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Bounding a European Option Price with Jensen’s Inequality

Article Quant Q&A · Author: M Smith
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Summary

The document works through a Black–Merton–Scholes example with zero interest, unit volatility, and an option payoff based on the squared log price. Under the risk-neutral model, the log stock price can be rewritten using Brownian motion, reducing the time-zero valuation to the expected value of the square root of one plus squared terminal Brownian motion.

The proposed bound follows by applying Jensen’s inequality to the concave square-root function: the square of the expected payoff is no greater than the expected value inside the root. Since terminal Brownian motion has variance equal to the expiry, this gives the stated upper bound. The response also corrects a time-index typo in the questioner's derivation. The argument depends on the stated model assumptions and risk-neutral pricing; it is an analytic bound for this payoff, not a general option valuation recipe.

Key ideas

  • Risk-neutral pricing reduces the option value to an expectation over terminal Brownian motion.
  • The log stock price cancels the time-dependent drift term inside the specified payoff.
  • Jensen’s inequality bounds the expected square root using the expectation of its argument.
  • The bound uses the terminal Brownian variance and the stated Black–Merton–Scholes assumptions.

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OptionsDerivatives pricingStatistics
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# How might I answer this past exam question relating to the value of a European option under the BMS market model?


# How might I answer this past exam question relating to the value of a European option under the BMS market model?












The following question, which is not homework, was taken from a past paper for a module I will soon be sitting:

> Consider a Black-Merton-Scholes stochastic market with drift $\mu = 1$, volatility $\sigma^2 = 1$ and interest rate $r = 0$. On this market there is a European option with a certain expiry $T > 0$ and a payoff $$ \sqrt{1 + \left( \frac{T}{2} + \ln S_T \right)^2} $$ where S stands for the stock price. Assuming $S_0 = 1$, show that the price of this option may not exceed $\sqrt{1 + T}$.

I am struggling to answer this question. My attempt so far is as follows:

Firstly, since we are considering the stock price under the BMS market model, I believe that $S_t$ will be given by $$ S_t = S_0 \exp \left( \left( r - \frac{\sigma^2}{2} \right)t + \sigma W_t \right) = \exp \left( - \frac{t}{2} + W_t \right) $$ Thus, denoting by $V_t$ the value of this option at time $t$, we have $$ V_t = \sqrt{1 + \left( \frac{t}{2} + \ln S_t \right)^2} = \sqrt{1 + \left( \frac{t}{2} + \left( - \frac{t}{2} + W_t \right) \right)^2} = \sqrt{1 + W_t^2} $$ and the price of this option at $t=0$ would thus be given by $$ V_0 = e^{-rT} \mathbb{E} [V_T] = \mathbb{E} \left[ \sqrt{1 + W_T^2} \right] $$

Assuming that I am correct thus far, I am now required to show that $$ \mathbb{E} \left[ \sqrt{1 + W_T^2} \right] \leq \sqrt{1+T} $$ How might I do this?

## Answer by Andrew (score 3, accepted)

https://quant.stackexchange.com/a/39931

You have a typo in the last two lines: $W_t$ should be $W_T$. Your approach is correct so far.

Use Jensen for the last step: $E[\sqrt{1+W_T^2})]^2\le E[1+W_T^2]=1+T$

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