The Evaluation of American Compound Option Prices Under Stochastic Volatility Using the Sparse Grid Approach

A compound option (the mother option) gives the holder the right, but not obligation to buy (long) or sell (short) the underlying option (the daughter option). In this paper, we demonstrate a partial differential equation (PDE) approach to pricing American-type compound options where the underlying dynamics follow Heston’s stochastic volatility model. This price is formulated as the solution to a two-pass free boundary PDE problem. A modified sparse grid approach is implemented to solve the PDEs, which is shown to be accurate and efficient compared with the results from Monte Carlo simulation combined with the Method of Lines.American compound option; stochastic volatility; free boundary problem; sparse grid; combination technique; Monte Carlo simulation; method of lines

Modelling and Estimating the Forward Price Curve in the Energy Market

The stochastic or random nature of commodity prices plays a central role in models for valuing financial contingent claims on commodities. In this paper, by enhancing a multifactor framework which is consistent not only with the market observable forward price curve but also the volatilities and correlations of forward prices, we propose a two factor stochastic volatility model for the evolution of the gas forward curve. The volatility is stochastic due to a hidden Markov Chain that causes it to switch between "on peak" and "off peak" states. Based on the structure functional forms for the volatility, we propose and implement the Markov Chain Monte Carlo (MCMC) method to estimate the parameters of the forward curve model. Applications to simulated data indicate that the proposed algorithm is able to accommodate more general features, such as regime switching and seasonality. Applications to the market gas forward data shows that the MCMC approach provides stable estimates.

Particle Filters for Markov Switching Stochastic Volatility Models

This paper proposes an auxiliary particle filter algorithm for inference in regime switching stochastic volatility models in which the regime state is governed by a first-order Markov chain. We proposes an ongoing updated Dirichlet distribution to estimate the transition probabilities of the Markov chain in the auxiliary particle filter. A simulation-based algorithm is presented for the method which demonstrated that we are able to estimate a class of models in which the probability that the system state transits from one regime to a different regime is relatively high. The methodology is implemented to analyze a real time series: the foreign exchange rate of Australian dollars vs South Korean won.Particle filters; Markov switching stochastic volatility models; Sequential Monte Carlo simulation

The Evaluation of Multiple Year Gas Sales Agreement with Regime Switching

A typical gas sales agreement (GSA) also called a gas swing contract, is an agreement between a supplier and a purchaser for the delivery of variable daily quantities of gas, between specified minimum and maximum daily limits, over a certain number of years at a specified set of contract prices. The main constraint of such an agreement that makes them difficult to value are that in each gas year there is a minimum volume of gas (termed take-or-pay or minimum bill) for which the buyer will be charged at the end of the year (or penalty date), regardless of the actual quantity of gas taken. We propose a framework for pricing such swing contracts for an underlying gas forward price curve that follows a regime-switching process in order to better capture the volatility behaviour in such markets. With the help of a recombing pentanonial tree, we are able to efficiently evaluate the prices of the swing contracts, find optimal daily decisions and optimaly early use of both the make-up bank and the carry forward bank at different regimes. We also show how the change of regime will affect the decisions.gas sales agreement; swing contract; take-or-pay; make-up; carry forward; forward price curve; regime switching volatility; recombing pentanomial tree

The Evaluation Of Barrier Option Prices Under Stochastic Volatility

This paperc onsiders the problem o fnumerically evaluating barrier option prices when the dynamics of the underlying are driven by stochastic volatility following the square root process of Heston (1993). We develop a method of lines approach to evaluate the price as well as the delta and gamma of the option. The method is able to effciently handle bothc ontinuously monitored and discretely monitored barrier options and can also handle barrier options with early exercise features. In the latter case, we can calculate the early exercise boundary of an American barrier option in both the continuously and discretely monitored cases.barrier option; stochastic volatility; continuously monitored; discretely monitored; free boundary problem; method of lines; Monte Carlo simulation

The Evaluation of American Option Prices Under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines

This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston (1993), and by a Poisson jump process of the type originally introduced by Merton (1976). We develop a method of lines algorithm to evaluate the price as well as the delta and gamma of the option, thereby extending the method developed by Meyer (1998) for the case of jump-diffusion dynamics. The accuracy of the method is tested against two numerical methods that directly solve the integro-partial differential pricing equation. The first is an extension to the jump-diffusion situation of the componentwise splitting method of Ikonen & Toivanen (2007). The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation which is taken as the benchmark. The relative efficiency of these methods for computing the American call option price, delta, gamma and free boundary is analysed. If one seeks an algorithm that gives not only the price but also the delta and gamma to the same level of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered.American options; stochastic volatility; jump-diffusion processes; Volterra integral equations; free boundary problem; method of lines

The Evaluation of Multiple Year Gas Sales Agreement with Regime Switching

A typical gas sales agreement (GSA), also called a gas swing contract, is an agreement between a supplier and a purchaser for the delivery of variable daily quantities of gas, between specified minimum and maximum daily limits, over a certain number of years at a specified set of contract prices. The main constraint of such an agreement that makes them difficult to value is that in each gas year there is a minimum volume of gas (termed take-or-pay or minimum bill) for which the buyer will be charged at the end of the year (or penalty date), regardless of the actual quantity of gas taken. We propose a framework for pricing such swing contracts for an underlying gas forward price curve that follows a regime-switching process in order to better capture the volatility behaviour in such markets. With the help of a recombining pentanomial tree, we are able to efficiently evaluate the prices of the swing contracts, find optimal daily decisions and optimal yearly use of both the make-up bank and the carry forward bank at different regimes. We also show how the change of regime will affect the decisions

Stochastic target hitting time and the problem of early retirement

We consider a problem of optimal control of a “retirement investment fund” over a finite time horizon with a target hitting time criteria. That is, we wish to decide, at each stage, what percentage of the current retirement fund to allocate into the limited number of investment options so that a decision maker can maximize the probability that his or her wealth exceeds a target prior to his or her retirement. We use Markov decision processes with probability criteria to model this problem and give an example based on data from certain options available in an Australian retirement fund

Economic determinants of oil futures volatility: a term structure perspective

To assess the economic determinants of oil futures volatility, we firstly develop and estimate a multi-factor oil futures pricing model with stochastic volatility that is able to disentangle long-term, medium-term and short-term variations in commodity markets volatility. The volatility estimates reveal that in line with theory, the volatility factors are unspanned, persistent and carry negative market price of risk, while crude oil markets are becoming more integrated with financial markets. After 2004, short-term volatility is driven by industrial production, term and credit spreads, the S&P 500 and the US dollar index, along with the traditional drivers including hedging pressure and VIX. Medium-term volatility is consistently related to open interest and credit spreads, while after 2004 oil sector variables such as inventory and consumption also impact this part of the term structure. Interest rates mostly matter for long-term futures price volatility