Boda Kang

The evaluation of barrier option prices under stochastic volatility

This paper considers the problem of numerically evaluating barrier option prices when the dynamics of the underlying are driven by stochastic volatility following the square root process of Heston (1993) [7]. 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 efficiently handle both continuously 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.

The Return-Volatility Relation in Commodity Futures Markets

By employing a continuous time multi‐factor stochastic volatility model, the dynamic relation between returns and volatility in the commodity futures markets is analyzed. The model is estimated by using an extensive database of gold and crude oil futures and futures options. A positive relation in the gold futures market and a negative relation in the crude oil futures market subsist, especially over periods of high volatility principally driven by market‐wide shocks. The opposite relation holds over quiet periods typically driven by commodity‐specific effects. According to the proposed convenience yield effect, normal (inverted) commodity futures markets entail a negative (positive) relation.

The evaluation of American compound option prices under stochastic volatility and stochastic interest rates

A compound option (the mother option) gives the holder the right, but not the obligation, to buy (long) or sell (short) the underlying option (the daughter option). In this paper, we consider the problem of pricing American-type compound options when the underlying dynamics follow Heston’s stochastic volatility and with stochastic interest rate driven by Cox–Ingersoll–Ross processes. We use a partial differential equation (PDE) approach to obtain a numerical solution. The problem is formulated as the solution to a two-pass free-boundary PDE problem, which is solved via a sparse grid approach and is found to be accurate and efficient compared with the results from a benchmark solution based on a least-squares Monte Carlo simulation combined with the projected successive over-relaxation method.

Humps in the Volatility Structure of the Crude Oil Futures Market: New Evidence

This paper analyzes the volatility structure of commodity derivatives markets. The model encompasses stochastic volatility that may be unspanned by futures contracts. A generalized hump-shaped volatility specification is assumed that entails a finite-dimensional affine model for the commodity futures curve and quasi-analytical prices for options on commodity futures. An empirical study of the crude oil futures volatility structure is carried out using an extensive database of futures prices as well as futures option prices spanning 21 years. The study supports a hump-shaped, partially spanned stochastic volatility specification. Factor hedging, which takes into account shocks to both the volatility processes and the futures curve, depicts the presence of unspanned components in the volatility of commodity futures and the outperformance of the hump-shaped volatility in comparison to the more popular exponential decaying volatility. This hump shaped feature is more pronounced when the market is volatile.

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.

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.

A comparative study on time-efficient methods to price compound options in the Heston model

The primary purpose of this paper is to provide an in-depth analysis of a number of structurally different methods to numerically evaluate European compound option prices under Hestons stochastic volatility dynamics. Therefore, we first outline several approaches that can be used to price these type of options in the Heston model: a modified sparse grid method, a fractional fast Fourier transform technique, a (semi-)analytical valuation formula using Greens function of logarithmic spot and volatility and a Monte Carlo simulation. Then we compare the methods on a theoretical basis and report on their numerical properties with respect to computational times and accuracy. One key element of our analysis is that the analyzed methods are extended to incorporate piecewise time-dependent model parameters, which allows for a more realistic compound option pricing. The results in the numerical analysis section are important for practitioners in the financial industry to identify under which model prerequisites (for instance, Heston model where Feller condition is fulfilled or not, Heston model with piecewise time-dependent parameters or with stochastic interest rates) it is preferable to use and which of the available numerical methods.

The Evaluation of Gas Swing Contracts with Regime Switching

A typical 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 period at a specified set of contract prices. The main constraint of such an agreement that makes them difficult to value are that 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 period (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 behavior in such markets. With the help of a recombining pentanomial tree, we are able to efficiently evaluate the prices of the swing contracts and find optimal daily decisions in different regimes. We also show how the change of regime will affect the decisions.

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 behavior 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.

Pricing an American Call Under Stochastic Volatility and Interest Rates

This chapter discusses the problem of pricing an American call option when the underlying dynamics follow the Heston’s stochastic volatility and the Cox-Ingersoll-Ross (CIR) stochastic interest rate. We use a partial differential equation (PDE) approach to obtain a numerical solution. The call is formulated as a free boundary PDE problem on a finite computational domain with appropriate boundary conditions. It is solved with the time discrete method of lines which is found to be accurate and efficient in producing option prices, early exercise boundaries and option hedge ratios like delta and gamma. The method of lines results are compared with those from a finite difference approximation of the corresponding linear complementarity formulation which were obtained with PSOR and the Sparse Grid approach.