Retrieving Risk Neutral Moments and Expected Quadratic Variation from Option Prices

Abstract:     
This paper derives exact formulas for retrieving risk neutral moments of future payoffs of any order from generic European-style option prices. It also provides an exact formula for retrieving the expected quadratic variation of the stock market implied by European option prices, which nowadays is used as an estimate of the implied volatility, and a formula approximating the jump component of this measure of variation. To implement the above formulas to discrete sets of option prices, the paper suggests a numerical procedure and provides upper bounds of its approximation errors. The performance of this procedure is evaluated through a Monte Carlo exercise. The paper provides clear cut evidence that ignoring the jump component of the underlying asset can lead to seriously biased estimates of the new volatility index suggested by the Chicago Board Options Exchange (CBOE). This is also confirmed by an empirical exercise based on market option prices written on the S&P 500 index, which shows that the jump component of quadratic variation is significant and varies substantially during financial crises.

Number of Pages in PDF File: 56

Keywords: Risk neutral moments, characteristic function, expected quadratic variation

JEL Classification: C14, G11, G12

working papers series