Abstract
Estimates of risk-neutral densities of future asset returns have been commonly used for pricing new financial derivatives, detecting profitable opportunities, and measuring central bank policy impacts. We develop a new nonparametric approach for estimating the risk-neutral density of asset prices and reformulate its estimation into a double-constrained optimization problem. We evaluate our approach using the S&P 500 market option prices from 1996 to 2015. A comprehensive cross-validation study shows that our approach outperforms the existing nonparametric quartic B-spline and cubic spline methods, as well as the parametric method based on the normal inverse Gaussian distribution. As an application, we use the proposed density estimator to price long-term variance swaps, and the model-implied prices match reasonably well with those of the variance future downloaded from the Chicago Board Options Exchange website.
Highlights
A financial derivative, such as option, swap, future, or forward contract, is an asset that is contingent on an underlying asset
We first provide a brief review of the cubic splines, quartic B-splines, and normal inverse Gaussian (NIG) approaches in the literature for recovering the risk-neutral density (RND)
Our study shows that our fair prices match the market prices of long-term variance swaps reasonably well
Summary
A financial derivative, such as option, swap, future, or forward contract, is an asset that is contingent on an underlying asset. Its fair price can be obtained by calculating the expected future payoff under a risk-neutral probability distribution. The problem of pricing a derivative can be addressed via estimating the risk-neutral density (RND) of the future payoff of the underlying asset. The estimated prices, known as fair prices, may help companies and investors to avoid financial risk or detect profitable opportunities, especially on over-the-counter (OTC) securities. Breeden and Litzenberger (1978) [4] were among the first to use option prices to estimate the riskneural probability distribution of the future payoff of the underlying asset. Among the financial products that can be used to recover the RND, European options are the most common ones, which give the investors rights to trade assets at a preagreed price (i.e., strike price) at the maturity date. Among all the underlying assets that options are written on, Standard and Poor’s 500 Index (S&P 500) is a popular one, which aggregates the values of stocks of 500 large companies traded on American stock exchanges and provides a credible view of the American stock market for investors
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