Pricing Partial Differential Equation Research Articles

High-order Gaussian RBF-FD methods for real estate index derivatives with stochastic volatility

The 2008 financial crisis showed the necessity for managing risk associated with the price movements of real estate assets. Appropriate financial instruments for hedging real estate portfolios include options on a real estate index. Based on empirical evidence that real estate returns exhibit time varying volatility, a real estate index derivative pricing model with stochastic volatility is proposed. An efficient procedure using Gaussian radial basis functions-finite differences is developed for the numerical solution of the two-dimensional pricing partial differential equation. Fourth-order convergence rates on uniform meshes for European options are numerically demonstrated and the algorithm is also shown to compute accurate American option prices. A numerical study of the stability of the scheme reveals that errors remain small.

Pricing American options with stochastic volatility and small nonlinear price impact: A PDE approach

In this paper, transaction costs as small nonlinear price impact are introduced into American option pricing under the Heston stochastic volatility model, forming a gap between option prices of the holder and writer. Through the use of a dynamic hedging strategy together with a known option independent of transaction costs, we derive two different nonlinear pricing partial differential equation systems for the holder and writer, respectively. A numerical algorithm is designed to solve the systems so that American option prices as well as the optimal exercise boundary can be simultaneously obtained. Examples are presented to illustrate the effect of transaction costs on both option prices and optimal exercise prices.

Dynamic Fund Protection for Property Markets

This article aims to investigate, from an academic perspective, a potential application of dynamic fund protection to protect a mortgagor of a property against the downside risk due to falling property price. The valuation of the dynamic fund protection is discussed through modeling the property price and interest rate, which may be considered to be two key factors having a material impact on the mortgagor. Specifically, a mean-reverting process is used to describe the property price and the Heath-Jarrow-Morton theory is used to model the interest rate. The valuation is done via the use of a forward measure approach. The numerical solution to the pricing partial differential equation is obtained via applying the finite difference method. Numerical results with some model parameters being estimated from the data on an Australian residential property index and Australian zero-coupon yields and forward rates are provided. The implications of the numerical results for the potential implementation of the dynamic fund protection are discussed.

Optimal Surrender Policy of Guaranteed Minimum Maturity Benefits in Variable Annuities with Regime-Switching Volatility

This study investigates valuation of guaranteed minimum maturity benefits (GMMB) in variable annuity contract in the case where the guarantees can be surrendered at any time prior to the maturity. In the event of the option being exercised early, early surrender charges will be applied. We model the underlying mutual fund dynamics under regime-switching volatility. The valuation problem can be reduced to an American option pricing problem, which is essentially an optimal stopping problem. Then, we obtain the pricing partial differential equation by a standard Markovian argument. A detailed discussion shows that the solution of the problem involves an optimal surrender boundary. The properties of the optimal surrender boundary are given. The regime-switching Volterra-type integral equation of the optimal surrender boundary is derived by probabilistic methods. Furthermore, a sensitivity analysis is performed for the optimal surrender decision. In the end, we adopt the trinomial tree method to determine the optimal strategy.

On the bond pricing partial differential equation in a convergence model of interest rates with stochastic correlation

Abstract Convergence models of interest rates are used to model a situation, where a country is going to enter a monetary union and its short rate is affected by the short rate in the monetary union. In addition, Wiener processes which model random shocks in the behaviour of the short rates can be correlated. In this paper we consider a stochastic correlation in a selected convergence model. A stochastic correlation has been already studied in different contexts in financial mathematics, therefore we distinguish differences which come from modelling interest rates by a convergence model. We provide meaningful properties which a correlation model should satisfy and afterwards we study the problem of solving the partial differential equation for the bond prices. We find its solution in a separable form, where the term coming from the stochastic correlation is given in its series expansion for a high value of the correlation.

A Quantum Algorithm for Linear PDEs Arising in Finance

We propose a hybrid quantum-classical algorithm, originated from quantum chemistry, to price European and Asian options in the Black-Scholes model. Our approach is based on the equivalence between the pricing partial differential equation and the Schrodinger equation in imaginary time. We devise a strategy to build a shallow quantum circuit approximation to this equation, only requiring few qubits. This constitutes a promising candidate for the application of Quantum Computing techniques (with large number of qubits affected by noise) in Quantitative Finance.

Modification terms to the Black–Scholes model in a realistic hedging strategy with discrete temporal steps

ABSTRACTOption pricing models generally require the assumption that stock prices are described by continuous-time stochastic processes. Although the time-continuous trading is easy to conceive theoretically, it is practically impossible to execute in real markets. One reason is because real markets are not perfectly liquid and purchase or sell any amount of an asset would change the asset price drastically. A realistic hedging strategy needs to consider trading that happens at discrete instants of time. This paper focuses on the impact and effect due to temporal discretization on the pricing partial differential equation (PDE) for European options. Two different aspects of temporal discretization are considered and used to derive the modification or correction source terms to the continuous pricing PDE. First the finite difference discretization of the standard Black–Scholes PDE and its modification due to discrete trading. Second the discrete trading leads to a discrete time re-balancing strategy that only cancels risks on average by using a discrete analogy of the stochastic process of the underlying asset. In both cases high order terms in the Taylor series expansion are used and the respective correction source terms are derived.

Valuation of GMWB under stochastic volatility

In this paper, we consider the pricing of the variable annuities with Guaranteed Minimum Withdrawal Benefit (GMWB) options, which are deferred, fund-linked annuity contracts, usually with a single premium payment up front. We use the Heston model for the financial market, which assumes the risky asset dynamic follows a stochastic volatility model. The structure of the GMWB policy enables us to adopt the pricing problem of a barrier option. We derive a GMWB pricing partial differential equation (PDE) and the insurance fee is computed by solving an optimization problem. The computed insurance fee is found to be underpriced in the market with a stochastic volatility model. A sensitivity analysis is performed to see the impacts of various parameters on the value of the policy, and the sensitivity of the pricing function with respect to the market risk.

Valuation of guaranteed minimum maturity benefits in variable annuities with surrender options

We present a numerical approach to the pricing of guaranteed minimum maturity benefits embedded in variable annuity contracts in the case where the guarantees can be surrendered at any time prior to maturity that improves on current approaches. Surrender charges are important in practice and are imposed as a way of discouraging early termination of variable annuity contracts. We formulate the valuation framework and focus on the surrender option as an American put option pricing problem and derive the corresponding pricing partial differential equation by using hedging arguments and Itô’s Lemma. Given the underlying stochastic evolution of the fund, we also present the associated transition density partial differential equation allowing us to develop solutions. An explicit integral expression for the pricing partial differential equation is then presented with the aid of Duhamel’s principle. Our analysis is relevant to risk management applications since we derive an expression of the delta for the sensitivity analysis of the guarantee fees with respect to changes in the underlying fund value. We provide algorithms for implementing the integral expressions for the price, the corresponding early exercise boundary and the delta of the surrender option. We quantify and assess the sensitivity of the prices, early exercise boundaries and deltas to changes in the underlying variables including an analysis of the fair insurance fees.

Valuation of Guaranteed Minimum Maturity Benefits in Variable Annuities with Surrender Options

We present a numerical approach to the pricing of guaranteed minimum maturity benefits embedded in variable annuity contracts in the case where the guarantees can be surrendered at any time prior to maturity that improves on current approaches. Surrender charges are important in practice and are imposed as a way of discouraging early termination of variable annuity contracts. We formulate the valuation framework and focus on the surrender option as an American put option pricing problem and derive the corresponding pricing partial differential equation by using hedging arguments and Ito's Lemma. Given the underlying stochastic evolution of the fund, we also present the associated transition density partial differential equation allowing us to develop solutions. An explicit integral expression for the pricing partial differential equation is then presented with the aid of Duhamel's principle. Our analysis is relevant to risk management applications since we derive an expression for the sensitivity of the guarantee fees with respect to changes in the underlying fund value (called the "delta"). We provide algorithms for implementing the integral expressions for the price, the corresponding early exercise boundary and the delta of the surrender option. We quantify and assess the sensitivity of the prices, early exercise boundaries and deltas to changes in the underlying variables including an analysis of the fair insurance fees.

Pricing Parisian down-and-in options

In this paper, we price American-style Parisian down-and-in call options under the Black–Scholes framework. Usually, pricing an American-style option is much more difficult than pricing its European-style counterpart because of the appearance of the optimal exercise boundary in the former. Fortunately, the optimal exercise boundary associated with an American-style Parisian knock-in option only appears implicitly in its pricing partial differential equation (PDE) systems, instead of explicitly as in the case of an American-style Parisian knock-out option. We also recognize that the “moving window” technique developed by Zhu and Chen (2013) for pricing European-style Parisian up-and-out call options can be adopted to price American-style Parisian knock-in options as well. In particular, we obtain a simple analytical solution for American-style Parisian down-and-in call options and our new formula is written in terms of four double integrals, which can be easily computed numerically.

A Homotopy Analysis Method for the Option Pricing PDE in Post-Crash Markets

Abstract We investigate a solution for the option pricing partial differential equation (PDE) in a market suffering from a financial crisis. The post-crash model assumes that the volatility is stochastic. It is an extension of the famous Black and Scholes model. Therefore, the option pricing PDE for the crisis model is a generalization of the Black and Scholes PDE. However, to the best knowledge, it does not have a closed form solution for the general case. In this paper, we provide a solution for the pricing PDE of a European option during financial crisis using the homotopy analysis method.

Pricing and Hedging Contingent Claims with Liquidity Costs and Market Impact

We study the influence of taking liquidity costs and market impact into account when hedging a contingent claim, first in the discrete time setting, then in continuous time. In the latter case and in a complete market, we derive a fully non-linear pricing partial differential equation, and characterizes its parabolic nature according to the value of a numerical parameter naturally interpreted as a relaxation coefficient for market impact. We then investigate the more challenging case of stochastic volatility models, and prove the parabolicity of the pricing equation in a particular case.

Pricing European options on deferred annuities

This paper considers the pricing of European call options written on pure endowment and deferred life annuity contracts, also known as guaranteed annuity options. These contracts provide a guaranteed value at the maturity of the option. The contract valuation is dependent on the stochastic interest rate and mortality processes. We assume single-factor stochastic square-root processes for both the interest rate and mortality intensity, with mortality being a time-inhomogeneous process. We then derive the pricing partial differential equation (PDE) and the corresponding transition density PDE for options written on the pure endowment and deferred annuity contracts. The general solution of the pricing PDE is derived as a function of the transition density function. We solve the transition density PDE by first transforming it to a system of characteristic PDEs using Laplace transform techniques and then applying the method of characteristics. Once an explicit expression for the density function is found, we then use sparse grid quadrature techniques to generate European call option prices on the pure endowment and deferred annuity contracts. This approach can easily be generalised to other contracts which are driven by similar stochastic processes presented in this paper. We test the sensitivity of the option prices by varying independent parameters in our model. As option maturity increases, the corresponding option prices significantly increase. The effect of mispricing the guaranteed annuity value is analysed, as is the benefit of replacing the whole-life annuity with a term annuity to remove volatility of the old age population.

Method of Lines Approach for Pricing American Spread Options

A numerical technique for the evaluation of American spread call options where the underlying asset dynamics evolve under the influence of a single stochastic variance process of the Heston (1993) type is presented. The numerical algorithm involves extending to the multi-dimensional setting the method of lines approach first presented in the option pricing framework by Meyer and van der Hoek (1997) when pricing the standard American put option. We transform the pricing partial differential equation to a corresponding system of ordinary differential equations with the aid of the Riccati transformation. We use the implicit trapezoidal rule to solve the resulting Riccati equations. Numerical results are presented outlining the effectiveness of the algorithm. The effects of stochastic volatility are explored by making comparisons with the geometric Brownian motion results.

Pricing European Options on Deferred Insurance Contracts

This paper considers the pricing of European call options written on pure endowment and deferred life annuity contracts, also known as guaranteed annuity options. These contracts provide a guaranteed value at the maturity of the option. The contract valuation is dependent on the stochastic interest rate and mortality processes. We assume single-factor stochastic square-root processes for both the interest rate and mortality intensity, with mortality being a time-inhomogeneous process. We then derive the pricing partial differential equation (PDE) and the corresponding transition density PDE for options written on the pure endowment and deferred annuity contracts. The general solution of the pricing PDE is derived as a function of the transition density function. We solve the transition density PDE by first transforming it to a system of characteristic PDEs using Laplace transform techniques and then applying the method of characteristics. Once an explicit expression for the density function is found, we then use sparse grid quadrature techniques to generate European call option prices on the pure endowment and deferred annuity contracts. This approach can easily be generalised to other contracts which are driven by similar stochastic processes presented in this paper. We test the sensitivity of the option prices by varying independent parameters in our model. As option maturity increases, the corresponding option prices significantly increase. The effect of mispricing the guaranteed annuity value is analysed, as is the benefit of replacing the whole-life annuity with a term annuity to remove volatility of the old age population.

Advances in pricing commodity futures: Multifactor models

The purpose of this paper is to obtain the risk-neutral drift of the state variables directly from market data in a multifactor commodity futures model. Note that the risk-neutral drift is a key factor in the general asset pricing theory but it is not observable.In this paper, we derive some exact results which relate the risk-neutral drifts to the slope of the commodity futures price jointly with the factors. This fact allows us to estimate some of the coefficients of the pricing partial differential equation directly from the futures data available in the markets. Moreover, we do not have to estimate either the physical drift or the market price of risk. Therefore we considerably reduce the number of functions to estimate and, as a consequence, we reduce the computational cost as well as the misspecification error. In order to investigate the finite sample properties of this approach we carry out some numerical experiments. Finally, an application to crude oil and natural gas futures contracts traded at the New York Mercantile Exchange (NYMEX) is also illustrated.

Partial differential equation representations of derivatives with bilateral counterparty risk and funding costs

We derive a partial differential equation (PDE) representation for the value of financial derivatives with bilateral counterparty risk and funding costs. The model is very general in that the funding rate may be different for lending and borrowing and the mark-to-market value at default can be specified exogenously. The buying back of a party's own bonds is a key part of the delta hedging strategy; we discuss how the cash account of the replication strategy provides sufficient funds for this. First, we assume that the mark-to-market value at default is given by the total value of the derivative, which includes counterparty risk.We find that the resulting pricing PDE becomes nonlinear, except in special cases, when the nonlinear terms vanish and a Feynman-Kac representation of the total value can be obtained. In these cases, the total value of the derivative can be decomposed into the default-free value plus a bilateral credit-valuation and funding adjustment. Second, we assume that the mark-to-market value at default is given by the counterparty-risk-free value of the derivative. This time, the resulting PDE is linear and the corresponding Feynman-Kac representation is used to decompose the total value of the derivative into the default-free value plus bilateral creditvaluation and funding-cost adjustments.Anumerical example shows that the effect on the valuation adjustments of a nonzero funding spread can be significant.

Interest rate risk premium and equity valuation

The authors employ the recent stochastic-control-based approach to financial mathematics to solve a problem of determination of the risk premium for a stochastic interest rate model, and the corresponding problem of equity valuation. The risk premium is determined explicitly, by means of solving a corresponding partial differential equation (PDE), in two forms: one, time-dependent, corresponding to a finite time contract expiration, and the simpler version corresponding to perpetual contracts. As stocks are perpetual contracts, when solving the problem of equity valuation, the latter form of the risk premium is used. By means of solving the general pricing PDE, an efficient equity valuation method was developed that is a combination of some sophisticated explicit formulas, and a numerical procedure.

A Closed-Form Approximation to the Stochastic-Volatility Jump-Diffusion Option Pricing Model

The risk-neutral pricing partial differential equation from a stochastic-volatility jump-diffusion model for a European call option is solved approximately using a regular perturbation and Green's function method. The closed-form approximate solution provides the same accuracy as the Heston model in in-sample and out-of-sample tests of S&P500 option pricing, with one to two orders of magnitude less computation time and much ease of numerical convergence.