Fast estimation of true bounds on Bermudan option prices under jump-diffusion processes

Fast pricing of American-style options has been a difficult problem since it was first introduced to the financial markets in 1970s, especially when the underlying stocks’ prices follow some jump-diffusion processes. In this paper, we extend the ‘true martingale algorithm’ proposed by Belomestny et al. [Math. Finance, 2009, 19, 53–71] for the pure-diffusion models to the jump-diffusion models, to fast compute true tight upper bounds on the Bermudan option price in a non-nested simulation manner. By exploiting the martingale representation theorem on the optimal dual martingale driven by jump-diffusion processes, we are able to explore the unique structure of the optimal dual martingale and construct an approximation that preserves the martingale property. The resulting upper bound estimator avoids the nested Monte Carlo simulation suffered by the original primal–dual algorithm, therefore significantly improving the computational efficiency. Theoretical analysis is provided to guarantee the quality of the martingale approximation. Numerical experiments are conducted to verify the efficiency of our algorithm.     

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.     

Improved lower and upper bound algorithms for pricing American options by simulation

This paper introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American-style options. For simulation algorithms that compute lower bounds of American option values, we apply martingale control variates and introduce the local policy enhancement, which adopts a local simulation to improve the exercise policy. For duality-based upper bound methods, specifically the primal–dual simulation algorithm, we have developed two improvements. One is sub-optimality checking, which saves unnecessary computation when it is sub-optimal to exercise the option along the sample path; the second is boundary distance grouping, which reduces computational time by skipping computation on selected sample paths based on the distance to the exercise boundary. Numerical results are given for single asset Bermudan options, moving window Asian options and Bermudan max options. In some examples the computational time is reduced by a factor of several hundred, while the confidence interval of the true option value is considerably tighter than before the improvements.     

Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff

Giles (Oper. Res. 56:607–617, 2008) introduced a multi-level Monte Carlo method for approximating the expected value of a function of a stochastic differential equation solution. A key application is to compute the expected payoff of a financial option. This new method improves on the computational complexity of standard Monte Carlo. Giles analysed globally Lipschitz payoffs, but also found good performance in practice for non-globally Lipschitz cases. In this work, we show that the multi-level Monte Carlo method can be rigorously justified for non-globally Lipschitz payoffs. In particular, we consider digital, lookback and barrier options. This requires non-standard strong convergence analysis of the Euler–Maruyama method.      

A pure martingale dual for multiple stopping

In this paper, we present a dual representation for the multiple stopping problem, hence multiple exercise options. As such, it is a natural generalization of the method in Rogers (Math. Finance 12:271–286, 2002) and Haugh and Kogan (Oper. Res. 52:258–270, 2004) for the standard stopping problem for American options. We term this representation a ‘pure martingale’ dual as it is solely expressed in terms of an infimum over martingales rather than an infimum over martingales and stopping times as in Meinshausen and Hambly (Math. Finance 14:557–583, 2004). For the multiple dual representation, we propose Monte Carlo simulation methods which require only one degree of nesting.     

Dual pricing of multi-exercise options under volume constraints

In this paper, we study the pricing problem of multi-exercise options under volume constraints. The volume constraint is modelled by an adapted process with values in the positive integers, which describes the maximal number of rights to be exercised at a given time. We derive a representation of the marginal value of an additional nth right as a standard single stopping problem with a modified cash-flow process. This representation then leads to a dual pricing formula, which generalizes a result by Meinshausen and Hambly (Math. Finance 14:557–583, 2004) from the standard multi-exercise option (with at most one right per time step) to general constraints. We also state an explicit Monte Carlo algorithm for computing confidence intervals for the price of multi-exercise options under volume constraints and present numerical results for the pricing of a swing contract in an electricity market.     

Computing the endogenous mortgage rate without iterations

A number of mortgage prepayment models require a specification of the mortgage rate process. Usually, ad-hoc models are used (e.g., a Treasury yield plus some constant). Recently, a number of papers have appeared where the authors have utilized a mortgage rate implied by the current yield curve (the so-called endogenous mortgage rate). However, the existing computational algorithms suffer from the curse of dimensionality and, consequently, are problematic to use for full-scale problems. A computational algorithm, proposed in this paper, is tractable in the sense that its complexity is equivalent to the problem of mortgage valuation. Moreover, the algorithm does not require iterations. The numerical example is based on a PDE computation. An implementation of a Monte Carlo method is also discussed.     

A computational definition of financial randomness

The aim of this study is to present an efficient and easy framework for the application of the Least Squares Monte Carlo methodology to the pricing of gas or power facilities as detailed in Boogert and de Jong [J. Derivatives, 2008, 15, 81–91]. As mentioned in the seminal paper by Longstaff and Schwartz [Rev. Financ. Stud. 2001, 113–147], the convergence of the Least Squares Monte Carlo algorithm depends on the convergence of the optimization combined with the convergence of the pure Monte Carlo method. In the context of the energy facilities, the optimization is more complex and its convergence is of fundamental importance in particular for the computation of sensitivities and optimal dispatched quantities. To our knowledge, an extensive study of the convergence, and hence of the reliability of the algorithm, has not been performed yet, in our opinion this is because the apparent infeasibility and complexity uses a very high number of simulations. We present then an easy way to simulate random trajectories by means of diffusion bridges in contrast to Dutt and Welke [J. Derivatives, 2008, 15 (4), 29–47] that considers time-reversal Itô diffusions and subordinated processes. In particular, we show that in the case of Cox-Ingersoll-Ross and Heston models, the bridge approach has the advantage to produce exact simulations even for non-Gaussian processes, in contrast to the time-reversal approach. Our methodology permits performing a backward dynamic programming strategy based on a huge number of simulations without storing the whole simulated trajectory. Generally, in the valuation of energy facilities, one is also interested in the forward recursion. We then design backward and forward recursion algorithms such that one can produce the same random trajectories by the use of multiple independent random streams without storing data at intermediate time steps. Finally, we show the advantages of our methodology for the valuation of virtual hydro power plants and gas storages.     

No-arbitrage bounds for the forward smile given marginals

We explore the robust replication of forward-start straddles given quoted (Call and Put options) market data. One approach to this problem classically follows semi-infinite linear programming arguments, and we propose a discretisation scheme to reduce its dimensionality and hence its complexity. Alternatively, one can consider the dual problem, consisting in finding optimal martingale measures under which the upper and the lower bounds are attained. Semi-analytical solutions to this dual problem were proposed by Hobson and Klimmek [Financ. Stochastics, 2015, 19, 189–214] and by Hobson and Neuberger [Math. Financ., 2012, 22, 31–56]. We recast this dual approach as a finite-dimensional linear program, and reconcile numerically, in the Black–Scholes and in the Heston model, the two approaches.     

Multilevel Monte Carlo for exponential Lévy models

We apply the multilevel Monte Carlo method for option pricing problems using exponential Lévy models with a uniform timestep discretisation. For lookback and barrier options, we derive estimates of the convergence rate of the error introduced by the discrete monitoring of the running supremum of a broad class of Lévy processes. We then use these to obtain upper bounds on the multilevel Monte Carlo variance convergence rate for the variance gamma, NIG and \(\alpha\)-stable processes. We also provide an analysis of a trapezoidal approximation for Asian options. Our method is illustrated by numerical experiments.     

Moments of renewal shot-noise processes and their applications

In this paper, we study the family of renewal shot-noise processes. The Feynmann–Kac formula is obtained based on the piecewise deterministic Markov process theory and the martingale methodology. We then derive the Laplace transforms of the conditional moments and asymptotic moments of the processes. In general, by inverting the Laplace transforms, the asymptotic moments and the first conditional moments can be derived explicitly; however, other conditional moments may need to be estimated numerically. As an example, we develop a very efficient and general algorithm of Monte Carlo exact simulation for estimating the second conditional moments. The results can be then easily transformed to the counterparts of discounted aggregate claims for insurance applications, and we apply the first two conditional moments for the actuarial net premium calculation. Similarly, they can also be applied to credit risk and reliability modelling. Numerical examples with four distribution choices for interarrival times are provided to illustrate how the models can be implemented.     

Hybrid scheme for Brownian semistationary processes

We introduce a simulation scheme for Brownian semistationary processes, which is based on discretizing the stochastic integral representation of the process in the time domain. We assume that the kernel function of the process is regularly varying at zero. The novel feature of the scheme is to approximate the kernel function by a power function near zero and by a step function elsewhere. The resulting approximation of the process is a combination of Wiener integrals of the power function and a Riemann sum, which is why we call this method a hybrid scheme. Our main theoretical result describes the asymptotics of the mean square error of the hybrid scheme, and we observe that the scheme leads to a substantial improvement of accuracy compared to the ordinary forward Riemann-sum scheme, while having the same computational complexity. We exemplify the use of the hybrid scheme by two numerical experiments, where we examine the finite-sample properties of an estimator of the roughness parameter of a Brownian semistationary process and study Monte Carlo option pricing in the rough Bergomi model of Bayer et al. (Quant. Finance 16:887–904, 2016), respectively.     

Unconditional return disturbances: A non-parametric simulation approach

Simulation methods are extensively used in Asset Pricing and Risk Management. The most popular of these simulation approaches, the Monte Carlo, requires model selection and parameter estimation. In addition, these approaches can be extremely computer intensive. Historical simulation has been proposed as a non-parametric alternative to Monte Carlo. This approach is limited to the historical data available.In this paper, we propose an alternative historical simulation approach. Given a historical set of data, we define a set of standardized disturbances and we generate alternative price paths by perturbing the first two moments of the original path or by reshuffling the disturbances. This approach is either totally non-parametric when constant volatility is assumed; or semi-parametric in presence of GARCH(1, 1) volatility. Without a loss in accuracy, it is shown to be much more powerful in terms of computer efficiency than the Monte Carlo approach. It is also extremely simple to implement and can be an effective tool for the valuation of financial assets.We apply this approach to simulate pay off values of options on the S&P 500 stock index for the period 1982–2003. To verify that this technique works, the common back-testing approach was used. The estimated values are insignificantly different from the actual S&P 500 options payoff values for the observed period.     

Approximation methods for multiple period Value at Risk and Expected Shortfall prediction

In this paper, we are interested in predicting multiple period Value at Risk and Expected Shortfall based on the so-called iterating approach. In general, the properties of the conditional distribution of multiple period returns do not follow easily from the one-period data generating process, rendering this a non-trivial task. We outline a framework that forms the basis for setting approximations and study four different approaches. Their performance is evaluated by means of extensive Monte Carlo simulations based on an asymmetric GARCH model, implying conditional skewness and excess kurtosis in the multiple period returns. This simulation-based approach was the best one, closely followed by that of assuming a skewed t-distribution for the multiple period returns. The approach based on a Gram–Charlier expansion was not able to cope with the implied non-normality, while the so-called Root-k approach performed poorly. In addition, we outline how the delta-method may be used to quantify the estimation error in the predictors and in the Monte Carlo study we found that it performed well. In an empirical illustration, we computed 10-day Value at Risk’s and Expected Shortfall for Brent Crude Oil, the EUR/USD exchange rate and the S&P 500 index. The Root-k approach clearly performed the worst and the other approaches performed quite similarly, with the simulation based approach and the one based on the skewed t-distribution somewhat better than the one based on the Gram–Charlier expansion.     

The stochastic collocation Monte Carlo sampler: highly efficient sampling from ‘expensive’ distributions

In this article, we propose an efficient approach for inverting computationally expensive cumulative distribution functions. A collocation method, called the Stochastic Collocation Monte Carlo sampler (SCMC sampler), within a polynomial chaos expansion framework, allows us the generation of any number of Monte Carlo samples based on only a few inversions of the original distribution plus independent samples from a standard normal variable. We will show that with this path-independent collocation approach the exact simulation of the Heston stochastic volatility model, as proposed in Broadie and Kaya [Oper. Res., 2006, 54, 217–231], can be performed efficiently and accurately. We also show how to efficiently generate samples from the squared Bessel process and perform the exact simulation of the SABR model.     

Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model

The rough Bergomi (rBergomi) model, introduced recently in Bayer et al. [Pricing under rough volatility. Quant. Finance, 2016, 16(6), 887–904], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet remarkably fits empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a time-consuming task. To overcome this issue, we have designed a novel, hierarchical approach, based on: (i) adaptive sparse grids quadrature (ASGQ), and (ii) quasi-Monte Carlo (QMC). Both techniques are coupled with a Brownian bridge construction and a Richardson extrapolation on the weak error. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method. They reach a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e. to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.     

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.     

On an efficient multiple time step Monte Carlo simulation of the SABR model

In this paper, we will present a multiple time step Monte Carlo simulation technique for pricing options under the Stochastic Alpha Beta Rho model. The proposed method is an extension of the one time step Monte Carlo method that we proposed in an accompanying paper Leitao et al. [Appl. Math. Comput. 2017, 293, 461–479], for pricing European options in the context of the model calibration. A highly efficient method results, with many very interesting and nontrivial components, like Fourier inversion for the sum of log-normals, stochastic collocation, Gumbel copula, correlation approximation, that are not yet seen in combination within a Monte Carlo simulation. The present multiple time step Monte Carlo method is especially useful for long-term options and for exotic options.     

Portfolio selection with qualitative input

We formulate a mean-variance portfolio selection problem that accommodates qualitative input about expected returns and provide an algorithm that solves the problem. This model and algorithm can be used, for example, when a portfolio manager determines that one industry will benefit more from a regulatory change than another but is unable to quantify the degree of difference. Qualitative views are expressed in terms of linear inequalities among expected returns. Our formulation builds on the Black-Litterman model for portfolio selection. The algorithm makes use of an adaptation of the hit-and-run method for Markov chain Monte Carlo simulation. We also present computational results that illustrate advantages of our approach over alternative heuristic methods for incorporating qualitative input.     

A generalization of reset options pricing formulae with stochastic interest rates

A generalization of reset call options with predetermined dates is derived in the case of time-dependent volatility and time-dependent interest rate by applying martingale method and change of nume?aire or change of probability measure. An analytical pricing formula for the reset call option is also obtained when the interest rate follows an extended Vasicek’s model. Numerical results show that the correlated coefficient between the stock price and interest rate is almost unacted on the price of reset call option with short maturity and Monte Carlo method is inefficient. Monte Carlo method should be only used if there is no closed-formed solution for option pricing.