Reconstructing the unknown local volatility function

Abstract

Abstract. Using market European option prices, a method for computing a smooth local volatility function in a 1-factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphasized that accurately approximating the true local volatility function is crucial in hedging even simple European options, and pricing exotic options. A spline functional approach is used: the local volatility function is represented by a spline whose values at chosen knots are determined by solving a constrained nonlinear optimization problem. The optimization formulation is amenable to various option evaluation methods; a partial differential equation implementation is discussed. Using a synthetic European call option example, we illustrate the capability of the proposed method in reconstructing the unknown local volatility function. Accuracy of pricing and hedging is also illustrated. Moreover, it is demonstrated that, using a different constant implied volatility for an option with different strike/maturity can produce erroneous hedge factors. In addition, real market European call option data on the S&P 500 stock index is used to compute the local volatility function; stability of the approach is demonstrated. * Presented at the first annual conference: Computational and Quantitative Finance '98, New York. This research was conducted using resources of the Cornell Theory Center, which is supported by Cornell University, New York State, the National Center for Research Resources at the National Institutes of Health, and members of the Corporate Partnership Program. 0 1. Introduction. An option pricing model establishes a relationship between the traded derivatives, the underlying asset and the market variables, e.g., volatility of the underlying asset [4,24]. Option pricing models are used in practice to price derivative securities given knowledge of the volatility and other market variables.The celebrated constant-volatility Black-Scholes model [4,24] is the most often used option pricing model in financial practice. This classical model assumes constant volatility; however, much recent evidence suggests that a constant volatility model is not adequate [27,26]. Indeed, numerically inverting the Black-Scholes formula on real data sets supports the notion of asymmetry with stock price (volatility skew), as well as dependence on time to expiration (volatility term structure). Collectively this dependence is often referred to as the volatility smile. The challenge is to accurately (and efficiently) model this volatility smile.In practice, the constant-volatility Black-Scholes model is often applied by simply using different volatility values for options with different strikes and maturities. In this paper, we refer to this approach as the constant implied volatility approach. Although this method works well for pricing European options, it is unsuitable for more complicated exotic options and options with ea...

Can anyone solve the smile problem?

“…Dupire's derivation essentially shows that any smile surface can be fitted by local volatility provided the model is non parametric, and it basically provides the non parametric formula. On the other hand, methods consisting in parameterizing the local volatility surface a priori (through spline functions or any other convenient representation), and in fitting the smile surface by minimization of a loss function (Coleman, Li, Verma (1999), Jackson, Sueli, Howison (1998)), suffer from the arbitrariness of the representation, particularly the arbitrariness of the behaviour of local volatility at the boundaries of the domain. Proponents of such approaches are always at pains trying to justify their favourite representation of the local volatility surface on grounds of its intuitive appeal or physical realism or what have you.…”

Numerical convergence properties of option pricing PDEs with uncertain volatility

“…More complicated models assume volatility surfaces across underlying asset prices and time (see, e.g. Andersen and Brotherton-Ratcliffe, 1998;Coleman et al, 1999, and references therein). These surfaces are often constructed by the implied volatilities under the Black and Scholes model for a variety of currently traded contracts.…”

Implied Volatility Function Approximation with Korean ELWs (Equity-Linked Warrants) via Gaussian Processes

“…Prior to simulations for real market data, we demonstrate the performance of the proposed method by using toy data set consisting of 22 European vanilla call options Coleman et al (1999). Let the initial stock price be 100, the riskless rate 5%, and the dividend rate 2%.…”