Derivative Pricing Models Research Articles

Local Volatility Pricing Models for Long-Dated FX Derivatives

We study the local volatility function in the Foreign Exchange market where both domestic and foreign interest rates are stochastic. This model is suitable to price long-dated FX derivatives. We derive the local volatility function and obtain several results that can be used for the calibration of this local volatility on the FX option's market. Then, we study two different extensions which allow the volatility of the spot FX rate to have stochastic behavior. First, we introduce a stochastic structure on the local volatility surface and show that local volatilities are risk-adjusted expectations of future instantaneous volatilities. The second extension is obtained by multiplying the FX spot local volatility with a stochastic volatility. Thanks to the Gyongy's mimicking property, we obtain a calibration method for the local volatility associated to this model.

Weather Index Derivatives in Risk Transfer for Agricultural Natural Hazards

Abstract The article begins by briefly reviewing agricultural risk management strategies used by farm households and some of the unique problems associated with agricultural insurance. Especially, the experience of developed countries is highlighted to make the case for why such approaches cannot work well in lower-income countries. The article introduces innovations that use index-based insurance products, and gives a pricing model for weather index derivatives which is deduced with utility-based pricing, and then, the article reviews both the advantages and disadvantages of index-based agricultural insurance and its applicable scope.

The pricing and hedging of structured notes with systematic jump risk: An analysis of the USD knock-out reversed swap

Most of the interest rate derivative pricing models are jump-diffusion models, where the jump risk is assumed diversifiable. In this paper, we propose a Heath–Jarrow–Morton model with systematic jump risk to derive the no-arbitrage condition using Esscher transformation. Based on the Heath–Jarrow–Morton model with systematic jump risk, the dynamic process of the LIBOR market model with systematic jump risk is then developed. By decomposing the USD knock-out reversed swap into three derivative components, i.e., interest rate swap, interest rate digital call (IRDC) and cap, the pricing of the swap can be obtained from the dynamic process of the LIBOR market model with systematic jump risk. We show how the swap issuers/investors can hedge the swap risk using these three derivative components. The numerical analyses are conducted to show the impact of jump risk on the values of IRDC, cap and swap.

Distribution of extreme changes in Asian currencies: tail index estimates and value-at-risk calculations

This study examines the distribution of extreme values in daily currency changes for nine Asian countries. Using an improved estimator, extreme changes in Asian currencies can generally be represented by Frechet distributions. Our results are robust to the choice of the numeraire currency, the Asian crises and the 1985 Plaza Agreement. These results are important as asset and derivative pricing models, and Value-at-Risk (VaR) calculations depend on accurate assessments of the distribution of extreme values. Indeed, VaRs based on our fitted extreme distributions are similar to VaRs based on historical distributions but are multiples of those based on normal distributions.

Testing Hedges under the Standard Tranched Credit Model

We examine the performance of the standard tranched credit derivative (or synthetic CDO) pricing model over most of the lifetime of these derivatives. As the market for these derivatives is quite liquid, we focus on the application of the model for hedging, rather than for absolute pricing. We investigate first whether the standard model provides demonstrably better hedges than a simple linear regression, and second whether any of the typical variations on the standard model outperforms the others. Our findings demonstrate that the standard model does produce better hedges than a simple benchmark, but that within the numerous variations of the standard model, the simplest one is clearly the most consistent and reliable.

An equilibrium pricing model for weather derivatives in a multi-commodity setting

Many industries are exposed to weather risk. Weather derivatives can play a key role in hedging and diversifying such risk because the uncertainty in a company's profit function can be correlated to weather condition which affects diverse industry sectors differently. Unfortunately the weather derivatives market is a classical example of an incomplete market that is not amenable to standard methodologies used for derivative pricing in complete markets. In this paper, we develop an equilibrium pricing model for weather derivatives in a multi-commodity setting. The model is constructed in the context of a stylized economy where agents optimize their hedging portfolios which include weather derivatives that are issued in a fixed quantity by a financial underwriter. The supply and demand resulting from hedging activities and the supply by the underwriter are combined in an equilibrium pricing model under the assumption that all agents maximize some risk averse utility function. We analyze the gains due to the inclusion of weather derivatives in hedging portfolios and examine the components of that gain attributable to hedging and to risk sharing.

A Consistent Pricing Model for Index Options and Volatility Derivatives

We propose and study a flexible modeling framework for the joint dynamics of an index and a set of forward variance swap rates written on this index, allowing options on forward variance swaps and options on the underlying index to be priced consistently. Our model reproduces various empirically observed properties of variance swap dynamics and allows for jumps in volatility and returns. An affine specification using L´evy processes as building blocks leads to analytically tractable pricing formulas for options on variance swaps as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index. The calibration of the model is done in two steps, first by matching VIX option prices and then by matching prices of options on the underlying

Enhancing hedging performance with the spanning polynomial projection

Statistical time-series approaches to hedging are difficult to beat, especially out-of-sample, and are capable of out-performing many theory-based derivative pricing model approaches to hedging commodity price risks using futures contracts. However, the vast majority of time-series approaches to hedging discussed in the literature are essentially linear statistical projections, whether univariate or multivariate. Little is known about the potential hedging capabilities of nonlinear methods. This study describes how least-squares orthogonal polynomial approximation methods based on the spanning polynomial projection (SPP) can be used to enhance standard (linear) optimal hedging methods and improve hedging performance for a hedger with a mean–variance objective. Empirical analyses show that the SPP can be used effectively for hedging and gives better out-of-sample hedging performance than the benchmark VEC-GARCH hedging model. Results are robust to the inclusion of transaction costs and risk-aversion assumptions.

Closed-Form Approximations for Spread Option Prices and Greeks

Beginning with Black and Scholes, it has been mathematically very convenient to model asset returns as logarithmic diffusions when developing pricing models for derivatives. But doing so causes a lot of trouble for instruments such as spread options, whose payoffs are a function of the arithmetic difference between two such assets, because the distribution of the price difference is not lognormal or any other familiar distribution. One could simply assume the difference is lognormal and obtain a simple model, but that would lead to an internal inconsistency with the models used for the component securities. Alternatively, one must develop numerical or approximation techniques to deal with this problem. In this article, the authors advance the technology by devising analytic formulas for spread option values, as well as their greeks, based on an approximation to the exercise boundary for one asset9s price, conditional on the other price. The result is much more efficient and accurate valuation than with previous approaches

Recovering Portfolio Default Intensities Implied by CDO Quotes

We propose a stable non-parametric algorithm for the calibration of pricing models for portfolio credit derivatives: given a set of observations of market spreads for CDO tranches, we construct a risk-neutral default intensity process for the portfolio underlying the CDO which matches these observations, by looking for the risk neutral loss process 'closest' to a prior loss process, verifying the calibration constraints. We formalize the problem in terms of minimization of relative entropy with respect to the prior under calibration constraints and use convex duality methods to solve the problem: the dual problem is shown to be an intensity control problem, characterized in terms of a Hamilton-Jacobi system of differential equations, for which we present an analytical solution. Given a set of observed CDO tranche spreads, our method allows to construct an implied intensity process consistent with the observed spreads. We illustrate our method on ITRAXX index data: our results reveal strong evidence for the dependence of loss transitions rates on the past number of defaults, thus offering quantitative evidence for contagion effects in the risk-neutral loss process.

A calibration algorithm for simulation-based pricing models

Derivative pricing models require calibration to market conditions in order to determine quantities such ashedging positions and the prices of other instruments. For stochastic models and/or complex derivativeswhose prices are not of an analytic form, prices must be computed via simulation and the calibration ismore difficult. A method to facilitate the calibration of simulation-based pricing models is proposed. Thealgorithm uses a statistically designed experiment to select the points at which simulations are performed.The method is quite general as it is independent of the stochastic model for the underlying and allows fordifferent objective functions that can incorporate information such as open interest and volume. Further-more, market prices from European- and/or American-style derivatives covering a range of strike pricesand maturities can be handled by this technique. Examples show the procedure is successful at calibratingwell-known asset pricing models to both simulated and market data.Keywords: derivatives pricing; calibration; simulation; experimental design; regression.

Diffusion Estimation and Structural Stability Tests of Short Term Interest Rate Models

This paper proposes a simple method for estimating the diffusion function of continuous-time diffusion processes without specifying the drift function structure. This estimation method is suitable for practitioners choosing an interest rate derivative pricing model among the short-term rate type. We estimate the diffusion function parameters by using one-month Euro rates; US Dollar, German Mark, and Japanese Yen. We conduct structural stability tests of the diffusion function parameters and find a statistically significant structural change in the German Mark process. We, however, do not detect structural changes in the US Dollar and the Japanese Yen processes during the same period.

Derivative Pricing Models with Regime Switching

Several different approaches to modeling returns processes with time–varying volatility have been developed in recent years. The affine jump–diffusion (AJD) class of stochastic processes, in which volatility can be a function of an arbitrary number of underlying stochastic state variables that follow correlated mean–reverting square root jump–diffusions, is perhaps the most tractable, since transform methods allow prices of derivatives tied to those processes to be expressed in (more or less) closed form. Without that, numerical solution techniques can entail a very heavy computational burden. But despite their flexibility, AJD processes still significantly restrict the behavior of the state variables. In this article, Edwards introduces a new regime–switching approach that can be adapted to a broad array of alternative processes. The key is to model the volatile regime using a stochastic time change.

Derivative pricing model and time-series approaches to hedging: A comparison

This research compares derivative pricing model and statistical time-series approaches to hedging. The finance literature stresses the former approach, while the applied economics literature has focused on the latter. We compare the out-of-sample hedging effectiveness of the two approaches when hedging commodity price risk using futures contracts. For various methods of parameter estimation and inference, we find that the derivative pricing models cannot out-perform a vector error-correction model with a GARCH error structure. The derivative pricing models’ unpalatable assumption of deterministically evolving futures volatility seems to impede

Daily volatility forecasts: reassessing the performance of GARCH models

AbstractVolatility plays a key role in asset and portfolio management and derivatives pricing. As such, accurate measures and good forecasts of volatility are crucial for the implementation and evaluation of asset and derivative pricing models in addition to trading and hedging strategies. However, whilst GARCH models are able to capture the observed clustering effect in asset price volatility in‐sample, they appear to provide relatively poor out‐of‐sample forecasts. Recent research has suggested that this relative failure of GARCH models arises not from a failure of the model but a failure to specify correctly the ‘true volatility’ measure against which forecasting performance is measured. It is argued that the standard approach of using ex post daily squared returns as the measure of ‘true volatility’ includes a large noisy component. An alternative measure for ‘true volatility’ has therefore been suggested, based upon the cumulative squared returns from intra‐day data. This paper implements that technique and reports that, in a dataset of 17 daily exchange rate series, the GARCH model outperforms smoothing and moving average techniques which have been previously identified as providing superior volatility forecasts. Copyright © 2004 John Wiley & Sons, Ltd.

Hamiltonian and potentials in derivative pricing models: exact results and lattice simulations

The pricing of options, warrants and other derivative securities is one of the great success of financial economics. These financial products can be modeled and simulated using quantum mechanical instruments based on a Hamiltonian formulation. We show here some applications of these methods for various potentials, which we have simulated via lattice Langevin and Monte Carlo algorithms, to the pricing of options. We focus on barrier or path dependent options, showing in some detail the computational strategies involved.

Drift Estimation of Generalized Security Price Processes from High Frequency Derivative Prices

This paper presents a framework for using high frequency derivative prices to estimate the drift of generalized security price processes. This work may be seen more generally as a quasi-likelihood approach to estimating continuous-time parameters of derivative pricing models using discrete option data. We develop a generalized derivative-based estimator for the drift where the underlying security price process follows any arbitrary state-time separable diffusion process (including arithmetic and geometric Brownian motion as special cases). The framework provides a method to measure premia in derivative prices, test for risk-neutral pricing and leads to a new empirical approach to pricing derivative contingent claims. A sufficient condition for the asymptotic consistency of the generalized estimator is also obtained. A study based on generating the S&P500 index and calls shows that the estimator can correctly estimate the drift parameter.

On modelling and pricing weather derivatives

The main objective of the work described is to find a pricing model for weather derivatives with payouts depending on temperature. Historical data are used to suggest a stochastic process that describes the evolution of the temperature. Since temperature is a non-tradable quantity, unique prices of contracts in an incomplete market are obtained using the market price of risk. Numerical examples of prices of some contracts are presented, using an approximation formula as well as Monte Carlo simulations.

Comparison Study of Different Implementations of Derivative Pricing Models

A common problem that instructors encounter in teaching financial modeling classes is how to implement these models on a PC. In this project, we survey three different ways to implement derivative pricing models and provide a detailed how-to, extending from writing computer codes to applying the models. The three programming tools chosen are Visual Basic for Applications (VBA), Matlab, and C. The user interface is as important as the speed of execution in justifying our choices. Because of the popularity and user-friendliness of MS Excel, we use Excel as our unified user interface and look for tools that provide an easy connection. In this regard, VBA is an obvious choice because it is the embedded macro language for Excel. C is known for its performance but is relatively difficult to code. Excel provides an interface to work with C (C API), so the user can combine the power of C and the user-friendliness of Excel. Matlab is used by engineers around the world for high power matrix computations. Its built-in library provides many functions that are essential for derivative modeling. In addition, it has a toolbox (Excel Link) that provides a linkage to Excel. This Web site provides both the online and printable hard copies of the user guide for the individual methods. All of the source codes and sample spreadsheets are categorized by programming languages and pricing models for easy downloading. New models are added to the Web site on an ongoing basis. We hope this open-source approach to implementation of pricing models will provide a cyber community for others to share their computer codes for teaching and research purposes.

Comparison Study of Different Implementations of Derivative Pricing Models

Comparison Study of Different Implementations of Derivative Pricing Models