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...

A penalty method for American options with jump diffusion processes

“…In order to match observed market prices for options, traders use a matrix of implied volatilities [33], or generate a volatility surface [8]. However, as discussed in [3], volatility surfaces tend not to be very stable as a function of time.…”

Jacobian-Free Implicit Inner-Iteration Preconditioner for Nonlinear Least Squares Problems

“…The first two problems are arising in data fitting. The third problem is a volatility surface construction problem arising in finance [8], whose Jacobian matrices are dense. The performance of nine methods is compared on all these testing problems.…”

“…For the third problem, we consider a more complex and realistic nonlinear least squares problem arising in finance [8]. The corresponding Jacobian matrices in this problem are dense and do not have any structures.…”

“…The function σ (S t , t) is called the local volatility function. Given the function σ (S, t), initial price S 0 , strike price K , expiry time T and μ(S, t) is the difference between risk free interest rate and dividend, r − q, which is constant, the European call option of an underlying asset satisfying (4.2) can be priced by the following generalized Black-Scholes equation [8] ∂ V ∂t…”

“…We can use Crank-Nicholson finite difference solution strategy for solving (4.3) based on discretization on a uniform grid [8]. In practice, it is hard to have the explicit form of the local volatility function σ (S, t), but we can observe the trading prices of these European options with various strikes and expiries from the market.…”