Abstract
Most of the models leading to an analytical expression for option prices are based on the assumption that underlying asset returns evolve according to a Brownian motion with drift. For some asset classes like commodities, a Brownian model does not fit empirical covariance and autocorrelation structures. This failure to replicate the covariance introduces a bias in the valuation of calendar spread exchange options. As the payoff of these options depends on two asset values at different times, particular care must be taken for the modeling of covariance and autocorrelation. This article proposes a simple alternative model for asset prices with sub-exponential, exponential and hyper-exponential autocovariance structures. In the proposed approach, price processes are seen as conditional Gaussian fields indexed by the time. In general, this process is not a semi-martingale, and therefore, we cannot rely on stochastic differential calculus to evaluate options. However, option prices are still calculable by the technique of the change of numeraire. A numerical illustration confirms the important influence of the covariance structure in the valuation of calendar spread exchange options for Brent against WTI crude oil and for gold against silver.
Highlights
Gaussian fields have been used for several decades in the analysis of spatial statistics
We refer the interested reader to the books of Cressie (1993), Adler (1981) or Matern (1986) for a detailed presentation of spatial statistics and the theory of Gaussian fields
In order to emphasize the role played by the correlation on option prices, we evaluate two exotic derivatives with a payoff depending on asset prices at different times
Summary
Gaussian fields have been used for several decades in the analysis of spatial statistics. Generalized the work of Kennedy (1994) and proposed a model for interest rates based on a two-dimensional random field. Modeled the intensity of mortality as a random field, in order to capture cross-generation (risk class) effects induced by the on-going management of portfolios of policies Based on this framework, Biagini et al (2017) studied the pricing and hedging of life insurance liabilities. Gaussian fields based on the spectral decomposition of the covariance function Within this approach, a simulated sample path is known continuously at all times. The second derivative is an Asian calendar spread exchange option with geometric averages This product pays the positive difference between the geometric average returns of two assets, calculated over different time intervals. Our results confirm that the misspecification of the autocorrelation function has a strong impact on the prices of calendar spread exchange options
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