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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?

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  • $\begingroup$ BSM not BMS. And it is not called a market model. $\endgroup$ – Dom May 24 '18 at 5:03
  • $\begingroup$ In my lecture notes it is Black-Merton-Scholes (BMS). $\endgroup$ – M Smith May 24 '18 at 8:42
  • $\begingroup$ Your lecturer is wrong. $\endgroup$ – Dom May 24 '18 at 16:43
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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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