How Fat Tails Can Affect Call Option Values

Summary

The question considers how a call’s value might change when stock returns have fatter tails than a log-normal model. It gives a numerical setup with zero interest rate, specified volatility and maturity, and an initial stock price equal to the strike. The questioner calculates a lower stock level at a stated percentile and considers two competing effects: the percentile drop may be larger under a fat-tailed distribution, while a greater chance of high terminal prices could increase the call’s expected payoff.

The brief answer says that, all else equal, greater likelihood of extreme outcomes raises effective volatility and therefore the option price. That is an intuition, not a general proof that the call must be worth more for every fat-tailed distribution. The comparison depends on how distributions are matched, including their means, variance, and risk-neutral pricing assumptions. The response does not analyze the percentile-drop calculation or establish which effect dominates under the stated setup.

Key ideas

• A call’s value depends on the risk-neutral distribution of its terminal payoff.
• Fatter tails can increase the chance of unusually high outcomes that benefit a call holder.
• A larger downside percentile move alone does not determine the value of a call.
• The claim that fat tails raise option value requires a clear basis for comparing distributions and other assumptions.

Tags

# Call Value After 98 percentile drop in Stock Price


# Call Value After 98 percentile drop in Stock Price












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?

## Answer by SD_ (score 2)

https://quant.stackexchange.com/a/75161

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