Analytic Approximation of Finite-Maturity Timer Option Prices

We develop an approximation technique for pricing finite-maturity timer options under Heston-like stochastic volatility models. By approximating the distributions of the accumulated variance and the random variance budget exceeding time, we obtain analytic expressions for timer option prices under zero correlation. For nonzero correlation, we use a simple linear combination approximation which matches the asymptotic correlation behavior. Numerical analysis using the Heston model shows that the method is fairly accurate, especially when the volatility of variance is small or the maximum maturity is large

A stochastic dynamic programming approach for pricing options on stock-index futures

The aim of this thesis is to price options on equity index futures with an application to standard options on S&P 500 futures traded on the Chicago Mercantile Exchange. Our methodology is based on stochastic dynamic programming, which can accommodate European as well as American options. The model accommodates dividends from the underlying asset. It also captures the optimal exercise strategy and the fair value of the option. This approach is an alternative to available numerical pricing methods such as binomial trees, finite differences, and ad-hoc numerical approximation techniques. Our numerical and empirical investigations demonstrate convergence, robustness, and efficiency. We use this methodology to value exchange-listed options. The European option premiums thus obtained are compared to Black's closed-form formula. They are accurate to four digits. The American option premiums also have a similar level of accuracy compared to premiums obtained using finite differences and binomial trees with a large number of time steps. The proposed model accounts for deterministic, seasonally varying dividend yield. In pricing futures options, we discover that what matters is the sum of the dividend yields over the life of the futures contract and not their distribution

Path integral approach to the pricing of timer options with the Duru-Kleinert time transformation

In this paper, a time substitution as used by Duru and Kleinert in their treatment of the hydrogen atom with path integrals is performed to price timer options under stochastic volatility models. We present general pricing formulas for both the perpetual timer call options and the finite time-horizon timer call options. These general results allow us to find closed-form pricing formulas for both the perpetual and the finite time-horizon timer options under the 3/2 stochastic volatility model as well as under the Heston stochastic volatility model. For the treatment of timer option under the 3/2 model we will rely on the path integral for the Morse potential, with the Heston model we will rely on the Kratzer potential

American Option Valuation Methods

This paper implements and compares eight American option valuation methods: binomial, trinomial, explicit finite difference, implicit finite difference and quadratic approximation methods. And three Monte Carlo methods: bundling technique of Tilley (1993), simulated tree (ST) of Broadie, Glasserman, and Jain (1997), and least square regression method (LSM) of Longstaff and Schwartz (2001). Methods are compared in terms of computation efficiency and price accuracy. The findings suggest that binomial is the best performing numerical method in terms of accuracy and efficiency. LSM beats the other two simulation methods in terms of efficiency, accuracy and number of discrete exercise opportunities

Semi-analytic pricing of American options in some time-dependent jump-diffusion models

In this paper we propose a semi-analytic approach to pricing American options for some time-dependent jump-diffusions models. The idea of the method is to further generalize our approach developed for pricing barrier, [Itkin et al., 2021], and American, [Carr and Itkin, 2021; Itkin and Muravey, 2023], options in various time-dependent one factor and even stochastic volatility models. Our approach i) allows arbitrary dependencies of the model parameters on time; ii) reduces solution of the pricing problem for American options to a simpler problem of solving an algebraic nonlinear equation for the exercise boundary and a linear Fredholm-Volterra equation for the the option price; iii) the options Greeks solve a similar Fredholm-Volterra linear equation obtained by just differentiating Eq. (25) by the required parameter.Comment: 18 pages, 1 table, 2 figure

Essays on Economic Sentiment Dynamics and Asymmetric Multifractal Models of Financial Volatility

In the first chapter of the dissertation an estimation of the continuous-time Markov chain (CTMC) model of experts' sentiment index is considered in the case of incomplete data. Particularly, three estimation approaches based on a discrete-time sample are presented: the EM algorithm and two versions of the maximum likelihood estimation method. The first approach for the estimation of the considered model is iterative and leads to massive recursive computations of matrices. The most crucial part of the second and third approaches is the numerical computation of the matrix exponential of the intensity matrix. In particular, the second approach is based on the eigendecomposition of the intensity matrix and the corresponding well-known property of matrix exponential for such decomposition. In order to increase the effectiveness of the method in the third approach the fact that the intensity matrix has a lower Hessenberg form is used. All three approaches are based on numerical optimization using the nonlinear conjugate optimizer. The second chapter is dedicated to the development of the methods of calibration and estimation of the model belonging to the asset price class of models. Two variants of the generalization of the Markov Switching Multifractal (MSM) model, called the Asymmetric Markov-Switching Multifrequency, are considered. The modifications are aimed to reproduce such a phenomenon of asset returns as leverage effect. Other features of the model, namely the long memory stylized fact for different frequencies and degrees of persistence, the mean reversion of volatility, and the volatility clustering, are investigated and proven. The option pricing theory based on risk-neutral measure is developed for this model. In-sample and out-of-sample performance are tested

On the Acceleration of Explicit Finite Difference Methods for Option Pricing

Implicit finite difference methods are conventionally preferred over their explicit counterparts for the numerical valuation of options. In large part the reason for this is a severe stability constraint known as the Courant–Friedrichs–Lewy (CFL) condition which limits the latter class’s efficiency. Implicit methods, however, are difficult to implement for all but the most simple of pricing models, whereas explicit techniques are easily adapted to complex problems. For the first time in a financial context, we present an acceleration technique, applicable to explicit finite difference schemes describing diffusive processes with symmetric evolution operators, called Super-Time-Stepping. We show that this method can be implemented as part of a more general approach for non-symmetric operators. Formal stability is thereby deduced for the exemplar cases of European and American put options priced under the Black–Scholes equation. Furthermore, we introduce a novel approach to describing the efficiencies of finite difference schemes as semi-empirical power laws relating the minimal real time required to carry out the numerical integration to a solution with a specified accuracy. Tests are described in which the method is shown to significantly ameliorate the severity of the CFL constraint whilst retaining the simplicity of the underlying explicit method. Degrees of acceleration are achieved yielding comparable, or superior, efficiencies to a set of benchmark implicit schemes. We infer that the described method is a powerful tool, the explicit nature of which makes it ideally suited to the treatment of symmetric and non-symmetric diffusion operators describing complex financial instruments including multi-dimensional systems requiring representation on decomposed and/or adaptive meshes

Pricing swaptions and credit default swaptions in the quadratic Gaussian factor model

University of Technology, Sydney. Faculty of Business.In this thesis we show how the multi-factor quadratic Gaussian model can be used to price default free and defaultable securities. The mathematical tools used include the theory of stochastic processes, the theory of matrix Riccati equations, the change of measure technique, Ito's formula, use of Fourier Transforms in swaption valuation and approximation methods based on replacing the values of some stochastic processes by their time zero values. The first chapter of the thesis deals with the derivation of efficient closed form formulas for the price of zero coupon bonds in the multi-factor quadratic Gaussian model and the calibration of the multi-factor quadratic Gaussian model to the domestic and foreign forward rate term structures through closed form formulas. In the second chapter of the thesis, we derive approximations for the price of default free swaptions which are based on log-quadratic Gaussian processes. Using numerical experiments, we show the limitations of these approximations. We also give some numerical results for the pricing of a default free swaption using moment-based density approximants of the probability density function of the swaption's payoff. The third chapter of the thesis deals with the calibration of a quadratic Gaussian reduced form model of credit risk to the default free forward rate curve and to the survival probability of an obligor. We also consider different approximations for the price of credit default swaptions. Using numerical experiments, we show the limitations of the approximations. The final chapter of this thesis considers a two country reduced form model of credit risk. We examine the relationship between the domestic forward credit spread and the foreign forward credit spread of an obligor and provide quanto adjustment formulas for the probability of survival of an obligor. In the final part of this chapter, we show that the valuation of a quanto default swap is tractable in a contagion type reduced form model of credit risk which assumes that underlying processes are modelled by quadratic Gaussian processes

Applications of Laplace transform for evaluating occupation time options and other derivatives

The present thesis provides an analysis of possible applications of the Laplace Transform (LT) technique to several pricing problems. In Finance this technique has received very little attention and for this reason, in the first chapter we illustrate with several examples why the use of the LT can considerably simplify the pricing problem. Observed that the analytical inversion is very often difficult or requires the computation of very complicated expressions, we illustrate also how the numerical inversion is remarkably easy to understand and perform and can be done with high accuracy and at very low computational cost. In the second and third chapters we investigate the problem of pricing corridor derivatives, i.e. exotic contracts for which the payoff at maturity depends on the time of permanence of an index inside a band (corridor) or below a given level (hurdle). The index is usually an exchange or interest rate. This kind of bond has evidenced a good popularity in recent years as alternative instruments to common bonds for short term investment and as opportunity for investors believing in stable markets (corridor bonds) or in non appreciating markets (hurdle bonds). In the second chapter, assuming a Geometric Brownian dynamics for the underlying asset and solving the relevant Feynman-Kac equation, we obtain an expression for the Laplace transform of the characteristic function of the occupation time. We then show how to use a multidimensional numerical inversion for obtaining the density function. In the third chapter, we investigate the effect of discrete monitoring on the price of corridor derivatives and, as already observed in the literature for barrier options and for lookback options, we observe substantial differences between discrete and continuous monitoring. The pricing problem with discrete monitoring is based on an appropriate numerical scheme of the system of PDE's. In the fourth chapter we propose a new approximation for pricing Asian options based on the logarithmic moments of the price average

Some topics in mathematical finance

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