The evaluation of European compound option prices under stochastic volatility using Fourier transform techniques

Compound options are not only sensitive to future movements of the underlying asset price, but also to future changes in volatility levels. Because the Black–Scholes analytical valuation formula for compound options is not able to incorporate the sensitivity to volatility, the aim of this paper is to develop a numerical pricing procedure for this type of option in stochastic volatility models, specifically focusing on the model of Heston. For this, the compound option value is represented as the difference of its exercise probabilities, which depend on three random variables through a complex functional form. Then the joint distribution of these random variables is uniquely determined by their characteristic function and therefore the probabilities can each be expressed as a multiple inverse Fourier transform. Solving the inverse Fourier transform with respect to volatility, we can reduce the pricing problem from three to two dimensions. This reduced dimensionality simplifies the application of the fast Fourier transform (FFT) method developed by Dempster and Hong when transferred to our stochastic volatility framework. After combining their approach with a new extension of the fractional FFT technique for option pricing to the two-dimensional case, it is possible to obtain good approximations to the exercise probabilities. The resulting upper and lower bounds are then compared with other numerical methods such as Monte Carlo simulations and show promising results.     

A simple efficient approximation to price basket stock options with volatility smile

This paper develops a new approach to obtain the price and risk sensitivities of basket options which have a volatility smile. Using this approach, the Black–Scholes model and the Stochastic Volatility Inspired model have been used to obtain an approximate analytical pricing formula for basket options with a volatility smile. It is found that our approximate formula is quite accurate by comparing it with Monte Carlo simulations. It is also proved the option value of our approach is consistent with the option value generated by Levy’s and Gentle’s approaches for typical ranges of volatility. Further, we give a theoretical proof that the option values from Levy’s and Gentle’s works are the upper bound and the lower bound, respectively, for our option value. The calibration procedure and a practical example are provided. The main advantage of our approach is that it provides accurate and easily implemented basket option prices with volatility smile and hedge parameters and avoids the need to use time-consuming numerical procedures such as Monte Carlo simulation.     

Approximate pricing of swaptions in affine and quadratic models

This paper proposes new bounds on the prices of European-style swaptions for affine and quadratic interest rate models. These bounds are computable whenever the joint characteristic function of the state variables is known. In particular, our lower bound involves the computation of a one-dimensional Fourier transform independently of the swap length. In addition, we control the error of our method by providing a new upper bound on swaption price that is applicable to all considered models. We test our bounds on different affine models and on a quadratic Gaussian model. We also apply our procedure to the multiple curve framework. The bounds are found to be accurate and computationally efficient.     

Pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes

We derive efficient and accurate analytical pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes. By extending the conditioning variable approach, we derive the lower bound on the Asian option price and construct an upper bound based on the sharp lower bound. We also consider the general partially exact and bounded (PEB) approximations, which include the sharp lower bound and partially conditional moment matching approximation as special cases. The PEB approximations are known to lie between a sharp lower bound and an upper bound. Our numerical tests show that the PEB approximations to discrete arithmetic Asian option prices can produce highly accurate approximations when compared to other approximation methods. Our proposed approximation methods can be readily applied to pricing Asian options under most common types of underlying asset price processes, like the Heston stochastic volatility model nested in the class of time-changed Lévy processes with the leverage effect.     

Unbiased and efficient Greeks of financial options

The price of a derivative security equals the discounted expected payoff of the security under a suitable measure, and Greeks are price sensitivities with respect to parameters of interest. When closed-form formulas do not exist, Monte Carlo simulation has proved very useful for computing the prices and Greeks of derivative securities. Although finite difference with resimulation is the standard method for estimating Greeks, it is in general biased and suffers from erratic behavior when the payoff function is discontinuous. Direct methods, such as the pathwise method and the likelihood ratio method, are proposed to differentiate the price formulas directly and hence produce unbiased Greeks (Broadie and Glasserman, Manag. Sci. 42:269–285, 1996). The pathwise method differentiates the payoff function, whereas the likelihood ratio method differentiates the densities. When both methods apply, the pathwise method generally enjoys lower variances, but it requires the payoff function to be Lipschitz-continuous. Similarly to the pathwise method, our method differentiates the payoff function but lifts the Lipschitz-continuity requirements on the payoff function. We build a new but simple mathematical formulation so that formulas of Greeks for a broad class of derivative securities can be derived systematically. We then present an importance sampling method to estimate the Greeks. These formulas are the first in the literature. Numerical experiments show that our method gives unbiased Greeks for several popular multi-asset options (also called rainbow options) and a path-dependent option.     

Approximation methods for piecewise deterministic Markov processes and their costs

In this paper, we analyse piecewise deterministic Markov processes (PDMPs), as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of PDMPs. There one is interested in computing certain quantities of interest such as the probability of ruin or the value of an insurance company. Instead of explicitly solving the related integro-(partial) differential equation (an approach which can only be used in few special cases), we adapt the problem in a manner that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules; this is in contrast to applying random integration algorithms such as Monte Carlo. To this end, we reformulate a general cost functional as a fixed point of a particular integral operator, which allows for iterative approximation of the functional. Furthermore, we introduce a smoothing technique which is applied to the integrands involved, in order to use error bounds for deterministic cubature rules. We prove a convergence result for our PDMPs approximation, which is of independent interest as it justifies phase-type approximations on the process level. We illustrate the smoothing technique for a risk-theoretic example, and compare deterministic and Monte Carlo integration.     

Smoothing the payoff for efficient computation of Basket option prices

We consider the problem of pricing basket options in a multivariate Black–Scholes or Variance-Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high-dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse-grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster than Monte Carlo or Quasi Monte Carlo methods in dimensions up to 35.     

Primal–dual quasi-Monte Carlo simulation with dimension reduction for pricing American options

The pricing of American options is one of the most challenging problems in financial engineering due to the involved optimal stopping time problem, which can be solved by using dynamic programming (DP). But applying DP is not always practical, especially when the state space is high dimensional. However, the curse of dimensionality can be overcome by Monte Carlo (MC) simulation. We can get lower and upper bounds by MC to ensure that the true price falls into a valid confidence interval. During the recent decades, progress has been made in using MC simulation to obtain both the lower bound by least-squares Monte Carlo method (LSM) and the upper bound by duality approach. However, there are few works on pricing American options using quasi-Monte Carlo (QMC) methods, especially to compute the upper bound. For comparing the sample variances and standard errors in the numerical experiments, randomized QMC (RQMC) methods are usually used. In this paper, we propose to use RQMC to replace MC simulation to compute both the lower bound (by the LSM) and the upper bound (by the duality approach). Moreover, we propose to use dimension reduction techniques, such as the Brownian bridge, principal component analysis, linear transformation and the gradients based principle component analysis. We perform numerical experiments on American–Asian options and American max-call options under the Black–Scholes model and the variance gamma model, in which the options have the path-dependent feature or are written on multiple underlying assets. We find that RQMC in combination with dimension reduction techniques can significantly increase the efficiency in computing both the lower and upper bounds, resulting in better estimates and tighter confidence intervals of the true price than pure MC simulation.     

New analytical option pricing models with Weyl–Titchmarsh theory

Abstract

In this paper we consider the valuation of Bermudan callable derivatives with multiple exercise rights. We present in this context a new primal–dual linear Monte Carlo algorithm that allows for efficient simulation of the lower and upper price bounds without using nested simulations (hence the terminology). The algorithm is essentially an extension of the primal–dual Monte Carlo algorithm for standard Bermudan options proposed by Schoenmakers et al. [SIAM J. Finance Math., 2013, 4, 86–116] to the case of multiple exercise rights. In particular, the algorithm constructs upwardly a system of dual martingales to be plugged into the dual representation of Schoenmakers. At each level, the respective martingale is constructed via a backward regression procedure starting at the last exercise date. The thus constructed martingales are finally used to compute an upper price bound. The algorithm also provides approximate continuation functions that may be used to construct a price lower bound. The algorithm is applied to the pricing of flexible caps in a Hull and White model setup. The simple model choice allows for comparison of the computed price bounds with the exact price obtained by means of a trinomial tree implementation. As a result, we obtain tight price bounds for the considered application. Moreover, the algorithm is generically designed for multi-dimensional problems and is tractable to implement.     

Implied stopping rules for American basket options from Markovian projection

This work addresses the problem of pricing American basket options in a multivariate setting, which includes among others, the Bachelier and Black–Scholes models. In high dimensions, nonlinear PDE methods for solving the problem become prohibitively costly due to the curse of dimensionality. Instead, this work proposes to use a stopping rule that depends on the dynamics of a low-dimensional Markovian projection of the given basket of assets. From a numerical analysis point of view, we split the given non-smooth high-dimensional problem into two subproblems, namely one dealing with a smooth high-dimensionality integration in the parameter space and the other dealing with a low-dimensional, non-smooth optimal stopping problem in the projected state space. Assuming that we know the density of the forward process and using the Laplace approximation, we first efficiently evaluate the diffusion coefficient corresponding to the low-dimensional Markovian projection of the basket. Then, we approximate the optimal early exercise boundary of the option by solving an HJB PDE in the projected, low-dimensional space. The resulting near-optimal early exercise boundary is used to produce an exercise strategy for the high-dimensional option, thereby providing a lower bound for the price of the American basket option. A corresponding upper bound is also provided. These bounds allow one to assess the accuracy of the proposed pricing method. Indeed, our approximate early exercise strategy provides a straightforward lower bound for the American basket option price. Following a duality argument due to Rogers, we derive a corresponding upper bound solving only the low-dimensional optimal control problem. Numerically, we show the feasibility of the method using baskets with dimensions up to 50. In these examples, the resulting option price relative errors are only of the order of few percent.     

Tight bounds on American option prices

In contrast to the constant exercise boundary assumed by Broadie and Detemple (1996) [Broadie, M., Detemple, J., 1996. American option valuation: New bounds, approximations, and comparison of existing methods. Review of Financial Studies 9, 1211–1250], we use an exponential function to approximate the early exercise boundary. Then, we obtain lower bounds for American option prices and the optimal exercise boundary which improve the bounds of Broadie and Detemple (1996). With the tight lower bound for the optimal exercise boundary, we further derive a tight upper bound for the American option price using the early exercise premium integral of Kim (1990) [Kim, I.J., 1990. The analytic valuation of American options. Review of Financial Studies 3, 547–572]. The numerical results show that our lower and upper bounds are very tight and can improve the pricing errors of the lower bound and upper bound of Broadie and Detemple (1996) by 83.0% and 87.5%, respectively. The tightness of our upper bounds is comparable to some best accurate/efficient methods in the literature for pricing American options. Moreover, the results also indicate that the hedge ratios (deltas and gammas) of our bounds are close to the accurate values of American options.     

Riding on the smiles

Using a data set of vanilla options on the major indexes we investigate the calibration properties of several multi-factor stochastic volatility models by adopting the fast Fourier transform as the pricing methodology. We study the impact of the penalizing function on the calibration performance and how it affects the calibrated parameters. We consider single-asset as well as multiple-asset models, with particular emphasis on the single-asset Wishart Multidimensional Stochastic Volatility model and the Wishart Affine Stochastic Correlation model, which provides a natural framework for pricing basket options while keeping the stylized smile–skew effects on single-name vanillas. For all models we give some option price approximations that are very useful for speeding up the pricing process. In addition, these approximations allow us to compare different models by conveniently aggregating the parameters, and they highlight the ability of the Wishart-based models to control separately the smile and the skew effects. This is extremely important from a risk-management perspective of a book of derivatives that includes exotic as well as basket options.     

American options under stochastic volatility: control variates,maturity randomization & multiscale asymptotics

American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of an American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow-scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price an American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where the scaling parameter value is equal to unity, fast and slow-scale approximations are equally accurate.     

American option valuation: new bounds, approximations, and a comparison of existing methods

We develop lower and upper bounds on the prices of Americancall and put options written on a dividend-paying asset. Weprovide two option price approximations one based on the lowerbound (termed LBA) and one based on both bounds (termed LUBA).The LUBA approximation has an average accuracy comparable toa l,000-step binomial tree. We introduce a modification of thebinomial method (termed BBSR) that is very simple to implementand performs remarkably well. We also conduct a careful large-scaleevaluation of many recent methods for computing American optionprices.     

General approximation schemes for option prices in stochastic volatility models

In this paper we develop a general method for deriving closed-form approximations of European option prices and equivalent implied volatilities in stochastic volatility models. Our method relies on perturbations of the model dynamics and we show how the expansion terms can be calculated using purely probabilistic methods. A flexible way of approximating the equivalent implied volatility from the basic price expansion is also introduced. As an application of our method we derive closed-form approximations for call prices and implied volatilities in the Heston [Rev. Financial Stud., 1993, 6, 327–343] model. The accuracy of these approximations is studied and compared with numerically obtained values.     

Pricing exotic options using MSL-MC

Today, better numerical approximations are required for multi-dimensional SDEs to improve on the poor performance of the standard Monte Carlo pricing method. With this aim in mind, this paper presents a method (MSL-MC) to price exotic options using multi-dimensional SDEs (e.g. stochastic volatility models). Usually, it is the weak convergence property of numerical discretizations that is most important, because, in financial applications, one is mostly concerned with the accurate estimation of expected payoffs. However, in the recently developed Multilevel Monte Carlo path simulation method (ML-MC), the strong convergence property plays a crucial role. We present a modification to the ML-MC algorithm that can be used to achieve better savings. To illustrate these, various examples of exotic options are given using a wide variety of payoffs, stochastic volatility models and the new Multischeme Multilevel Monte Carlo method (MSL-MC). For standard payoffs, both European and Digital options are presented. Examples are also given for complex payoffs, such as combinations of European options (Butterfly Spread, Strip and Strap options). Finally, for path-dependent payoffs, both Asian and Variance Swap options are demonstrated. This research shows how the use of stochastic volatility models and the θ scheme can improve the convergence of the MSL-MC so that the computational cost to achieve an accuracy of O(ε) is reduced from O(ε^?3) to O(ε^?2) for a payoff under global and non-global Lipschitz conditions.     

Robust animal spirits

In a real business cycle model, an agent's fear of model misspecification interacts with stochastic volatility to induce time varying worst case scenarios. These time varying worst case scenarios capture a notion of animal spirits where the probability distributions used to evaluate decision rules and price assets do not necessarily reflect the fundamental characteristics of the economy. Households entertain a pessimistic view of the world and their pessimism varies with the overall level of volatility in the economy, implying an amplification of the effects of volatility shocks. By using perturbation methods and Monte Carlo techniques we extend the class of models analyzed with robust control methods to include the sort of nonlinear production-based DSGE models that are popular in academic research and policymaking practice.     

最优多因子仿射利率期限结构模型选择

在利用NS模型估计出市场即期利率的基础上,采用卡尔曼滤波方法对多因子Vasieck和CIR模型进行参数估计,最后运用蒙特卡罗模拟方法对交易所国债价格进行模拟,并与实际价格进行比较,进而确定了符合我们国债市场的最优多因子仿射利率期限结构模型。研究结果表明：多因子CIR模型对数据的拟合效果及对国债价格模拟效果要明显优于多因子Vasicek模型;对于多因子CIR模型而言,因子个数增加并没有提高模型的价格模拟效果;两因子CIR模型具有最优的国债价格模拟效果。     

Pitfalls in backtesting Historical Simulation VaR models

Historical Simulation (HS) and its variant, the Filtered Historical Simulation (FHS), are the most popular Value-at-Risk forecast methods at commercial banks. These forecast methods are traditionally evaluated by means of the unconditional backtest. This paper formally shows that the unconditional backtest is always inconsistent for backtesting HS and FHS models, with a power function that can be even smaller than the nominal level in large samples. Our findings have fundamental implications in the determination of market risk capital requirements, and also explain Monte Carlo and empirical findings in previous studies. We also propose a data-driven weighted backtest with good power properties to evaluate HS and FHS forecasts. A Monte Carlo study and an empirical application with three US stocks confirm our theoretical findings. The empirical application shows that multiplication factors computed under the current regulatory framework are downward biased, as they inherit the inconsistency of the unconditional backtest.     

Chebyshev interpolation for parametric option pricing

Recurrent tasks such as pricing, calibration and risk assessment need to be executed accurately and in real time. We concentrate on parametric option pricing (POP) as a generic instance of parametric conditional expectations and show that polynomial interpolation in the parameter space promises to considerably reduce run-times while maintaining accuracy. The attractive properties of Chebyshev interpolation and its tensorized extension enable us to identify broadly applicable criteria for (sub)exponential convergence and explicit error bounds. The method is most promising when the computation of the prices is most challenging. We therefore investigate its combination with Monte Carlo simulation and analyze the effect of (stochastic) approximations of the interpolation. For a wide and important range of problems, the Chebyshev method turns out to be more efficient than parametric multilevel Monte Carlo. We conclude with a numerical efficiency study.