As it is well-known, the centrepiece of model calibration is regularization which plays an import... more As it is well-known, the centrepiece of model calibration is regularization which plays an important role in transforming an ill-posed calibration problem into a stable and well-formulated one. Empirically, this realm of research has not been explored in much details in the literature. The goal of this paper is to understand and give an answer to a question concerning the pricing accuracy using the parameters resulted from correctly posed calibration problem in comparison to the ones inferred from a relaxed calibration. Our empirical findings indicate that regularized calibration is only recommended when considering out-of-sample pricing for a long time horizon.
The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 1994
A three-dimensional barotropic and baroclinic model is developed to simulate currents and tempera... more A three-dimensional barotropic and baroclinic model is developed to simulate currents and temperature changes induced by tropical cyclones traversing the continental shelf and slope region of the Australian North West Shelf. The model is based on a layered, explicit, finite difference formulation using the nonlinear primitive equations with an embedded entrainment scheme; a mixed-surface-layer interface is defined, which is allowed to shift from one interface to another, depending on the strength of a storm. The model has been tested by simulating the currents and temperature changes induced by tropical cyclones Orson and Ian. The modelled currents and temperatures agreed well with the available measured records except near the seabed. It has been found that the pre-storm currents have very little influence on the peak of the storm-induced currents and the currents in the wake of a tropical cyclone. The model contained no coefficients which must be calibrated for a particular applic...
While a classic result by Merton (1973,Bell J. Econ. Manage. Sci., 141–183) is that one should ne... more While a classic result by Merton (1973,Bell J. Econ. Manage. Sci., 141–183) is that one should never exercise an American call option just before expiration if the underlying stock pays no dividends, the conclusion of a very recent empirical study conducted by Jensen and Pedersen (2016,J. Financ. Econ.121(2), 278–299) suggests that one should ‘never say never’. This paper complements Jensen and Pedersen's empirical study by presenting a theoretical study on how to price American call options under a hard-to-borrow stock model proposed by Avellaneda and Lipkin (2009,Risk22(6), 92–97). Our study confirms that it is the lending fee that results in the early exercise of American call options and we shall also demonstrate to what extent lending fees have affected the early exercise decision.
In this paper, a numerical method with the combined Laplace transform and the Dual Reciprocity Me... more In this paper, a numerical method with the combined Laplace transform and the Dual Reciprocity Method proposed by Zhu et al. [I] is extended to solve time-dependent diffusion equations with nonlinear source terms. The nonlinear source terms are linearised first and then an iteration is carried out until a convergent value of the unknown function at a desired time level is obtained. We shall demonstrate, through the preliminary results of our numerical experiments, that the method allows an accurate numerical solution at any desired observation time to be calculated directly and efficiently. It is believed that such a high efficiency is achieved as a direct consequence of the fact that no domain integral is involved in the calculation and the time variable is also temporarily removed from the problem.
Guo and Zhu (2017) recently proposed an equal-risk pricing approach to the valuation of contingen... more Guo and Zhu (2017) recently proposed an equal-risk pricing approach to the valuation of contingent claims when short selling is completely banned and two elegant pricing formulae are derived in some special cases. In this paper, we establish a unified framework for this new pricing approach so that its range of application can be significantly expanded. The main contribution of our framework is that it not only recovers the analytical pricing formula derived by Guo and Zhu (2017) when the payoff is monotonic, but also numerically produces equal-risk prices for contingent claims with non-monotonic payoffs, a task which has not been accomplished before. Furthermore, we demonstrate how a short selling ban affects the valuation of contingent claims by comparing equal-risk prices with Black-Scholes prices.
Under mean-variance-utility framework, we propose a new portfolio selection model, which allows w... more Under mean-variance-utility framework, we propose a new portfolio selection model, which allows wealth and time both have influences on risk aversion in the process of investment. We solved the model under a game theoretic framework and analytically derived the equilibrium investment (consumption) policy. The results conform with the facts that optimal investment strategy heavily depends on the investor's wealth and future income-consumption balance as well as the continuous optimally consumption process is highly dependent on the consumption preference of the investor.
In this article, we derive a closed-form pricing formula for catastrophe equity put options under... more In this article, we derive a closed-form pricing formula for catastrophe equity put options under a stochastic interest rate framework. A distinguishing feature of the proposed solution is its simplified form in contrast to several recently published formulae that require evaluating several layers of infinite sums of $n$ -fold convoluted distribution functions. As an application of the proposed formula, we consider two different frameworks and obtain the closed-form formula for the joint characteristic function of the asset price and the losses, which is the only required ingredient in our pricing formula. The prices obtained by the newly derived formula are compared with those obtained using Monte-Carlo simulations to show the accuracy of our formula.
International Journal of Financial Engineering, 2018
In this paper, an analytical solution for the well-known Hamilton–Jacobi–Bellman (HJB) equation t... more In this paper, an analytical solution for the well-known Hamilton–Jacobi–Bellman (HJB) equation that arises from the Merton problem subject to general utility functions is presented for the first time. The solution presented in this paper is written in the form of a Taylor’s series expansion and constructed through the homotopy analysis method (HAM). The fully nonlinear HJB equation is decomposed into an infinite series of linear PDEs which can be solved analytically. Four examples are presented with the first two cases showing the accuracy of our solution approach; while the last two demonstrating its versatility.
International Journal of Theoretical and Applied Finance, 2019
In this paper, the pricing problem of variance and volatility swaps is discussed under a two-fact... more In this paper, the pricing problem of variance and volatility swaps is discussed under a two-factor stochastic volatility model. This model can be treated as a two-factor Heston model with one factor following the CIR process and another characterized by a Markov chain, with the motivation originating from the popularity of the Heston model and the strong evidence of the existence of regime switching in real markets. Based on the derived forward characteristic function of the underlying price, analytical pricing formulae for variance and volatility swaps are presented, and numerical experiments are also conducted to compare swap prices calculated through our formulae and those obtained under the Heston model to show whether the introduction of the regime switching factor would lead to any significant difference.
In this paper, we propose an integral equation approach for pricing an American-style Parisian up... more In this paper, we propose an integral equation approach for pricing an American-style Parisian up-and-out call option under the Black–Scholes framework. The main difficulty of pricing this option lies in the determination of its optimal exercise price, which is a three-dimensional surface, instead of a two-dimensional (2-D) curve as is the case for a “one-touch” barrier option. In our approach, we first reduce the 3-D pricing problem to a 2-D one by using the “moving window” technique developed by Zhu and Chen (2013, Pricing Parisian and Parasian options analytically. Journal of Economic Dynamics and Control, 37(4): 875-896), then apply the Fourier sine transform to the 2-D problem to obtain two coupled integral equations in terms of two unknown quantities: the option price at the asset barrier and the optimal exercise price. Once the integral equations are solved numerically by using an iterative procedure, the calculation of the option price and the hedging parameters is straightf...
As it is well-known, the centrepiece of model calibration is regularization which plays an import... more As it is well-known, the centrepiece of model calibration is regularization which plays an important role in transforming an ill-posed calibration problem into a stable and well-formulated one. Empirically, this realm of research has not been explored in much details in the literature. The goal of this paper is to understand and give an answer to a question concerning the pricing accuracy using the parameters resulted from correctly posed calibration problem in comparison to the ones inferred from a relaxed calibration. Our empirical findings indicate that regularized calibration is only recommended when considering out-of-sample pricing for a long time horizon.
The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 1994
A three-dimensional barotropic and baroclinic model is developed to simulate currents and tempera... more A three-dimensional barotropic and baroclinic model is developed to simulate currents and temperature changes induced by tropical cyclones traversing the continental shelf and slope region of the Australian North West Shelf. The model is based on a layered, explicit, finite difference formulation using the nonlinear primitive equations with an embedded entrainment scheme; a mixed-surface-layer interface is defined, which is allowed to shift from one interface to another, depending on the strength of a storm. The model has been tested by simulating the currents and temperature changes induced by tropical cyclones Orson and Ian. The modelled currents and temperatures agreed well with the available measured records except near the seabed. It has been found that the pre-storm currents have very little influence on the peak of the storm-induced currents and the currents in the wake of a tropical cyclone. The model contained no coefficients which must be calibrated for a particular applic...
While a classic result by Merton (1973,Bell J. Econ. Manage. Sci., 141–183) is that one should ne... more While a classic result by Merton (1973,Bell J. Econ. Manage. Sci., 141–183) is that one should never exercise an American call option just before expiration if the underlying stock pays no dividends, the conclusion of a very recent empirical study conducted by Jensen and Pedersen (2016,J. Financ. Econ.121(2), 278–299) suggests that one should ‘never say never’. This paper complements Jensen and Pedersen's empirical study by presenting a theoretical study on how to price American call options under a hard-to-borrow stock model proposed by Avellaneda and Lipkin (2009,Risk22(6), 92–97). Our study confirms that it is the lending fee that results in the early exercise of American call options and we shall also demonstrate to what extent lending fees have affected the early exercise decision.
In this paper, a numerical method with the combined Laplace transform and the Dual Reciprocity Me... more In this paper, a numerical method with the combined Laplace transform and the Dual Reciprocity Method proposed by Zhu et al. [I] is extended to solve time-dependent diffusion equations with nonlinear source terms. The nonlinear source terms are linearised first and then an iteration is carried out until a convergent value of the unknown function at a desired time level is obtained. We shall demonstrate, through the preliminary results of our numerical experiments, that the method allows an accurate numerical solution at any desired observation time to be calculated directly and efficiently. It is believed that such a high efficiency is achieved as a direct consequence of the fact that no domain integral is involved in the calculation and the time variable is also temporarily removed from the problem.
Guo and Zhu (2017) recently proposed an equal-risk pricing approach to the valuation of contingen... more Guo and Zhu (2017) recently proposed an equal-risk pricing approach to the valuation of contingent claims when short selling is completely banned and two elegant pricing formulae are derived in some special cases. In this paper, we establish a unified framework for this new pricing approach so that its range of application can be significantly expanded. The main contribution of our framework is that it not only recovers the analytical pricing formula derived by Guo and Zhu (2017) when the payoff is monotonic, but also numerically produces equal-risk prices for contingent claims with non-monotonic payoffs, a task which has not been accomplished before. Furthermore, we demonstrate how a short selling ban affects the valuation of contingent claims by comparing equal-risk prices with Black-Scholes prices.
Under mean-variance-utility framework, we propose a new portfolio selection model, which allows w... more Under mean-variance-utility framework, we propose a new portfolio selection model, which allows wealth and time both have influences on risk aversion in the process of investment. We solved the model under a game theoretic framework and analytically derived the equilibrium investment (consumption) policy. The results conform with the facts that optimal investment strategy heavily depends on the investor's wealth and future income-consumption balance as well as the continuous optimally consumption process is highly dependent on the consumption preference of the investor.
In this article, we derive a closed-form pricing formula for catastrophe equity put options under... more In this article, we derive a closed-form pricing formula for catastrophe equity put options under a stochastic interest rate framework. A distinguishing feature of the proposed solution is its simplified form in contrast to several recently published formulae that require evaluating several layers of infinite sums of $n$ -fold convoluted distribution functions. As an application of the proposed formula, we consider two different frameworks and obtain the closed-form formula for the joint characteristic function of the asset price and the losses, which is the only required ingredient in our pricing formula. The prices obtained by the newly derived formula are compared with those obtained using Monte-Carlo simulations to show the accuracy of our formula.
International Journal of Financial Engineering, 2018
In this paper, an analytical solution for the well-known Hamilton–Jacobi–Bellman (HJB) equation t... more In this paper, an analytical solution for the well-known Hamilton–Jacobi–Bellman (HJB) equation that arises from the Merton problem subject to general utility functions is presented for the first time. The solution presented in this paper is written in the form of a Taylor’s series expansion and constructed through the homotopy analysis method (HAM). The fully nonlinear HJB equation is decomposed into an infinite series of linear PDEs which can be solved analytically. Four examples are presented with the first two cases showing the accuracy of our solution approach; while the last two demonstrating its versatility.
International Journal of Theoretical and Applied Finance, 2019
In this paper, the pricing problem of variance and volatility swaps is discussed under a two-fact... more In this paper, the pricing problem of variance and volatility swaps is discussed under a two-factor stochastic volatility model. This model can be treated as a two-factor Heston model with one factor following the CIR process and another characterized by a Markov chain, with the motivation originating from the popularity of the Heston model and the strong evidence of the existence of regime switching in real markets. Based on the derived forward characteristic function of the underlying price, analytical pricing formulae for variance and volatility swaps are presented, and numerical experiments are also conducted to compare swap prices calculated through our formulae and those obtained under the Heston model to show whether the introduction of the regime switching factor would lead to any significant difference.
In this paper, we propose an integral equation approach for pricing an American-style Parisian up... more In this paper, we propose an integral equation approach for pricing an American-style Parisian up-and-out call option under the Black–Scholes framework. The main difficulty of pricing this option lies in the determination of its optimal exercise price, which is a three-dimensional surface, instead of a two-dimensional (2-D) curve as is the case for a “one-touch” barrier option. In our approach, we first reduce the 3-D pricing problem to a 2-D one by using the “moving window” technique developed by Zhu and Chen (2013, Pricing Parisian and Parasian options analytically. Journal of Economic Dynamics and Control, 37(4): 875-896), then apply the Fourier sine transform to the 2-D problem to obtain two coupled integral equations in terms of two unknown quantities: the option price at the asset barrier and the optimal exercise price. Once the integral equations are solved numerically by using an iterative procedure, the calculation of the option price and the hedging parameters is straightf...
Uploads
Papers by Song-ping Zhu