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The Call Option Valuation Wedge (movable)

The following picture is the the wedge-shaped solid, the volume of which is used as the basis for the valuation of the call option according to the method of Black-Scholes and Merton. You can rotate this with the mouse, and zoom by holding shift.

Notice that this displays two wedges, one corresponding to a safer stock, with a taller, more peaked probability distribution (and thinner tails) and one corresponding to a riskier stock, that has a flatter distribution of prices and fatter tails. 

The point of the graph is to illustrate that calls on riskier stocks are more valuable. This is a counterintuitive notion in abstract, particularly since riskier goods are less desirable, hence less valuable. Seeing the call-valuation wedge should indicate why the call on the riskier stock is more valuable: Its fatter tail adds a lot to its volume compared to the volume that is added because of the thinner tail of the safe stock because of the large option payouts that correspond to this region. The higher peak of the safe stock stock does not compensate for the lost volume because this exists in the thin part of the wedge, where the option payouts are small.

If you are perplexed because the two distributions appear to not be located at the same mean, you are very observant! The peaks of the two distributions do not coincide, although their means are the same, because the distribution used is the LogNormal, which is not symmetric. Since only one tail exists to become fatter as standard deviation increases, the peak moves away from the tail to maintain the mean constant.

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