Chapter 13Probability Density Functions

A volatility smile at a given maturity can be converted into an equivalent probability density function (pdf). The probability density function contains useful information because integrating an area under the curve gives the likelihood of spot being within the given range at maturity.

Starting with the simplest case, Exhibit 13.1 shows a 1yr volatility smile with 10% volatility for all strikes (i.e., pure Black-Scholes world).

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Exhibit 13.1 Volatility smile with flat 10% implied volatility

This volatility smile generates the standard log-normal bell-shaped pdf shown in Exhibit 13.2. The method of generating pdfs from option prices is explored in Practical G.

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Exhibit 13.2 Probability density function from flat 10% volatility smile

Exhibit 13.3 shows implied volatility rising to 15% for all strikes. Increased volatility widens the distribution and the pdf extends out on both sides as shown in Exhibit 13.4.

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Exhibit 13.3 Volatility smiles with flat 10% and flat 15% implied volatility

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Exhibit 13.4 Probability density functions from ...

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