A systematic and efficient simulation scheme for the Greeks of financial derivatives

Greeks are the price sensitivities of financial derivatives and are essential for pricing, speculation, risk management, and model calibration. Although the pathwise method has been popular for calculating them, its applicability is problematic when the integrand is discontinuous. To tackle this problem, this paper defines and derives the parameter derivative of a discontinuous integrand of certain functional forms with respect to the parameter of interest. The parameter derivative is such that its integration equals the differentiation of the integration of the aforesaid discontinuous integrand with respect to that parameter. As a result, unbiased Greek formulas for a very broad class of payoff functions and models can be systematically derived. This new method is applied to the Greeks of (1) Asian options under two popular Lévy processes, i.e. Merton's jump-diffusion model and the variance-gamma process, and (2) collateralized debt obligations under the Gaussian copula model. Our Greeks outperform the finite-difference and likelihood ratio methods in terms of accuracy, variance, and computation time.     

An alternative approach to the valuation of American options and applications

In this paper we examine the structure of American option valuation problems and derive the analytic valuation formulas under general underlying security price processes by an alternative but intuitive method. For alternative diffusion processes, we derive closed-form analytic valuation formulas and analyze the implications of asset price dynamics on the early exercise premiums of American options. In this regard, we introduce useful and interesting diffusion processes into American option-pricing literature, thus providing a wide range of choices of pricing models for various American-type derivative assets. This work offers a useful analytic framework for empirical testing and practical applications such as the valuation of corporate securities and examining the impact of options trading on market micro-structure.     

Computing deltas without derivatives

A well-known application of Malliavin calculus in mathematical finance is the probabilistic representation of option price sensitivities, the so-called Greeks, as expectation functionals that do not involve the derivative of the payoff function. This allows numerically tractable computation of the Greeks even for discontinuous payoff functions. However, while the payoff function is allowed to be irregular, the coefficients of the underlying diffusion are required to be smooth in the existing literature, which for example already excludes simple regime-switching diffusion models. The aim of this article is to generalise this application of Malliavin calculus to Itô diffusions with irregular drift coefficients, where we focus here on the computation of the delta, which is the option price sensitivity with respect to the initial value of the underlying. To this end, we first show existence, Malliavin differentiability and (Sobolev) differentiability in the initial condition for strong solutions of Itô diffusions with drift coefficients that can be decomposed into the sum of a bounded, but merely measurable, and a Lipschitz part. Furthermore, we give explicit expressions for the corresponding Malliavin and Sobolev derivatives in terms of the local time of the diffusion, respectively. We then turn to the main objective of this article and analyse the existence and probabilistic representation of the corresponding deltas for European and path-dependent options. We conclude with a small simulation study of several regime-switching examples.     

A New Approach to International Arbitrage Pricing

This paper uses a nonlinear arbitrage-pricing model, a conditional linear model, and an unconditional linear model to price international equities, bonds, and forward currency contracts. Unlike linear models, the nonlinear arbitrage-pricing model requires no restrictions on the payoff space, allowing it to price payoffs of options, forward contracts, and other derivative securities. Only the nonlinear arbitrage-pricing model does an adequate job of explaining the time series behavior of a cross section of international returns.     

Testing Greeks and price changes in the S&P 500 options and futures contract: A regression analysis

We use a regression model to test observed price changes with Greeks as regressors. Greeks are computed using implied volatility, price-change implied volatility and historical volatility. We find sufficient evidence to reject model Greeks as unbiased responses to underlying price as well as sufficient evidence that the American version of binomial model results in biased estimates of price changes. We use options on the S&P 500 futures contracts and their underlying. We also evaluate the frequency of “wrong signs.” Call prices and their underlying move in the opposite direction almost 10 percent of the time.     

Pricing and hedging power options

In this paper we study the pricing and hedging of options whose payoff is a polynomial function of the underlying price at expiration; so-called ‘power options’. Working in the well-known Black and Scholes (1973) framework we derive closed-form formulas for the prices of general power calls and puts. Parabola options are studied as a special case. Power options can be hedged by statically combining ordinary options in such a way that their payoffs form a piecewise linear function which approximates the power option's payoff. Traditional delta hedging may subsequently be used to reduce any residual risk.     

A reduced PDE method for European option pricing under multi-scale,multi-factor stochastic volatility

The number of tailor-made hybrid structured products has risen more prominently to fit each investor’s preferences and requirements as they become more diversified. The structured products entail synthetic derivatives such as combinations of bonds and/or stocks conditional on how they are backed up by underlying securities, stochastic volatility, stochastic interest rates or exchanges rates. The complexity of these multi-asset structures yields lots of difficulties of pricing the products. Because of the complexity, Monte-Carlo simulation is a possible choice to price them but it may not produce stable Greeks leading to a trouble in hedging against risks. In this light, it is desirable to use partial differential equations with relevant analytic and numerical techniques. Even if the partial differential equation method would generate stable security prices and Greeks for single asset options, however, it may result in the curse of dimensionality when pricing multi-asset derivatives. In this study, we make the best use of multi-scale nature of stochastic volatility to lift the curse of dimensionality for up to three asset cases. Also, we present a transformation formula by which the pricing group parameters required for the multi-asset options in illiquid market can be calculated from the underlying market parameters.     

Pricing Derivatives on Financial Securities Subject to Credit Risk

This article provides a new methodology for pricing and hedging derivative securities involving credit risk. Two types of credit risks are considered. The first is where the asset underlying the derivative security may default. The second is where the writer of the derivative security may default. We apply the foreign currency analogy of Jarrow and Turnbull (1991) to decompose the dollar payoff from a risky security into a certain payoff and a “spot exchange rate.” Arbitrage-free valuation techniques are then employed. This methodology can be applied to corporate debt and over the counter derivatives, such as swaps and caps.     

A Nonparametric Approach to Pricing and Hedging Derivative Securities Via Learning Networks

We propose a nonparametric method for estimating the pricing formula of a derivative asset using learning networks. Although not a substitute for the more traditional arbitrage-based pricing formulas, network-pricing formulas may be more accurate and computationally more efficient alternatives when the underlying asset's price dynamics are unknown, or when the pricing equation associated with the no-arbitrage condition cannot be solved analytically. To assess the potential value of network pricing formulas, we simulate Black-Scholes option prices and show that learning networks can recover the Black-Scholes formula from a two-year training set of daily options prices, and that the resulting network formula can be used successfully to both price and delta-hedge options out-of-sample. For comparison, we estimate models using four popular methods: ordinary least squares, radial basis function networks, multilayer perceptron networks, and projection pursuit. To illustrate the practical relevance of our network pricing approach, we apply it to the pricing and delta-hedging of S&P 500 futures options from 1987 to 1991.     

Spanning and completeness with options

The role of ordinary options in facilitating the completionof securities markets is examined in the context of a modelof contigent claims sufficiently general to accommodate thecontinuous distributions of asset pricing theory and optionpricing theory. In this context, it is shown that call optionswritten on a single security approximately span all contingentclaims written on this security and that call options writtenon portfolios of call options on individual primitive securitiesapproximately span all contingent claims that can be writtenon these primitive securities. In the case of simple options,explicit formulas are given for the approximating options andportfolios of options. These results are applied to the pricingof contingent claims by arbitrage and to irrelevance propositionsin corporate finance.     

Risk measures for derivatives with Markov-modulated pure jump processes

We consider a regime-switching HJB approach to evaluate risk measures for derivative securities when the price process of the underlying risky asset is governed by the exponential of a pure jump process with drift and a Markov switching compensator. The pure jump process is flexible enough to incorporate both the infinite, (small), jump activity and the finite, (large), jump activity. The drift and the compensator of the pure jump process switch over time according to the state of a continuous-time hidden Markov chain representing the state of an economy. The market described by our model is incomplete. Hence, there is more than one pricing kernel and there is no perfect hedging strategy for a derivative security. We derive the regime-switching HJB equations for coherent risk measures for the unhedged position of derivative securities, including standard European options and barrier options. For measuring risk inherent in the unhedged option position, we first need to mark the position into the market by valuing the option. We employ a well-known tool in actuarial science, namely, the Esscher transform to select a pricing kernel for valuation of an option and to generate a family of real-world probabilities for risk measurement. We also derive the regime-switching HJB-variational inequalities for coherent risk measures for American-style options.     

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.      

Rainbow trend options: valuation and applications

Asset selection and timing decisions are major investment concerns. To resolve these issues simultaneously, a new class of rainbow trend options is proposed. The diversification effect of rainbow options can reduce the importance of asset selection decisions and trend options can mitigate unfavorable effects on market entry and exit decisions. We consider a general framework to facilitate the derivation of analytic pricing formulas for simple, pure, and Asian rainbow trend options using the martingale pricing method. The properties of these options and their Greeks are analyzed. We also investigate the performance of the dynamic delta hedging strategy for issuers of rainbow trend options. Last, this paper explores the applications of rainbow trend options for hedging price risks, designing executive stock options, modifying countercyclical capital buffer proposed by Basel Committee, and acting as control variates of the Monte Carlo simulation.     

Jackknifing Bond Option Prices

Prices of interest rate derivative securities depend cruciallyon the mean reversion parameters of the underlying diffusions.These parameters are subject to estimation bias when standardmethods are used. The estimation bias can be substantial evenin very large samples and much more serious than the discretizationbias, and it translates into a bias in pricing bond optionsand other derivative securities that is important in practicalwork. This article proposes a very general and computationallyinexpensive method of bias reduction that is based on Quenouille's(1956; Biometrika, 43, 353–360) jackknife. We show howthe method can be applied directly to the options price itselfas well as the coefficients in the models. We investigate itsperformance in a Monte Carlo study. Empirical applications toU.S. dollar swap rates highlight the differences between bondand option prices implied by the jackknife procedure and thoseimplied by the standard approach. These differences are largeand suggest that bias reduction in pricing options is importantin practical applications.     

Financial Modeling in a Fast Mean-Reverting Stochastic Volatility Environment

We present a derivative pricing and estimation methodology for a class of stochastic volatility models that exploits the observed 'bursty' or persistent nature of stock price volatility. Empirical analysis of high-frequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tick-by- tick fluctuations of the index value, but it is fast mean- reverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter provides a simple procedure to 'fit the skew' from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for European-style securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the log- moneyness-to-maturity-ratio. The results considerably simplify the estimation procedure. The remaining parameters, including the growth rate of the underlying, the correlation between asset price and volatility shocks, the rate of mean-reversion of the volatility and the market price of volatility risk are not needed for the asymptotic pricing formulas for European derivatives, and we derive the formula for a knock-out barrier option as an example. The extension to American and path-dependent contingent claims is the subject of future work. This revised version was published online in August 2006 with corrections to the Cover Date.     

Valuing Fixed-Income Options and Mortgage-Backed Securities with Alternative Term Structure Models

To value mortgage-backed securities and options on fixed-income securities, it is necessary to make assumptions regarding the term structure of interest rates. We assume that the multi-factor fixed parameter term structure model accurately represents the actual term structure of interest rates, and that the values of mortgage-backed securities and discount bond options derived from such a term structure model are correct. Differences in the prices of interest rate derivative securities based on single-factor term structure models are therefore due to pricing bias resulting from the term structure model. The price biases that result from the use of single-factor models are compared and attributed to differences in the underlying models and implications for the selection of alternative term structure models are considered.     

Sensitivity estimates for portfolio credit derivatives using Monte Carlo

Portfolio credit derivatives are contracts that are tied to an underlying portfolio of defaultable reference assets and have payoffs that depend on the default times of these assets. The hedging of credit derivatives involves the calculation of the sensitivity of the contract value with respect to changes in the credit spreads of the underlying assets, or, more generally, with respect to parameters of the default-time distributions. We derive and analyze Monte Carlo estimators of these sensitivities. The payoff of a credit derivative is often discontinuous in the underlying default times, and this complicates the accurate estimation of sensitivities. Discontinuities introduced by changes in one default time can be smoothed by taking conditional expectations given all other default times. We use this to derive estimators and to give conditions under which they are unbiased. We also give conditions under which an alternative likelihood ratio method estimator is unbiased. We illustrate the application and verification of these conditions and estimators in the particular case of the multifactor Gaussian copula model, but the methods are more generally applicable.