# Distribution of pay-off of an exotic option

Can any assumptions be made about the pay-off of an exotic option? For example, might we say the distribution of the pay-off a vanilla option would be Normal?

I have built a valuation tool that estimates the price of a replicating delta-hedging strategy through Monte Carlo methods by trading the structure of a forward curve. It seems that a histogram of the pay-offs have two relative maximums. Can anyone explain this?

Are options (/ real options) prices logNormally distributed, or does the standard assumption not hold given convexity?

Thanks

In general, an option payoff cannot be normal, as the payoff is generally positive, while a normal variable can be negative.

For a standard call option, the distribution function can be computed from the distribution of the underlying stock. Specifically, consider the vanilla European option payoff $X=(S_T-K)^+$. Then, for $x < 0$, \begin{align*} P(X \le x) = 0, \end{align*} while for $x>0$, \begin{align*} P(X \le x) &= P(S_T \le K+x), \end{align*} which can be computed, if the distribution of $S_T$ is given.

I think you are confused with what's exactly log-normally distributed. The distribution of option prices can't be normal or log-normal because the prices can't be negative.

In general, we don't model option prices, we model the underlying stochastic processes (i.e: geometric brownian motion, mean-reverting etc). We then use the distribution of those processes to derive the payoff.

Exotic option is really not much different to vanilla option. More complicated payoff structure, but similar idea.

Let's take a call option on a stock with exercise price $K$. What is the risk-neutral probability of a payoff $x$?
$$P(x=0) = P(\text{stock} \le K)$$
Also we have $P(\text{payoff} = x > 0) = P(\text{stock}=x+K)$. Hence the required density function $f$ has two parts,
(a) an accumulation point at zero representing the probability of being out of the money and
(b) a truncated part of the stock distribution.

This will indeed give a bimodal distribution if the call is out of the money.