The Effect of Non-Smooth Payoffs on the Penalty Approximation of American Options

This article combines various methods of analysis to draw a comprehensive picture of penalty approximations to the value, hedge ratio, and optimal exercise strategy of American options. While convergence of the penalised solution for sufficiently smooth obstacles is well established in the literature, sharp rates of convergence and particularly the effect of gradient discontinuities (i.e., the omni-present `kinks' in option payoffs) on this rate have not been fully analysed so far. This effect becomes important not least when using penalisation as a numerical technique. We use matched asymptotic expansions to characterise the boundary layers between exercise and hold regions, and to compute first order corrections for representative payoffs on a single asset following a diffusion or jump-diffusion model. Furthermore, we demonstrate how the viscosity theory framework in [Jakobsen, 2006] can be applied to this setting to derive upper and lower bounds on the value. In a small extension to [Bensoussan & Lions, 1982], we derive weak convergence rates also for option sensitivities for convex payoffs under jump-diffusion models. Finally, we outline applications of the results, including accuracy improvements by extrapolation.Comment: 34 Pages, 10 Figure

On a free boundary problem for an American put option under the CEV process

We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free bondary satisfies a nonlinear integral equation, and analyze it in the limit of small $\rho$ = $2r/ \sigma^2$, where $r$ is the interest rate and $\sigma$ is the volatility. We use perturbation methods to find that the free boundary behaves differently for five ranges of time to expiry.Comment: 14 pages, 0 figure

On nonlinear models of markets with finite liquidity: Some cautionary notes

The recent financial crisis and related liquidity issues have illuminated an urgent need for a better understanding of the effects of limited liquidity on all aspects of the financial system. This paper considers such effects on the Black-Scholes-Merton financial model, which for the most part result in highly nonlinear partial differential equations (PDEs). We investigate in detail a model studied by Schönbucher and Wilmott (2000) which incorporates the price impact of option hedging strategies. First, we consider a first-order feedback model, which leads to the exceptional case of a linear PDE. Numerical results, and more particularly an asymptotic approach close to option expiry, reveal subtle differences from the Black-Scholes-Merton model. Second, we go on to consider a full-feedback model in which price impact is fully incorporated into the model. Here, standard numerical techniques lead to spurious results in even the simplest cases. An asymptotic approach, valid close to expiry, is mounted, and a robust numerical procedure, valid for all times, is developed, revealing two distinct classes of behavior. The first may be attributed to the infinite second derivative associated with standard option payoff conditions, for which it is necessary to admit solutions with discontinuous first derivatives; perhaps even more disturbingly, negative option values are a frequent occurrence. The second failure (applicable to smoothed payoff functions) is caused by a singularity in the coefficient of the diffusion term in the option-pricing equation. Our conclusion is that several classes of model in the literature involving permanent price impact irretrievably break down (i.e., there is insufficient "financial modeling" in the pricing equation). Our analysis should provide the information necessary to avoid such pitfalls in the future. © 2010 Society for Industrial and Applied Mathematics

High dimensional American options

Pricing single asset American options is a hard problem in mathematical finance. There are no closed form solutions available (apart from in the case of the perpetual option), so many approximations and numerical techniques have been developed. Pricing multi–asset (high dimensional) American options is still more difficult. We extend the method proposed theoretically by Glasserman and Yu (2004) by employing regression basis functions that are martingales under geometric Brownian motion. This results in more accurate Monte Carlo simulations, and computationally cheap lower and upper bounds to the American option price. We have implemented these models in QuantLib, the open–source derivatives pricing library. The code for many of the models discussed in this thesis can be downloaded from quantlib.org as part of a practical pricing and risk management library. We propose a new type of multi–asset option, the “Radial Barrier Option” for which we find analytic solutions. This is a barrier style option that pays out when a barrier, which is a function of the assets and their correlations, is hit. This is a useful benchmark test case for Monte Carlo simulations and may be of use in approximating multi–asset American options. We use Laplace transforms in this analysis which can be applied to give analytic results for the hitting times of Bessel processes. We investigate the asymptotic solution of the single asset Black–Scholes–Merton equation in the case of low volatility. This analysis explains the success of some American option approximations, and has the potential to be extended to basket options

Derivative pricing and optimal execution of portfolio transactions in finitely liquid markets

In real markets, to some degree, every trade will incur a non-zero cost and will influence the price of the asset traded. In situations where a dynamic trading strategy is implemented these liquidity effects can play a significant role. In this thesis we examine two situations in which such trading strategies are inherent to the problem; that of pricing a derivative contingent on the asset and that of executing a large portfolio transaction in the asset. The asset's finite liquidity has been incorporated explicitly into its price dynamics using the Bakstein-Howison model [4]. Using this model we have derived the no-arbitrage price of a derivative on the asset and have found a true continuous-time equation when the bid-ask spread in the asset is neglected. Focussing on this pure liquidity case we then employ an asymptotic analysis to examine the price of a European call option near strike and expiry where the liquidity effects are shown to be most significant and closed-form expressions for the price are derived in this region. The asset price model is then extended to incorporate the empirical fact that an asset's liquidity mean reverts stochastically. In this situation the pricing equation is analyzed using the multiscale asymptotic technique developed by Fouque, Papanicolaou, and Sircar [22] and a simplified pricing and calibration framework is developed for an asset possessing liquidity risk. Finally, the derivative pricing framework (both with and without liquidity risk) is applied to a new contract termed the American forward which we present as a possible hedge against an asset's liquidity risk. In the second part of the thesis we investigate how to optimally execute a large transaction of a finitely liquid asset. Using stochastic dynamic programming and attempting only to minimize the transaction's cost, we first find that the optimal strategy is static and contains the naive strategy found in previous studies, but with an extra term to account for interest rates neglected by those studies. Including time risk into the optimization procedure we find expressions for the optimal strategy in the extreme cases when the trader's aversion to this risk is very small and very large. In the former case the optimal strategy is simply the cost-minimization strategy perturbed by a small correction proportional to the trader's level of risk aversion. In the latter case the problem is shown to be much more difficult; we analyze and derive implicit closed-form solutions to the much-simplified perfect liquidity case and show numerical results to demonstrate the agreement of the solution with our intuition

Pricing American Options by the Black-Scholes Equation with a Nonlinear Volatility Function

Doutoramento em Matemática Aplicada à Economia e à GestãoIn this thesis we are concerned with the study of American-style options in presence of variable transactions costs. This leads to consider some generalized Black-Scholes equations with a nonlinear volatility function depending on the product of the underlying asset price and the second derivative of the option price. Mathematically, this involves the study of a free boundary problem for a nonlinear parabolic equation. The fully nonlinear character of the corresponding differential operator induces increased difficulties. By overcoming adequately those difficulties, we obtain qualitative and quantitative results regarding both types of American-style options, that is put and call options, as described next. Firstly, we investigate the qualitative and quantitative behaviour of a solution to the problem of pricing American style perpetual put options. We assume the option price is a solution to a stationary generalized Black-Scholes equation with a nonlinear volatility function. We prove existence and uniqueness of a solution to the free boundary problem. We derive a single implicit integral equation for the free boundary position and a closed form formula for the option price. It is a generalization of the well-known explicit closed form solution derived by Merton for the case of constant volatility. We also present results of numerical computations for the free boundary position, option price and their dependence on model parameters. Secondly, we analyze a nonlinear generalization of the Black{Scholes equation for pricing American-style call options, with nonlinear volatility. This model generalizes the well-known Leland model with constant transaction costs. Due to the fully nonlinear nature of the differential operator that appears in the model, the direct computation of the nonlinear complementarity problem becomes harder and unstable. Therefore, we propose a new approach to reformulate the nonlinear complementarity problem in terms of the new transformed variable for which the differential operator has the form of a quasilinear parabolic operator. We derive the nonlinear complementarity problem for the transformed variable in order to apply the Gamma transformation for American style options. We then solve the variacional problem by means of the modi ed projected successive over relaxation (PSOR) for constructing an effective numerical scheme for discretization of the Gamma variacional inequality. Finally, we present several computational examples of the nonlinear Black- Scholes equation for pricing American-style call options in the presence of variable transaction costs.Esta dissertação incide sobre o estudo de opções americanas admitindo a existência de custos de transação variáveis. Tal estudo leva-nos a considerar equações de Black-Scholes generalizadas, com uma função de volatilidade não linear que depende do produto do preço do ativo subjacente e da segunda derivada do preço da opção, o que, do ponto de vista matemático, implica a análise de um problema de fronteira livre para uma equação parabólica não linear. O carácter não linear do operador diferencial correspondente gera dificuldades acrescidas. Contudo, um estudo adequado a condição de não linearidade permite-nos estabelecer resultados qualitativos e quantitativos sobre os dois tipos de opções americanas, mas precisamente, opções de venda e de compra, conforme descrito a seguir. Em primeiro lugar, investigamos o comportamento qualitativo e quantitativo de uma solução do problema de apreçamento de opções de venda perpétuas do tipo americano. Assumimos que o preço da opção é uma solução para uma equação de Black- Scholes generalizada estacionária com uma função de volatilidade não linear. Provamos existência e unicidade de uma solução do problema da fronteira livre. Derivamos uma equação integral implícita para o valor de fronteira livre e uma solução de forma fechada para o preço da opção. É uma generalização da conhecida solução de forma fechada explícita derivada por Merton para o caso de volatilidade constante. Também apresentamos resultados de cálculo numérico para o valor de fronteira livre, assim como para preço da opção e sua dependência dos parâmetros do modelo. Em segundo lugar, analisamos uma generalização não linear da equação de Black-Scholes para o apreçamento de opções de compra de tipo americano, com volatilidade não linear. Este modelo generaliza o conhecido modelo de Leland com custos de transação constantes. Devido à natureza totalmente não linear do operador diferencial que aparece no modelo, o cálculo direto do problema de complementaridade não linear torna-se mais difícil e instável. Portanto, propomos uma nova abordagem para reformular o problema de complementaridade não linear em termos de uma nova variável para a qual o operador diferencial tem a forma de um operador parabólico quase-linear. Derivamos o problema de complementaridade não linear para a variável transformada afim de aplicar a transformação Gama para opções de tipo americano. Em seguida, resolvemos o problema variacional por meio do relaxamento projetado sucessivo modificado (PSOR) para construir um esquema numérico eficaz para discretização da desigualdade variacional Gama. Finalmente, apresentamos vários exemplos computacionais da equação não linear de Black-Scholes para apreçamento de opções de compra no tipo americano em presença de custos de transação variáveis.info:eu-repo/semantics/publishedVersio

A non-arbitrage liquidity model with observable parameters for derivatives

We develop a parameterised model for liquidity effects arising from the trading in an asset. Liquidity is defined via a combination of a trader's individual transaction cost and a price slippage impact, which is felt by all market participants. The chosen definition allows liquidity to be observable in a centralised order-book of an asset as is usually provided in most non-specialist exchanges. The discrete-time version of the model is based on the CRR binomial tree and in the appropriate continuous-time limits we derive various nonlinear partial differential equations. Both versions can be directly applied to the pricing and hedging of options; the nonlinear nature of liquidity leads to natural bid-ask spreads that are based on the liquidity of the market for the underlying and the existence of (super-)replication strategies. We test and calibrate our model set-up empirically with high-frequency data of German blue chips and discuss further extensions to the model, including stochastic liquidity

Pricing and hedging american options analytically: A perturbation method

This paper studies the critical stock price of American options with continuous dividend yield. We solve the integral equation and derive a new analytical formula in a series form for the critical stock price. American options can be priced and hedged analytically with the help of our critical-stock-price formula. Numerical tests show that our formula gives very accurate prices. With the error well controlled, our formula is now ready for traders to use in pricing and hedging the S&P 100 index options and for the Chicago Board Options Exchange to use in computing the VXO volatility index. © 2010 Wiley Periodicals, Inc.postprin