Pricing Of Exotic Options Research Articles

Static hedging and pricing of exotic options with payoff frames

AbstractWe develop a general framework for statically hedging and pricing European‐style options with nonstandard terminal payoffs, which can be applied to mixed static–dynamic and semistatic hedges for many path‐dependent exotic options including variance swaps and barrier options. The goal is achieved by separating the hedging and pricing problems to obtain replicating strategies. Once prices have been obtained for a set of basis payoffs, the pricing and hedging of financial securities with arbitrary payoff functions is accomplished by computing a set of “hedge coefficients” for that security. This method is particularly well suited for pricing baskets of options simultaneously, and is robust to discontinuities of payoffs. In addition, the method enables a systematic comparison of the value of a payoff (or portfolio) across a set of competing model specifications with implications for security design.

Machine Learning for Quantitative Finance: Fast Derivative Pricing, Hedging and Fitting

In this paper, we show how we can deploy machine learning techniques in the context of traditional quant problems. We illustrate that for many classical problems, we can arrive to speed-ups of several orders of magnitude by deploying machine learning techniques based on Gaussian process regression. The price we have to pay for this extra speed is some loss of accuracy. However, we show that this reduced accuracy is often well within reasonable limits and hence very acceptable from a practical point of view. The concrete examples concern fitting and estimation. In the fitting context, we fit sophisticated Greek profiles and summarize implied volatility surfaces. In the estimation context, we reduce computation times for the calculation of vanilla option values under advanced models, the pricing of American options and the pricing of exotic options under models beyond the Black-Scholes setting.

American and exotic option pricing with jump diffusions and other Levy processes

In general, no analytical formulas exist for pricing discretely monitored exotic options, even when a geometric Brownian motion governs the risk-neutral underlying. While specialized numerical algorithms exist for pricing particular contracts, few can be applied universally with consistent success and with general Lévy dynamics. This paper develops a general methodology for pricing early exercise and exotic financial options by extending the recently developed PROJ method. We are able to efficiently obtain accurate values for complex products including Bermudan/ American options, Bermudan barrier options, survival probabilities and credit default swaps by value recursion; European barrier and lookback/hindsight options by density recursion; and arithmetic Asian options by characteristic function recursion. This paper presents a unified approach to tackling these and related problems. Algorithms are provided for each option type, along with a demonstration of convergence. We also provide a large set of reference prices for exotic, American and European options under Black–Scholes–Merton, normal inverse Gaussian, Kou’s double exponential jump diffusion, Carr–Madan–Geman–Yor (also known as KoBoL) and Merton’s jump-diffusion models.

Constrained Wiener Processes and Their Financial Applications

The extrema of Wiener processes are relevant to the pricing of so-called exotic options, which have many financial applications. The probability densities of such extrema are well known for one dimensional Wiener processes. We employ elementary methods to derive analytical expressions for the densities for multidimensional Wiener processes, with multiple extrema. These take the form of (possibly infinite) series expansions of Gaussian densities. This is undertaken using the characterization of the Wiener process by the heat equation, a well known connection in mathematical physics.

On the interchangeability of barrier option pricing models

An important question when modelling option prices is which of the multitude of option pricing models to use. In this paper, the calculation of barrier option prices is considered. These exotic options are found in many financial markets the world over. It is demonstrated numerically that it is possible to replicate (with a high degree of accuracy) the barrier option prices obtained from one model by making use of a different model; these models are referred to as ‘interchangeable’. Tests for the interchangeability of barrier option pricing models are developed and applied. However, the tests developed are not specific to barrier option pricing models and can also be applied to the prices of other exotic options.

Extrema of Wiener Processes

The extrema of Wiener processes are relevant to the pricing of so-called exotic options, which have many financial applications. The probability den-sities of such extrema are well known for one dimensional Wiener processes. We employ elementary methods to derive analytical expressions for the den-sities for multidimensional Wiener processes, with multiple extrema. These take the form of (possibly infinite) series expansions of Gaussian densities. This is undertaken using the characterization of the Wiener process by the heat equation, a well known connection in mathematical physics.

The Hybrid Stochastic-Local Volatility Model with Applications in Pricing FX Options

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

Gaussian Quadrature Method for Pricing American and Exotic Options in a Jump-Diffusion Process

In this paper we propose a Gaussian quadrature method to study American and exotic option pricing under the jump-diffusion model of Merton (1976). Our numerical experiments show that the Gaussian quadrature method, compared to several existing methods in the literature, including the fast Gauss transform method (Broadie and Yamamoto, 2003), the bivariate tree approach (Hilliard and Schwartz, 2005), and the extrapolation approach (Feng and Linetsky, 2008), is accurate for valuing American options. In addition to American options, we also show that the Gaussian quadrature method performs well for the valuation of exotic options under the jump-diffusion model. Overall, the Gaussian quadrature method is highly accurate and suitable for the valuation of price options with early exercise features under a jump-diffusion process.

A backward Monte Carlo approach to exotic option pricing

We propose a novel algorithm which allows to sample paths from an underlying price process in a local volatility model and to achieve a substantial variance reduction when pricing exotic options. The new algorithm relies on the construction of a discrete multinomial tree. The crucial feature of our approach is that – in a similar spirit to the Brownian Bridge – each random path runs backward from a terminal fixed point to the initial spot price. We characterize the tree in two alternative ways: (i) in terms of the optimal grids originating from the Recursive Marginal Quantization algorithm, (ii) following an approach inspired by the finite difference approximation of the diffusion's infinitesimal generator. We assess the reliability of the new methodology comparing the performance of both approaches and benchmarking them with competitor Monte Carlo methods.

A Modified Stochastic Volatility Model for Levy Process Based on Gamma Ornstein-Uhlenbeck Process and Option Pricing.

Stochastic volatility model of the Gamma Ornstein-Uhlenbeck possess authentic capability of both capturing some stylized features of financial time series and pricing European options. In this work we modify the Gamma OU model from the viewpoint of Monte Carlo simulation, which is crucial in both model inference and exotic option pricing. We discuss topics related to the measure transformation between objective and risk-neutral measures, arbitrage-free and market incompleteness of the new model. Furthermore, we investigate the performance of this model in European options pricing and an empirical application is presented.

Calibration Risk for Exotic Options

Option pricing models are calibrated to market data of plain vanillas by minimization of an error functional. From the economic viewpoint, there are several possibilities to measure the error between the market and the model. These different specifications of the error give rise to different sets of calibrated model parameters and the resulting prices of exotic options vary significantly. These price differences often exceed the usual profit margin of exotic options.We provide evidence for this calibration risk in a time series of DAX implied volatility surfaces from April 2003 to March 2004. We analyze in the Heston and in the Bates model factors influencing these price differences of exotic options and finally recommend an error functional. Moreover, we determine the model risk of these two stochastic volatility models for the time series and consider its relation to calibration risk.

Optimal execution of accelerated share repurchase contracts with fixed notional

Whether it be to take advantage of stock undervaluation or in order to distribute part of their profits to shareholders, firms may buy back their own shares. One of the ways they do this is by including accelerated share repurchases as part of their repurchase programs. We study the pricing and optimal execution strategy of an accelerated share repurchase contract with a fixed notional. In such a contract the firm pays a fixed notional F to the bank and receives in exchange a number of shares corresponding to the ratio of F to the average stock price over the purchase period (the duration of this period being decided upon by the bank). From a mathematical point of view, the problem is related to both optimal execution and exotic option pricing.

Quantization Goes Polynomial

Quantization algorithms have been recently successfully adopted in option pricing problems to speed up Monte Carlo simulations thanks to the high convergence rate of the numerical approximation. In particular, recursive marginal quantization has been proven a flexible and versatile tool when applied to stochastic volatility processes. In this paper we apply for the first time these techniques to the family of polynomial processes, by exploiting, whenever possible, their peculiar properties. We derive theoretical results to assess the approximation errors, and we describe in numerical examples practical tools for fast exotic option pricing.

A modified stochastic volatility model based on Gamma Ornstein–Uhlenbeck process and option pricing

Stochastic volatility model of the Gamma Ornstein–Uhlenbeck possess authentic capability of both capturing some stylized features of financial time series and pricing European options. In this work we modify the Gamma OU model from the viewpoint of Monte Carlo simulation, which is crucial in both model inference and exotic option pricing. We discuss topics related to the measure transformation between objective and risk-neutral measures, arbitrage-free and market incompleteness of the new model. Furthermore, we investigate the performance of this model in European options pricing and an empirical application is presented.

Incorporating Implied Volatility in Pricing Options using Binomial Tree

Background/Objectives: The main objective of this paper is to present an n-step binomial model which can be used to price an option under any exotic conditions. Methods/Statistical Analysis: Mathematical models have been presented using which an n-step binomial model can be developed. The model can be used for estimating price of options under n of number exotic conditions that influence the option price. Findings: Pricing of exotic options like Asian, American etc., undertaken through Binomial Trees using only one-step considering maximum and minimum values that can be taken by the underling at the maturity leads to a rough approximation of the option price. The approximation is possible by assuming stock price movements to be in one or two binomial steps during the life of the option. A binomial tree extended to an N-step Model can be used to price various exotic options. A study of the convergence in European option price with respect to Number of steps (N) and variation in price of Asian and American options with respect to confidence factor (k) (proxy for implied volatility) using the maximum and minimum boundaries on the value of k gives the investors the ability to change the value of k so that they can have their own opinions concerning the risk-neutral probability distribution.

Lattice Boltzmann methods for solving partial differential equations of exotic option pricing

This paper establishes a lattice Boltzmann method (LBM) with two amending functions for solving partial differential equations (PDEs) arising in Asian and lookback options pricing. The time evolution of stock prices can be regarded as the movement of randomizing particles in different directions, and the discrete scheme of LBM can be interpreted as the binomial models. With the Chapman-Enskog multi-scale expansion, the PDEs are recovered correctly from the continuous Boltzmann equation and the computational complexity is O(N), where N is the number of space nodes. Compared to the traditional LBM, the coefficients of equilibrium distribution and amending functions are taken as polynomials instead of constants. The stability of LBM is studied via numerical examples and numerical comparisons show that the LBM is as accurate as the existing numerical methods for pricing the exotic options and takes much less CPU time.

State price densities implied from weather derivatives

A State Price Density (SPD) is the density function of a risk neutral equivalent martingale measure for option pricing, and is indispensable for exotic option pricing and portfolio risk management. Many approaches have been proposed in the last two decades to calibrate a SPD using financial options from the bond and equity markets. Among these, non and semiparametric methods were preferred because they can avoid model mis-specification of the underlying. However, these methods usually require a large data set to achieve desired convergence properties. One faces the problem in estimation by e.g., kernel techniques that there are not enough observations locally available. For this situation, we employ a Bayesian quadrature method because it allows us to incorporate prior assumptions on the model parameters and hence avoids problems with data sparsity. It is able to compute the SPD of both call and put options simultaneously, and is particularly robust when the market faces the data sparsity issue. As illustration, we calibrate the SPD for weather derivatives, a classical example of incomplete markets with financial contracts payoffs linked to non-tradable assets, namely, weather indices. Finally, we study related weather derivatives data and the dynamics of the implied SPDs.

Applying the Wiener-Hopf Monte Carlo Simulation Technique for Lévy Processes to Path Functionals

In this paper we apply the recently established Wiener-Hopf Monte Carlo simulation technique for Lévy processes from Kuznetsov et al. (2011) to path functionals; in particular, first passage times, overshoots, undershoots, and the last maximum before the passage time. Such functionals have many applications, for instance, in finance (the pricing of exotic options in a Lévy model) and insurance (ruin time, debt at ruin, and related quantities for a Lévy insurance risk process). The technique works for any Lévy process whose running infimum and supremum evaluated at an independent exponential time can be sampled from. This includes classic examples such as stable processes, subclasses of spectrally one-sided Lévy processes, and large new families such as meromorphic Lévy processes. Finally, we present some examples. A particular aspect that is illustrated is that the Wiener-Hopf Monte Carlo simulation technique (provided that it applies) performs much better at approximating first passage times than a ‘plain’ Monte Carlo simulation technique based on sampling increments of the Lévy process.

Applying the Wiener-Hopf Monte Carlo Simulation Technique for Lévy Processes to Path Functionals

In this paper we apply the recently established Wiener-Hopf Monte Carlo simulation technique for Lévy processes from Kuznetsov et al. (2011) to path functionals; in particular, first passage times, overshoots, undershoots, and the last maximum before the passage time. Such functionals have many applications, for instance, in finance (the pricing of exotic options in a Lévy model) and insurance (ruin time, debt at ruin, and related quantities for a Lévy insurance risk process). The technique works for any Lévy process whose running infimum and supremum evaluated at an independent exponential time can be sampled from. This includes classic examples such as stable processes, subclasses of spectrally one-sided Lévy processes, and large new families such as meromorphic Lévy processes. Finally, we present some examples. A particular aspect that is illustrated is that the Wiener-Hopf Monte Carlo simulation technique (provided that it applies) performs much better at approximating first passage times than a ‘plain’ Monte Carlo simulation technique based on sampling increments of the Lévy process.

Multilevel Richardson-Romberg Extrapolation

We propose and analyze a Multilevel Richardson-Romberg ($MLRR$) estimator which combines the higher order bias cancellation of the Multistep Richardson-Romberg ($MSRR$) method introduced in [Pag07] and the variance control resulting from the stratification in the Multilevel Monte Carlo ($MLMC$) method (see [Hei01, Gil08]). Thus we show that in standard frameworks like discretization schemes of diffusion processes an assigned quadratic error $\varepsilon$ can be obtained with our ($MLRR$) estimator with a global complexity of $\log(1/\varepsilon)/\varepsilon^2$ instead of $(\log(1/\varepsilon))^2/\varepsilon^2$ with the standard ($MLMC$) method, at least when the weak error $\mathbf{E}[Y_h]-\mathbf{E}[Y_0]$ of the biased implemented estimator $Y_h$ can be expanded at any order in $h$. We analyze and compare these estimators on two numerical problems: the classical vanilla and exotic option pricing by Monte Carlo simulation and the less classical Nested Monte Carlo simulation.