Pricing American Options Fitting the Smile

Abstract

This paper is a compendium of results-theoretical and computational-from a series of recent papers developing a new American option valuation technique based on linear programming (LP). Some further computational results are included for completeness. A proof of the basic analytical theorem is given, as is the analysis needed to solve the inverse problem of determining local (one-factor) volatility from market data. The ideas behind a fast accurate revised simplex method, whose performance is linear in time and space discretizations, are described and the practicalities of fitting the volatility smile are discussed. Numerical results are presented which show the LP valuation technique to be extremely fast-lattice speed with PDE accuracy. American options valued in the paper range from vanilla, through exotic with constant volatility, to exotic options fitting the volatility smile. Copyright Blackwell Publishers, Inc..

A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics

“…In local volatility pricers, the IVS is employed to calibrate the local volatility function to the market. Then, the BS partial differential equation with generalized volatility function is solved for pricing the exotic options, Andersen and Brotherton-Ratcliffe (1997) and Dempster and Richards (2000). Therefore, unlike to the BS model, prices depend on the entire IVS, and not simply on the implied volatility at a specific strike.…”

“…This is particularly obvious for the pricing systems relying on the local volatility models. Initially developed by Dupire (1994) and Der- man and Kani (1994), they are in wide-spread use in form of the efficient implementations by Andersen and Brotherton-Ratcliffe (1997) and Dempster and Richards (2000). Thus, refined statistical model building of the IVS determines vitally the accuracy of applications in trading and risk-management.…”

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

“…Since the option prices evaluated by Black-Scholes option pricing model are equal to the real market option prices with implied volatility, modeling the IVS directly becomes a major concern recently. Especially, A deterministic volatility function of the underlying price and time that was originally studied by Dupire (1994), Derman and Kani (1994), and Rubinstein (1994), and inserted into highly efficient pricing models by Andersen and BrothertonRatcliffe (1997), Dempster and Richards (2000), and Kim et al (2009) are very much dependent on an estimate of the IVS. Dumas et al (1998) proposed a few volatility functions as functions of both strike and time to expiration for S&P 500 options.…”

Arbitrage-free smoothing of the implied volatility surface

“…A crucial property of the implied volatility surface (IVS) is the absence of arbitrage. Especially, local volatility models that were initially proposed by Dupire (1994), Derman and Kani (1994), and Rubinstein (1994) and put into highly efficient pricing engines by Andersen and Brotherton-Ratcliffe (1997) and Dempster and Richards (2000) amongst others, heavily rely on an arbitrage-free estimate of the IVS: if there are arbitrage violations, negative transition probabilities and negative local volatilities ensue, which obstructs the convergence of the algorithm solving the underlying generalized Black Scholes partial differential equation. While occasional arbitrage violations may safely be overridden by some ad hoc approach, the algorithm breaks down, if the violations become too excessive.…”

“…Rather, the set of knots and the second-order derivatives is passed to the pricing engine, and can be evaluated directly on the desired grid. Thus, the method can be integrated into local volatility pricing engines such those proposed by Andersen and Brotherton-Ratcliffe (1997) and Dempster and Richards (2000).…”