The analysis was not extended into April as the possibility of negative prices violate some of the fundamental assumptions used. The above technique was repeated on the CLK2020 option chain for the four dates shown in the figure below to see how the option implied volatility RNDs reacted to the developing macro landscape including Covid-19, OPEC+, and crude oil physical storage capacity. Importantly, there are no further restrictions on the probabilistic nature of the underlying asset price process. Further implicit assumptions include constant interest rates, that the call option price is twice differentiable and that a (smooth) probability density function of the price of the underlying asset exists to start with. In fact, the absence of arbitrage is one of the few assumptions needed for the above mathematical derivation to hold. There are additional arbitrage conditions to consider on the fitted implied volatility smile but our distributions meet the above conditions nicely so will be sufficient for our analysis. – The strike weighted probability density function needs to integrate to the underlying futures price – The exercise-price delta needs to be monotone and bounded by and 0 – The probability density function needs to integrate to 1 and remain positive – The cumulative distribution needs to be bounded between 0 and 1 To evaluate the quality of the fit, we need to check a few conditions. It can be difficult to generate clean RNDs and through this process it became clear how cumbersome it would be to generate these over an extended time series. The fourth and final step, is to calculate using the second derivative of the call price with respect to the strike price and scaling it by. To compare these distributions across strike prices, using moneyness,, as the x-axis is helpful. This effectively corresponds to using a forward difference to approximate the partial derivative. Numpy’s diff function will numerically difference your call price function. The third step, following the mathematical derivation above, is to calculate using the first derivative of the call price with respect to the strike price. The second step is to calculate the call prices using the Black-Scholes formula for constant rates with your fitted implied volatility function, underlying price, time to expiration, and observed risk-free rate. ![]() ![]() Numpy’s polyfit is then used to interpolate and extrapolate the smile as needed. We constructed the smile using out-of-the-money calls and puts as these options tend to have more open interest than their in-the-money equivalents. To ensure that the prices and implied volatilities are clean, open interest can be used to weight or exclude certain strikes. We interpolate implied volatilities rather than prices because the former tend to be smoother and better behaved. The first step is thus to fit a smooth function to the Black-Scholes implied volatility smile of the option prices. ![]() In fact, Breeden and Litzenberger simply assume that options are traded at every positive strike price. To approximate the partial derivatives, we need option prices for a fine grid of strike prices. Albeit these are American options, the analysis is interesting nonetheless. Now we will present an overview of deriving these distributions numerically using the infamous May 2020 WTI Crude Oil contract that went negative in April. To arrive at the risk-neutral probability density function, one more derivative with respect to the strike price is needed. This result contains the risk-neutral cumulative distribution function, the probability that the underlying price at expiration will be or lower. In the absence of arbitrage, the European call option value can be related to the discounted expected terminal value under the risk-neutral distribution.ĭifferentiating the call option value with respect to the strike price gives what Malz refers to as the “exercise-price delta”. The derivation of the relationship is well presented in A Simple and Reliable Way to Compute Option-Based Risk-Neutral Distributions by Allan Malz which is summarized below. In 1978, Breeden and Litzenberger presented a method to derive this distribution for an underlying asset from observable option prices. Risk-neutral probability distributions (RND) are used to compute the fair value of an asset as a discounted conditional expectation of its future payoff.
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