Recovering Volatility from Option Prices by Evolutionary Optimization

Abstract

We propose a probabilistic approach for estimating parameters of an option pricing model from a set of observed option prices. Our approach is based on a stochastic optimization algorithm which generates a random sample from the set of global minima of the in-sample pricing error and allows for the existence of multiple global minima. Starting from an IID population of candidate solutions drawn from a prior distribution of the set of model parameters, the population of parameters is updated through cycles of independent random moves followed by "selection" according to pricing performance. We examine conditions under which such an evolving population converges to a sample of calibrated models. The heterogeneity of the obtained sample can then be used to quantify the degree of ill-posedness of the inverse problem: it provides a natural example of a coherent measure of risk, which is compatible with observed prices of benchmark ("vanilla") options and takes into account the model uncertainty resulting from incomplete identification of the model. We describe in detail the algorithm in the case of a diffusion model, where one aims at retrieving the unknown local volatility surface from a finite set of option prices, and illustrate its performance on simulated and empirical data sets of index options.

Option pricing model calibration using a real-valued quantum-inspired evolutionary algorithm

“…This is because the two parameters θ and ν have slightly offsetting effects. Cont and Tankov [3] also graph a similar objective function for the V G option pricing model using DAX index option data and illustrate the potential for gradient based optimisers to converge to a local rather than the global minimum.…”

“…One of the difficulties in model calibration is that the available market information may be insufficient to completely identify the parameters of a model [3]. If the model is sufficiently rich relative to the number of market prices available, a number of possible parameter vector combinations will be compatible with market prices and the objective function G (Θ) may not be convex function of Θ.…”

Model-free Representation of Pricing Rules as Conditional Expectations

“…The specication of an objective probability measure typically plays no role in this process. In fact, in most cases (Black-Scholes model, diusion models, stochastic volatility models,..) the probability measures (Q θ , θ ∈ E) are mutually singular so the model selection problem cannot be formulated as a search among martingale measures equivalent to a given measure P [2]. So, any characterization of absence of arbitrage in terms of equivalent martingale measure would appear as inconsistent with the practice of specifying and calibrating pricing rules in this way.…”

“…In fact, in most cases (Black-Scholes model, diusion models, stochastic volatility models,..) the probability measures (Q θ , θ ∈ E) are mutually singular: for example, if Q σ designates a Black-Scholes model with volatility parameter σ then σ 1 = σ 2 entails that Q σ 1 and Q σ 2 are mutually singular measures. So, the model selection problem cannot be formulated as a search among martingale measures equivalent to a given measure P [2].…”

Retrieving Lévy Processes from Option Prices: Regularization of an Ill-posed Inverse Problem

“…The number of observed options can be large ( 100 − 200 for index options) and the Black-Scholes model has to be replaced with models with richer structure such as nonlinear diffusion models [17] or models with jumps [12]. The inverse problem is ill-posed in these settings [13,33] and various methods have been proposed for solving it in a stable manner, mostly in the framework of diffusion models [3,4,5,7,14,17,25,32,33].We study in this paper the calibration problem for the class of option pricing models with jumps -exponential Lévy models-where the risk-neutral dynamics of the logarithm of the stock price is given by a Lévy process. The problem is then to choose the Lévy process -described by its Lévy measure-in a way compatible with a set of observed option prices.…”