Abstract
A technique known as calibration is often used when a given option pricing model is fitted to observed financial data. This entails choosing the parameters of the model so as to minimise some discrepancy measure between the observed option prices and the prices calculated under the model in question. This procedure does not take the historical values of the underlying asset into account. In this paper, the density function of the log-returns obtained using the calibration procedure is compared to a density estimate of the observed historical log-returns. Three models within the class of geometric Lévy process models are fitted to observed data; the Black-Scholes model as well as the geometric normal inverse Gaussian and Meixner process models. The numerical results obtained show a surprisingly large discrepancy between the resulting densities when using the latter two models. An adaptation of the calibration methodology is also proposed based on both option price data and the observed historical log-returns of the underlying asset. The implementation of this methodology limits the discrepancy between the densities in question.
Highlights
It is well known that the arbitrage free price of an option can be calculated as the discounted value of the expected payoff of the option in question
When calibration is used in order to fit these models to observed option prices, it is observed that the risk neutral measure closely resembles the objective measure in the case of the Black-Scholes model
If the Esscher transform is used in order to find a locally equivalent martingale measure (LEMM), Lt follows a N ◦ IG (α, β + γ∗, μ, δ) process, where γ = γ∗ is the solution to r = μ + δ α2 − (β + γ)2 − α2 − (β + γ + 1)2
Summary
It is well known that the arbitrage free price of an option can be calculated as the discounted value of the expected payoff of the option in question. When calibration is used in order to fit these models to observed option prices, it is observed that the risk neutral measure closely resembles the objective measure in the case of the Black-Scholes model. Even if a sufficient amount of historical data were available, the prices of options should be calculated under current market conditions This means that returns observed several years before are outdated and should not be taken into account when fitting the model. The models used in practice are often fitted using daily data and the parameters reported are shown in annual terms This mismatch in the time horisons considered can obscure some of the differences between the objective and risk neutral measures; this is demonstrated using a numerical example in the paper.
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