Jensen’s Inequality and Call Prices Under Random Volatility

Summary

The document asks whether a European call is worth more when volatility is random with a given mean than when volatility is fixed at that mean. It introduces Jensen’s inequality as the key tool: if the option price is a convex function of volatility, then averaging prices across a volatility distribution gives a value at least as large as the price at average volatility.

The argument reduces the pricing comparison to checking the curvature of the call price with respect to volatility. The excerpt poses that condition but does not provide the subsequent analysis or a general conclusion. The implication therefore depends on the option’s price being convex over the relevant volatility range and on holding other pricing inputs fixed. It gives no numerical example, empirical evidence, or treatment of how a volatility distribution is specified, so the central practical question remains only partly answered.

Key ideas

• Jensen’s inequality compares the average option price under random volatility with the price at mean volatility.
• The comparison favors random volatility when call price is convex in volatility.
• The key analytical step is determining where the call price has that convexity.
• The excerpt does not complete the curvature analysis or establish a universal pricing result.

Tags

# European call option on constant volatility or drawn from a volatility distribution


# European call option on constant volatility or drawn from a volatility distribution












Which is more expensive:

A European call option on constant volatility of 30% or or drawn from a random distribution of mean 30%?

The answer in A Practical Guide To Quantitative Finance Interviews, isnt very clear at all. The justification as to why stochastic volatility would make the price more or less expensive is not clear to me.

## Answer by Pontus Hultkrantz (score 3, accepted)

https://quant.stackexchange.com/a/59721

Jensen's inequality states that given a convex function $f(x)$ and random variable $X$ we have $$ \mathbb{E}[f(X)] \geq f(\mathbb{E}[X]). $$

Now $f$ is our call price, and $X$ is our random volatility with $\mathbb{E}[X]=\sigma_0$. The question now is, when is the call price convex in volatility?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.