Search

The Convertible Bonds (CBs) market has witnessed an unprecedented level of activity over the last few years not only in developed countries such as the United States but also in BRICK countries such as China. Exploring new properties of CBs or CBs with clauses becomes important for academia communities in financial mathematics. In this paper, we build two coupled partial differential equations (PDEs) for pricing a callable CB, and find a newly identified inherent property of this bond. The new property is that the conversion ratio will not affect the critical recall time indicating the time beyond the callability. Besides this property, we also find that solving the critical recall time separately and superimposing later using a non-callable CB is the same as the method of a hybrid free boundary (the critical recall time) and a moving boundary (the optimal conversion price) though the callability and the American-style conversion are nonlinearly coupled. [ABSTRACT FROM AUTHOR]

Parisian options are a useful real-option tool in risk management, particularly for corporate bankruptcy protection. However, in the past, only one barrier with the Parisian feature was studied in the literature, possibly due to the additional complication that arises with the co-existence of two barriers. In this paper, we present an analytic solution for the valuation of European-style double-barrier Parisian options by casting the pricing problem into three inter-coupled partial differential equations. These are then solved using a dimension reduction procedure and the 'moving window' technique. Our solution is in an explicit and analytical form that is written in terms of multiple integrals. This represents an important advantage over the purely numerical approaches previously published in the literature. [ABSTRACT FROM AUTHOR]

Regime-switching models have been heavily studied recently, as they have some clear advantages of over other non-constant volatility model to resolve the so-called smirk effect displayed when constant volatility models are used to price financial derivatives such as options. However, due to the increased model complexity, the associated computational effort usually increases as well, particularly when they are used to price American-style options. In this paper, a novel computational approach based on integral equations is presented. A distinctive feature of our approach, in comparison with other numerical approaches, is that the coupled partial differential equations (PDEs) in a PDE system have been decoupled in the Fourier space, resulting in a completely decoupled integral equation for each economical states, and thus has greatly reduced computational effort. Some examples with preliminary results for a two-state regime-switching model are used to demonstrate our approach. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an optimal asset allocation problem in a financial market consisting of one risk-free asset, one liquid risky asset and one illiquid risky asset. The liquidity risk stems from the asset that cannot be traded continuously, and the trading opportunities are captured by a Poisson process with constant intensity. Also, it is assumed that the interest rate is stochastically varying and follows the Cox–Ingersoll–Ross model. The performance functional of the decision maker is selected as the expected logarithmic utility of the total wealth at terminal time. The dynamic programming principle coupled with the Hamilton–Jacobi–Bellman equation has been adopted to solve this stochastic optimal control problem. In order to reduce the dimension of the problem, we introduce the proportion of the wealth invested in the illiquid risky asset and derive the semi-analytical form of the value function using a separation principle. A finite difference method is employed to solve the controlled partial differential equation satisfied by a function which is an important component of the value function. The numerical examples and their economic interpretations are then discussed. [ABSTRACT FROM AUTHOR]

A novel hybrid option pricing model using a deep learning neural network has been developed. The hybrid model keeps the traditional option pricing model with the same input parameters while simultaneously adjusting the model with neural network methods to improve accuracy when applied to real market data, especially in OTM options. The new hybrid model demonstrates superior accuracy compared to both traditional parametric and non-parametric option pricing models for both Call and Put options across all moneyness levels. The empirical results of the hybrid model provide an explanation for the deviation from the Put-Call parity observed in real market data. [ABSTRACT FROM AUTHOR]

In quantitative finance practice, model calibration is a key challenge. This is especially so when a local regime-switching model needs to be calibrated because designing an efficient and reliable algorithm to obtain local volatility values as a function of underlying price and time is important for the model to be successfully used in practice. Therefore, this paper proposes a new algorithm for calibrating local regime-switching models with observed option prices available for a particular market that is suitable for this type of model. The newly proposed algorithm is tested with calibrations performed on synthetic as well as real market data. Our empirical test results indicate that the algorithm has great potential to be used in financial risk management. [ABSTRACT FROM AUTHOR]

In this paper, the fair price of an American-style resettable convertible bond (CB) under the Black–Scholes model with a particular reset clause is calculated. This is a challenging problem because an unknown optimal conversion price needs to be determined together with the bond price. There is also an additional complexity that the value of the conversion ratio will change when the underlying price touches the reset price. Because of the additional reset clause, the bond price is not always a monotonically increasing function with the underlying price, which is impossible for other types of the CBs. Of course, the problem can be dealt with using the Monte-Carlo simulation. But, a partial differential equation (PDE)/integral equation approach is far superior in terms of computational efficiency. Fortunately, after establishing the PDE system governing the bond price, we are able to present an integral equation representation by applying the incomplete Fourier transform on the PDE system. [ABSTRACT FROM AUTHOR]

Utility-indifference approach is a useful approach to be adopted for pricing financial derivatives in an incomplete market and is an ongoing hot research topic in quantitative finance. One interesting question associated with this approach is whether or not it renders to the same option prices, degenerately, when the market becomes infinitesimally close to a complete market. The answer for such a question has been provided for European-style options as there is a well-documented theoretical proof in Davis et al. (1993). However, a theoretical proof for the case of pricing American-style options is unavailable at this stage and the answer for this question must be at least numerically confirmed before it can be comfortably used to price American-style options in incomplete markets. The contribution of this paper is to provide such a numerical verification. [ABSTRACT FROM AUTHOR]

An important issue in derivative pricing that hasn't been explored much until very recently is the impact of short selling to the price of an option. This paper extends a recent publication in this area to the case in which a ban of short selling of the underlying alone is somewhat less 'effective' than the extreme case discussed by Guo and Zhu [Equal risk pricing under convex trading constraints. J. Econ. Dyn. Control, 2017, 76, 136–151]. The case presented here is closer to reality, in which the effect of a ban on the underlying of an option alone may quite often be 'diluted' due to market interactions of the underlying asset with other correlated assets. Under a new assumption that there exists at least a correlated asset in the market, which is allowed to be short sold and thus can be used by traders for hedging purposes even though short selling of the underlying itself is banned, a new closed-form equal-risk pricing formula for European options is successfully derived. The new formula contains two distinguishable advantages; (a) it does not induce any significantly extra burden in terms of numerically computing option values, compared with the effort involved in using the Black–Scholes formula, which is still popularly used in finance industry today; (b) it remains simple and elegant as only one additional parameter beyond the Black–Scholes formula is introduced, to reflect the dilution effect to the ban as a result of market interactions. [ABSTRACT FROM AUTHOR]

In this paper, the pricing problem for the American-style convertible bonds with the Heston stochastic volatility and that with the Cox–Ingersoll–Ross (CIR) stochastic interest rate are both considered. Due to the complexity of both problems, resulting from an additional stochastic factor, it is almost impossible to find any analytical solution. Therefore, a predictor–corrector scheme is chosen as the numerical scheme to solve the partial differential equations (PDEs), with the Douglas–Rachford (D–R) method being utilized as one of the Alternating Direction Implicit (ADI) methods for the correction step to obtain the numerical solution. Finally, the accuracy of our approach is numerically verified, and different properties of convertible bond price and the optimal conversion price are also demonstrated and discussed through examples. [ABSTRACT FROM AUTHOR]

This paper studies European call option pricing problem under a hard-to-borrow stock model where stock price and buy-in rate are fully coupled. Avellaneda and Lipkin (2009) proposed a simplified solution approach with an independence assumption, and then derived a semi-explicit pricing formula. However, such an approach has limited its application to more general cases. In this paper, we propose a partial differential equation (PDE) approach for pricing European call options, regardless of the independence assumption. A two-dimensional PDE is derived first with a set of appropriate boundary conditions. Then, two numerical schemes are provided with different treatments of the jump term. Through our numerical results, we find that the semi-explicit formula is a good approximate solution when the coupling parameter is small. However, when the stock price and the buy-in rate are significantly coupled, the PDE approach is preferred to solve the option pricing problem under the full hard-to-borrow model. [ABSTRACT FROM AUTHOR]

We present a new tree-based numerical approach for options pricing under Heston's stochastic volatility model. The tree approach is simple to implement, computationally cheap and has a very clear financial interpretation. In addition, we build an equivalence between our tree method and an explicit finite difference method for the Heston model based on our tree approach. Such an equivalence is proven analytically and verified numerically. The numerical performance shows that our tree is robust and provides accurate results within very short computing time. [ABSTRACT FROM AUTHOR]

In this paper, we study the effects of cointegration on optimal investment and consumption strategies for an investor with exponential utility. A Hamilton-Jacobi-Bellman (HJB) equation is derived first and then solved analytically. Both the optimal investment and consumption strategies are expressed in closed form. A verification theorem is also established to demonstrate that the solution of the HJB equation is indeed the solution of the original optimization problem under an integrability condition. In addition, a simple and sufficient condition is proposed to ensure that the integrability condition is satisfied. Financially, the optimal investment and consumption strategies are decomposed into two parts: the myopic part and the hedging demand caused by cointegration. Discussions on the hedging demand are carried out first, based on analytical formulae. Then numerical results show that ignoring the information about cointegration results in a utility loss. [ABSTRACT FROM AUTHOR]

Convexity correction is a well-known approximation technique used in pricing volatility swaps and VIX futures. However, the accuracy of the technique itself and the validity condition of this approximation have hardly been addressed and discussed in the literature. This paper shows that, through both theoretical analysis and numerical examples, this type of approximations is not necessarily accurate and one should be very careful in using it. We also show that a better accuracy cannot be achieved by extending the convexity correction approximation from a second-order Taylor expansion to third-order or fourth-order Taylor expansions. We then analyze why and when it deteriorates, and provide a validity condition of applying the convexity correction approximation. Finally, we propose a new approximation, which is an extension of the convexity correction approximation, to achieve better accuracies. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present analytical pricing formulae for variance and volatility swaps, when both of the volatility and interest rate are assumed to be stochastic and follow a CIR (Cox–Ingersoll–Ross) process, forming a Heston–CIR hybrid model. The solutions are written in a series form with a theoretical proof of their convergence, ensuring the accuracy of the determined swap prices. The application of the formulae in practice is also demonstrated through the designed numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a closed-form pricing formula for European options in the form of an infinite series is derived under the Heston model with the interest rate being another random variable following the CIR (Cox–Ingersoll–Ross) model. One of the main advantages for the newly derived series solution is that we can provide a radius of convergence, which is complemented by some numerical experiments demonstrating its speed of convergence. To further verify our formula, option prices calculated through our formula are also compared with those obtained from Monte Carlo simulations. Finally, a set of pricing formulae are derived with the series expanded at different points so that the entire time horizon can be covered by converged solutions. [ABSTRACT FROM AUTHOR]

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, Risk 22 (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. [ABSTRACT FROM AUTHOR]

American-style puttable convertible bonds are often priced with various numerical solutions because the predominant complexity arises from the determination of the two free boundaries together with the bond price. In this paper, two forms of integral equations are derived to price a puttable convertible bond on a single underlying asset. The first form is obtained under the Black–Scholes framework by using an incomplete Fourier transform. However, this integral equation formulation possesses a discontinuity along both free boundaries. An even worse problem is that this representation contains two first-order derivatives of the unknown exercise prices, which demands a higher smoothness of the interpolation functions used in the numerical solution procedure. Thus, a second integral equation formulation is developed based on the first form to overcome those problems. Numerical experiments are conducted to show several interesting properties of puttable convertible bonds. [ABSTRACT FROM AUTHOR]

In this paper, a completely new integral equation for the price of an American put option as well as its optimal exercise price is successfully derived. Compared to existing integral equations for pricing American options, the new integral formulation has two distinguishable advantages: (i) it is in a form of one-dimensional integral, and (ii) it is in a form that is free from any discontinuity and singularities associated with the optimal exercise boundary at the expiry time. These rather unique features have led to a significant enhancement of the computational accuracy and efficiency as shown in the examples. [ABSTRACT FROM PUBLISHER]

In this paper, a modified Black–Scholes (B–S) model is proposed, based on a revised assumption that the range of the underlying price varies within a finite zone, rather than being allowed to vary in a semi-infinite zone as presented in the classical B–S theory. This is motivated by the fact that the underlying price of any option can never reach infinity in reality; a trader may use our new formula to adjust the option price that he/she is willing to long or short. To develop this modified option pricing formula, we assume that a trader has a view on the realistic price range of a particular asset and the log-returns follow a truncated normal distribution within this price range. After a closed-form pricing formula for European call options has been successfully derived, some numerical experiments are conducted. To further demonstrate the meaning of the proposed model, empirical studies are carried out to compare the pricing performance of our model and that of the B–S model with real market data. [ABSTRACT FROM AUTHOR]

We introduce an additional factor in the Heston-CIR model to form a new hybrid model in this paper. This new model features a more general correlation structure with the interest rate being correlated with the underlying asset, while still preserving the analytical tractability. We derive a series solution to the forward characteristic function when solving a time-dependent Riccati equation, after the measure transform is performed, so that variance and volatility swap prices can be finally written in an analytical form. The theoretical results are also accompanied with some numerical results demonstrating its potential of complementing the usage of the currently adopted non-correlated Heston-CIR model. • A new two-factor Heston-CIR model is introduced to capture correlation structure. • Analytical pricing formulae for variance and volatility swaps are derived. • The convergence of the series solution is theoretically proved. [ABSTRACT FROM AUTHOR]

This paper presents an alternative form of the Heston model that preserves an essential advantage of the Heston model, its analytic tractability, by imposing the necessary and sufficient conditions for the existence of a solution in affine form, while it is in a different form so that it offers certain advantages in parameter determination. To demonstrate this, we conducted some empirical studies, exploring if this new form does have certain advantages over the original version under certain market conditions. [ABSTRACT FROM AUTHOR]

In this paper, a closed-form pricing formula in the form of an infinite series for European call options is derived for the Heston stochastic volatility model under a chosen martingale measure. Given that markets with the stochastic volatility are incomplete, there exists a number of equivalent martingale measures and consequently investors face a problem of making a choice of appropriate measure when they price options. The one we adopt here is the so-called minimal entropy martingale measure shown to be related to the expected utility maximization theory (Frittelli 2000 Math. Finance10(1), 39–52) and the financial rationality for choosing this measure will be further illustrated in this paper. A great advantage of our newly-derived pricing formula is that the convergence of the solution in series form can be proved theoretically; such a proof of the convergence is also complemented by some numerical examples to demonstrate the speed of convergence. To further show the validity of our formula, a comparison of prices calculated through the newly derived formula is made with those obtained directly from the Monte Carlo simulation as well as those from solving the PDE (partial differential equation) with the finite difference method. [ABSTRACT FROM PUBLISHER]

In this paper, an integral equation approach is adopted to price American-style down-and-out calls. Instead of using the probability theory as used in the literature, we use the continuous Fourier sine transform to solve the partial differential equation system governing the option prices. As a way of validating our approach, we show that the “early exercise premium representation” for American-style down-and-out calls without rebate can be re-derived by using our approach. We then examine the case that time-dependent rebates are included in the contract of American-style down-and-out calls. As a result, a more general integral representation for the price of an American-style down-and-out call, with the presence of an extra term associated with the rebate, is obtained. Our numerical method based on the newly-derived integral representation appears to be efficient in computing the price and the hedging parameters for American-style down-and-out calls with rebates. In addition, significant effects of rebates on the option prices and the optimal exercise boundaries are illustrated through selected numerical results. [ABSTRACT FROM AUTHOR]

Papers focusing on analytically pricing discretely-sampled volatility swaps are rare in literature, mainly due to the inherent difficulty associated with the nonlinearity in the pay-off function. In this paper, we present a closed-form exact solution for the pricing of discretely-sampled volatility swaps, under the framework of Heston (1993) stochastic volatility model, based on the definition of the so-called average of realized volatility. By working out such a closed-form exact solution for discretely-sampled volatility swaps, this work represents a substantial progress in the field of pricing volatility swaps, as it has: (1) significantly reduced the computational time in obtaining numerical values for the discretely-sampled volatility swaps; (2) improved the computational accuracy of discretely-sampled volatility swaps, comparing with the continuous sampling approximation, especially when the time interval between sampling points is large; (3) enabled all the hedging ratios of a volatility swap to be analytically derived. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, a general approach is presented to price forward-start variance swaps with discrete sampling times, based on the Heston (1993)’s two-factor stochastic volatility model. Using this approach we work out two analytical closed-form formulae for the price of forward-start variance swap with the realized variance being defined by the actual-return realized variance and the log-return realized variance, respectively. The main features of this new approach and the developed formulae as a result include: (1) treating the pricing problem of variance swaps with different definitions of discretely-sampled realized variance in a highly unified way; (2) easily obtaining analytical closed-form solutions for forward-start variance swaps with two popularly-used definitions of discretely-sampled realized variance; (3) enabling the investigation of some important properties of the forward-start variance swaps, utilizing the elegant and simple form of the obtained solutions. With these advantages, we believe that the approach can be applied to price variance swaps based on other stochastic volatility models as well. [ABSTRACT FROM AUTHOR]

Farms of wave energy convertors are more likely to be deployed than widely separated individual devices so as to harness maximum available power and to facilitate installation and electrical power transmission. In this paper, the theory of the interaction between oscillating systems, developed by Falnes and McIver (1985), is for the first time applied and extended to an explicit study of the dynamic and energetic performance of a finite array of fixed OWC devices. The interactions between devices and air compressibility are taken into account. Following the method, a FEM model is applied to the study of a single OWC device and three different array configurations. It is demonstrated that the inner properties and the interaction between OWC devices are strongly dependent on the position of the devices in the array and should be taken into account when determining the optimum device parameters so as to increase the maximum power extraction of the system or the overall frequency power-capture band-width. [ABSTRACT FROM AUTHOR]

In this paper, the impact of liquidity on the underlying asset is taken into account when pricing European options through a discounting factor which depends on two factors, i.e., market liquidity risk modeled as a mean-reverting stochastic process and the sensitivity of the underlying to market liquidity. A closed-form pricing formula for liquidity-adjusted European options is derived in the form of an infinite series using Karhunen–Loève expansion for the Ornstein–Uhlenbeck process. The convergence of the series solution is theoretically proved to guarantee closedness so that market practitioners can adopt the new formula when they need to account in market liquidity risk. The speed of convergence is demonstrated through numerical experiments. Finally, the accuracy of the newly derived formula is shown by comparing option prices calculated with our formula and those obtained from Monte-Carlo simulation, and various properties of our formula are also investigated. • A closed-form pricing formula for liquidity-adjusted European options is derived. • The series solution is obtained with Karhunen–Loève expansion for the OU process. • The convergence of the series solution is theoretically proved. [ABSTRACT FROM AUTHOR]

We develop a simplified analytical approach for pricing discretely-sampled variance swaps with the realized variance, defined in terms of the squared log return of the underlying price. The closed-form formula obtained for Heston’s two-factor stochastic volatility model is in a much simpler form than those proposed in literature. Most interestingly, we discuss the validity of our solution as well as some other previous solutions in different forms in the parameter space. We demonstrate that market practitioners need to be cautious, making sure that their model parameters extracted from market data are in the right parameter subspace, when any of these analytical pricing formulae is adopted to calculate the fair delivery price of a discretely-sampled variance swap. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a new class of optimization problems, which maximize the terminal wealth and accumulated consumption utility subject to a mean–variance criterion controlling the final risk of the portfolio. The multiple-objective optimization problem is firstly transformed into a single-objective one by introducing the concept of overall 'happiness' of an investor defined as the aggregation of the terminal wealth under the mean–variance criterion and the expected accumulated utility, and then solved under a game-theoretic framework. We have managed to maintain analytical tractability; the closed-form solutions found for a set of special utility functions enable us to discuss some interesting optimal investment strategies that have not been revealed in the literature before. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of pricing perpetual American put options with volatility driven by two other processes. By using a perturbation approach, we obtain approximate but explicit closed-form pricing formulae for the option and optimal exercise prices, respectively, under a general multi-scale SV (stochastic volatility) model. A key feature of the expansion methodology employed here is to balance the two SV processes, while dealing with the free boundary conditions properly. It turns out that in the current formulae, the fast volatility factor does not play an explicit role, while the slow factor is quite crucial, a phenomenon that is shown to be quite reasonable through our discussions. [ABSTRACT FROM AUTHOR]

Abstract: Pricing variance swaps under stochastic volatility with discretely-sampled realized variance has been a hot subject pursued recently; quite a few papers have already been published (Zhu and Lian (2009, 2011, ); Swishchuk and Li (2011) ). In this paper, we present a simplified approach to price discretely-sampled variance swaps. Compared with the approach presented by Zhu and Lian (2011) , an important feature of our approach is that there is no need for the introduction of a new state variable and the utilization of the generalized Fourier transform. This has significantly simplified the solution procedure and will thus enable researchers to view this type of problems from a different angle. [Copyright &y& Elsevier]

Abstract: This paper is an extension to a recent paper by Zhu and Lian (2011) [1], in which a closed-form exact solution was presented for the price of variance swaps with a particular definition of the realized variance. Here, we further demonstrate that our approach is quite versatile and can be used for other definitions of the realized variance as well. In particular, we present a closed-form formula for the price of a variance swap with the realized variance in the payoff function being defined as a logarithmic return of the underlying asset at some pre-specified discretely sampling points. The simple formula presented here is a result of successfully finding an exact solution of the partial differential equation (PDE) system based on the Heston (1993)’s [2] two-factor stochastic volatility model. A distinguishable feature of this new solution is that the computational time involved in pricing variance swaps with discretely sampling time has been substantially improved. [Copyright &y& Elsevier]

Abstract: In this study, we derive a new exact solution for pricing European options in a two-state regime-switching economy. Two coupled Black–Scholes partial differential equations (PDEs) under the regime switching are solved using the Fourier Transform method. A key feature of the newly-derived solution is its simplicity in the form of a single integral with a real integrand, which leads to great computational efficiency in comparison with other closed-form solutions previously presented in the literature. Numerical examples are provided to demonstrate some interesting results obtained from our pricing formula. [Copyright &y& Elsevier]

Abstract: Prior research on wave energy devices such as Oscillating Water Column (OWC) systems has revealed the complex nature of the interactions between ocean waves and these systems. This is especially true when such devices are arranged in arrays. Within these arrays, the multiple scattered waves produced by each device interact and can lead to unexpected fluctuations of the power captured by each individual OWC. In the present study, a finite element model based on linear wave theory has been developed to study scattered waves around single and multiple oscillating water column wave energy conversion devices. The power capture efficiency of a single cylindrical OWC deduced from the finite element model is compared with the numerical approximation used by . The efficiency of the single OWC is then compared with that of an array of OWCs of the same dimensions, as a function of array spacing, pneumatic damping and direction of the incident waves. It is found that the presence of neighboring OWCs has a significant influence on the power capture efficiency of individual devices, even for large separations, and that the optimal pneumatic damping for OWCs in an array may differ from that of an isolated OWC of the same dimensions. [Copyright &y& Elsevier]

Abstract: In this paper, we present a “correction” to Merton’s (1973) well-known classical case of pricing perpetual American puts by considering the same pricing problem under a general fast mean-reverting SV (stochastic-volatility) model. By using the perturbation method, two analytic formulae are derived for the option price and the optimal exercise price, respectively. Based on the newly obtained formulae, we conduct a quantitative analysis of the impact of the SV term on the price of a perpetual American put option as well as its early exercise strategies. It shows that the presence of a fast mean-reverting SV tends to universally increase the put option price and to defer the optimal time to exercise the option contract, had the underlying been assumed to be falling. It is also noted that such an effect could be quite significant when the option is near the money. [Copyright &y& Elsevier]

Abstract: Based on the Legendre pseudospectral method, we propose a numerical treatment for pricing perpetual American put option with stochastic volatility. In this simple approach, a nonlinear algebraic equation system is first derived, and then solved by the Gauss–Newton algorithm. The convergence of the current scheme is ensured by constructing a test example similar to the original problem, and comparing the numerical option prices with those produced by the classical Projected SOR (PSOR) method. The results of our numerical experiments suggest that the proposed scheme is both accurate and efficient, since the spectral accuracy can be easily achieved within a small number of iterations. Moreover, based on the numerical results, we also discuss the impact of stochastic volatility term on the prices of perpetual American puts. [Copyright &y& Elsevier]

In this paper, a comprehensive review of the valuation of American options is presented. Various approaches to pricing American option contracts are discussed, with the pros and cons of each being briefly outlined. The paper aims to provide a literature review for those who are interested in this very fundamental research, that provides a basis for pricing any American-style derivatives. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce a new numerical scheme, based on the ADI (alternating direction implicit) method, to price American put options with a stochastic volatility model. Upon applying a front-fixing transformation to transform the unknown free boundary into a known and fixed boundary in the transformed space, a predictor–corrector finite difference scheme is then developed to solve for the optimal exercise price and the option values simultaneously. Based on the local von Neumann stability analysis, a stability requirement is theoretically obtained first and then tested numerically. It is shown that the instability introduced by the predictor can be damped, to some extent, by the ADI method that is used in the corrector. The results of various numerical experiments show that this new approach is fast and accurate, and can be easily extended to other types of financial derivatives with an American-style exercise. Another key contribution of this paper is the proposition of a set of appropriate boundary conditions, particularly in the volatility direction, upon realizing that appropriate boundary conditions in the volatility direction for stochastic volatility models appear to be controversial in the literature. A sound justification is also provided for the proposed boundary conditions mathematically as well as financially. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new numerical scheme, based on the finite difference method, to solve American put option pricing problems. Upon applying a Landau transform or the so-called front-fixing technique to the Black–Scholes partial differential equation, a predictor–corrector finite difference scheme is proposed to numerically solve the nonlinear differential system. Through the comparison with Zhu’s analytical solution , we shall demonstrate that the numerical results obtained from the new scheme converge well to the exact optimal exercise boundary and option values. The results of our numerical examples suggest that this approach can be used as an accurate and efficient method even for pricing other types of financial derivative with American-style exercise. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we apply singular perturbation techniques to price European puts with a stochastic volatility model, and derive a simple and elegant analytical formula as an approximation for the value of European put options. In contrast to the existing Heston’s semi-analytical formula, this approximation has the following unique feature: the latter only involves the standard normal distribution function, which is as fast and easy to implement as the Black–Scholes formula; whereas the former requires the evaluation of a logarithm with a complex argument during the involved Fourier inverse transform, which may sometimes result in numerical instability. Various numerical experiments suggest that our new formula can achieve a high order of accuracy for a large class of option derivatives with relatively short tenor. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, is the optimal exercise boundary being elegantly and temporarily removed in the solution process of each order, and, consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical and exact series-expansion solution for the optimal exercise boundary and the option price of American put options. [ABSTRACT FROM AUTHOR]