option payoffs and determine the expected payoff as E[P(S_T)]. The accurate option valuation method is based on the risk-neutral stock price. If payoff is convex,

E[P(S_T)] \geq P(E[S_T])

Using a Taylor series approximation around mean of S, we can estimate the difference between the left and right sides. Write

S = \overline{S}+\epsilon

where \overline {S} = E[S], so E[\epsilon] = 0. Then,

E[f(S)] = E[f(\overline {S} + \epsilon)] = E[f(\overline{S})+\epsilon {f}'(\overline{S}) + \dfrac{1}{2}\epsilon ^{2}{f}''(\overline{S})+ ...]
\approx f(\overline{S}) + \dfrac{1}{2}{f}''(\overline{S})E[\epsilon ^{2}]
= f (E[S]) + \dfrac{1}{2} {f}'' (E[S])E[\epsilon ^{2}]

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

\dfrac{1}{2} {f}'' (E[S])E[\epsilon ^{2}]

Conclusion

This shows the importance of two concepts

  • {f}'' (E[S]): Convexity of an option. Typically, this increases the value of a choice. Understanding linear contracts (forwards and futures) may not be applicable to non-linear products like options.
  • E[\epsilon ^{2}]: Randomness in the underlying and its variability. Modelling unpredictability is crucial for modelling alternatives.

When a contract contains convexity in a random variable or parameter, pricing should account for it. Identifying the level of convexity and unpredictability is necessary for accurate results.

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