Pricing perpetual options using Mellin transforms

AbstractWe have derived the expression for the free boundary and price of an American perpetual put as the limit of a finite-lived option

Pricing American options with Mellin transforms

Mellin transforms in option pricing theory were introduced by Panini and Srivastav (2004). In this contribution, we generalize their results to European power options. We derive Black-Scholes-Merton-like valuation formulas for European power put options using Mellin transforms. Thereafter, we restrict our attention to plain vanilla options on dividend-paying stocks and derive the integral equations to determine the free boundary and the price of American put options using Mellin transforms. We recover a result found by Kim (1990) regarding the optimal exercise price of American put options at expiry and prove the equivalence of integral representations herein, the representation derived by Kim (1990), Jacka (1991), and by Carr et al. (1992). Finally, we extend the results obtained in Panini and Srivastav (2005) and show how the Mellin transform approach can be used to derive the valuation formula for perpetual American put options on dividend-paying stocks. --Mellin transform,Power option,American put option,Free boundary,Integral representation

On modified Mellin transforms, Gauss-Laguerre quadrature, and the valuation of American call options

We extend a framework based on Mellin transforms and show how to modify the approach to value American call options on dividend paying stocks. We present a new integral equation to determine the price of an American call option and its free boundary using modi ed Mellin transforms. We also show how to derive the pricing formula for perpetual American call options using the new framework. A recovery of a result due to Kim (1990) regarding the optimal exercise price at expiry is also presented. Finally, we apply Gauss-Laguerre quadrature for the purpose of an efficient and accurate numerical valuation. --Modified Mellin transform,American call option,Integral representation

Pricing American Options with Mellin Transforms

Mellin transforms in option pricing theory were introduced by Panini and Srivastav (2004). In this contribution, we generalize their results to European power options. We derive Black-Scholes-Merton-like valuation formulas for European power put options using Mellin transforms. Thereafter, we restrict our attention to plain vanilla options on dividend-paying stocks and derive the integral equations to determine the free boundary and the price of American put options using Mellin transforms. We recover a result found by Kim (1990) regarding the optimal exercise price of American put options at expiry and prove the equivalence of integral representations herein, the representation derived by Kim (1990), Jacka (1991), and by Carr et al. (1992). Finally, we extend the results obtained in Panini and Srivastav (2005) and show how the Mellin transform approach can be used to derive the valuation formula for perpetual American put options on dividend-paying stocks

On Modified Mellin Transforms, Gauss-Laguerre Quadrature, and the Valuation of American Call Options

We extend a framework based on Mellin transforms and show how to modify the approach to value American call options on dividend paying stocks. We present a new integral equation to determine the price of an American call option and its free boundary using modied Mellin transforms. We also show how to derive the pricing formula for perpetual American call options using the new framework. A recovery of a result due to Kim (1990) regarding the optimal exercise price at expiry is also presented. Finally, we apply Gauss-Laguerre quadrature for the purpose of an efficient and accurate numerical valuation

On a New Technique for the Solution of the Black-Scholes Partial Differential Equation for European Call Option

This paper presents a new technique for the solution of the Black-Scholes partial differential equation for European call option using a method based on the modified Mellin transform. We also used the modified Mellin transform method to determine the price of European call option. The modified Mellin transform method is mutually consistent and agrees with the values of Black-Scholes model as shown i

경로에 의존하는 미국형 옵션의 해석적 가치 평가

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 8. 강명주.American options are type of options that can be exercised anytime during their life. Therefore, the valuation of such options is usually classified as optimal stopping problems or free boundary problems. I derive the analytic pricing formulas and integral equations of American chained options, Russian options with finite time horizon, American floating strike lookback options, and American maximum quanto options. To verify the derived pricing formula and the integral equation satisfied by the free boundary are correct, we numerically solve the derived integral equations using recursive integration method or simple iterative method.Chapter 1 Introduction 1 Chapter 2 Chained knock-in barrier option 3 2.1 Preliminaries 4 2.2 Analytic Valuation of Chained American Barrier Options 6 2.2.1 Crossing a single barrier 6 2.2.2 Crossing two barriers 10 2.3 Numerical results 14 2.4 Summary 19 Chapter 3 Russian option with finite time horizon 22 3.1 Model Formulation: Free Boundary Problem 23 3.2 Inhomogeneous Black-Scholes Partial Differential Equation:Mixed Boundary Problem 26 3.3 Integral Equation of Russian Options with Finite Time Horizon:Premium Decomposition 34 3.3.1 Case of r 6= q 35 3.3.2 Case of r = q 37 3.4 Valuing Russian Options: Perpetual Case 42 3.5 Numerical Results 47 3.5.1 Recursive Integration Method for Russian Option with Finite Time Horizon 47 3.5.2 Results : Qualitative analysis 51 3.5.3 Results : Comparison with Other Methods 51 3.6 Summary 59 Chapter 4 American floating strike lookback option 60 4.1 Model formulation 62 4.2 Inhomogeneous Black-Scholes equation with Neumann boundary condition 65 4.3 Integral equation representation of American floating strike lookback option 71 4.4 Perpetual American floating strike lookback option 78 4.5 Summary 85 Chapter 5 American maximum exchange rate quanto lookback option 87 5.1 Model Formulation : Free boundary problem 89 5.2 Derivation of analytic solution for two-dimensional inhomogeneous Black-Scholes PDE 93 5.2.1 Two-dimensional Inhomogeneous Black-Scholes parabolic PDE on Unrestricted Domain 93 5.2.2 Two-dimensional inhomogeneous Black-Scholes parabolic PDE : Dirichlet Boundary Conditions 95 5.2.3 Two-dimensional inhomogeneous Black-Scholes parabolic PDE : Mixed Boundary Conditions 98 5.3 Analytic Pricing of American Maximum Exchange-Rate Quanto Lookback Options 105 5.3.1 European Maximum Exchange Rate Quanto Lookback Options 106 5.3.2 American Maximum Exchange Rate Quanto Lookback Options 109 5.4 Numerical Results 114 5.4.1 An iterative method 114 5.4.2 Forward shooting grid method for two-state model 116 5.4.3 Implications 118 5.5 Summary 126 Appendix A Basic Properties of Mellin Transforms 133 A.1 Properties of Mellin transform 133 A.2 Properties of double Mellin transform 135 Appendix B Some useful lemmas 137 Abstract (in Korean) 138Docto

Black-Scholes Partial Differential Equation In The Mellin Transform Domain

Abstract: This paper presents Black-Scholes partial differential equation in the Mellin transform domain. The Mellin transform method is one of the most popular methods for solving diffusion equations in many areas of science and technology. This method is a powerful tool used in the valuation of options. We extend the Mellin transform method proposed by Panini and Srivasta

Analytic Solution of Black-Scholes-Merton European Power Put Option Model on Dividend Yield with Modified-Log-Power Payoff Function

This paper proposes a framework based on the celebrated transform of Mellin type (MT) for the analytic solution of the Black-Scholes-Merton European Power Put Option Model (BSMEPPOM) on Dividend Yield (DY) with Modified-Log-Power Payoff Function (MLPPF) under the geometric Brownian motion. The MT has the capability of tackling complex functions by means of its fundamental properties and it is closely related to other well-known transforms such as Laplace and Fourier types. The main goal of this paper is to use MT to obtain a valuation formula for the European Power Put Option (EPPO) which pays a DY with MLPPF. By means of MT and its inversion formula, the price of EPPO on DY was expressed in terms of integral equation. Moreover, the valuation formula of EPPO was obtained with the help of the convolution property of MT and final time condition. The MT was tested on an illustrative example in order to measure its performance, effectiveness and suitability. The MLPPF was compared with other existing payoff functions. Hence, the effect of DY on the pricing of EPPO with MLPPF was also investigated

Applications of the Scaled Laplace Transform in some Financial and Risk Models

In this work, we propose several approximations for the evaluation of some risk measures and option prices based on the inversion of the scaled version of the Laplace transform which was suggested by Mnatsakanov and Sarkisian (2013). The classical risk model is considered for the evaluation of probability of ultimate ruin. Approximations of the inverse function of the ruin probability is proposed and its natural extension to the computation of Value at Risk, a benchmark risk measure for insurance and finance sectors, is proposed. The recovery of the distributions of bivariate models and bivariate aggregate claims amount on insurance policies is suggested. The proposed method is also applied to the Black-Scholes model for the estimation of option prices. Simulation studies and results are presented to demonstrate the performance of the proposed method