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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 ...


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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 ...


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Let's take a call option on a stock with exercise price K. What is the risk-neutral probability of a payoff x? Probability (x=0) = p(stock <=K). Also we have prob(payoff = x>0) = probability (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 ...



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