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This question is from Joshi's quant book.

Assume r = 0, σ = 0.1, T-t = 1, X = 100 = S(t). Initially, the call is worth $3.99.

The first question asks what the value of the call is after a 98 percentile drop in S(t).

That was simple. Z = -2.05 is our critical value so plugging that into the distribution of S(T) (which I assumed to be log-normal), I get that the new S is 81.03 and the call is now worth $0.06.

The second question asks:

What if the stock had a distribution with fatter tails than a log-normal distribution. Would the value of the call be more or less?

My initial thoughts:

(1) Fatter tails means the 98 percentile will be further away compared to a lighter tail distribution so the 98 percentile drop in a fatter-tailed distribution will be lower compared to the 98 percentile drop in the log-normal.

(2) But, when calculating the call value using discounted expected payoff under the risk-neutral measure, a fatter-tail distribution will have greater probability of 'hitting' higher values that could cause it to finish ITM compared to a lighter-tailed distribution.

I'm not sure which effect wins out? Any guidance?

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I think that the Fatter tails => extreme events are more likely => vol increases => option price (keeping all other variables constant) increases

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