Density approximations for multivariate affine jump-diffusion processes

We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert spaces for random variables which possess all polynomial moments. We establish parametric conditions which guarantee existence and differentiability of transition densities of affine models and show how they naturally fit into the approximation framework. Empirical applications in option pricing, credit risk, and likelihood inference highlight the usefulness of our expansions. The approximations are extremely fast to evaluate, and they perform very accurately and numerically stable.     

Galerkin infinite element approximation for pricing barrier options and options with discontinuous payoff

Abstract We analyze the Galerkin infinite element method for pricing European barrier options and, more generally, options with discontinuous payoff. The infinite element method is a simple and efficient modification of the more common finite element method. It keeps the best features of finite elements, i.e., bandedness, ease of programming, accuracy. Three main aspects are considered: (i) the degeneracy of the pricing PDE models at hand; (ii) the presence of discontinuities at the barriers or in the payoff clause and their effects on the numerical approximation process; (iii) the need for resorting to suitable numerical methods for unbounded domains when appropriate asymptotic conditions are not specified. The numerical stability and convergence of the proposed method are proved. Mathematics Subject Classification (2000): 65N30, 65J10 Journal of Economic Literature Classification: G13, C63     

基于风险中性路径概率的三叉树期权定价模型

树方法是给经典期权进行定价的非常实用的数值方法,目前最流行的是二叉树模型。三叉树定价模型作为二叉树的一个扩展,其同样是在风险中性概率的基础上给经典期权进行定价,并且可以通过MATLAB实现。相比于二叉树而言,三叉树模型的定价结果具有更好的收敛性。除此之外,用三叉树模型对影响期权价格的一些因素进行敏感性分析,可以验证该模型的合理性。     

A moments and strike matching binomial algorithm for pricing American Put options

This paper is dedicated to a new binomial lattice method called Moments and Strike Matching (MSM) consistent with the Black–Scholes model in the limit of an infinite step number and such that the Strike K is equal to one of the final nodes of the tree. The method is very easy to implement, since the parameters are explicitly given. Asymptotic expansions are obtained for the MSM European Put price and delta, which motivates the use of Richardson extrapolation. A numerical comparison with the best lattice based numerical methods known in literature, shows the efficiency of the proposed algorithm for pricing and hedging American Put options.      

Pricing VIX options with stochastic volatility and random jumps

This study presents an analytical exact solution for the price of VIX options under stochastic volatility model with simultaneous jumps in the asset price and volatility processes. We shall demonstrate that our new pricing formula can be used to efficiently compute the numerical values of a VIX option. While we also show that the numerical results obtained from our formula consistently match those obtained from Monte Carlo simulation perfectly as a verification of the correctness of our formula, numerical evidence is offered to illustrate that the correctness of the formula proposed in Lin and Chang (J Futur Markets 29(6), 523–543, 2009) is in serious doubt. Moreover, some important and distinct properties of VIX options (e.g., put-call parity, hedging ratios) are also examined and discussed.     

Quadratic hedging schemes for non-Gaussian GARCH models

We propose different schemes for option hedging when asset returns are modeled using a general class of GARCH models. More specifically, we implement local risk minimization and a minimum variance hedge approximation based on an extended Girsanov principle that generalizes Duan׳s (1995) delta hedge. Since the minimal martingale measure fails to produce a probability measure in this setting, we construct local risk minimization hedging strategies with respect to a pricing kernel. These approaches are investigated in the context of non-Gaussian driven models. Furthermore, we analyze these methods for non-Gaussian GARCH diffusion limit processes and link them to the corresponding discrete time counterparts. A detailed numerical analysis based on S&P 500 European call options is provided to assess the empirical performance of the proposed schemes. We also test the sensitivity of the hedging strategies with respect to the risk neutral measure used by recomputing some of our results with an exponential affine pricing kernel.     

Asymmetry in the jump-size distribution of the S&P 500: Evidence from equity and option markets

This paper studies alternative distributions for the size of price jumps in the S&P 500 index. We introduce a range of new jump-diffusion models and extend popular double-jump specifications that have become ubiquitous in the finance literature. The dynamic properties of these models are tested on both a long time series of S&P 500 returns and a large sample of European vanilla option prices. We discuss the in- and out-of-sample option pricing performance and provide detailed evidence of jump risk premia. Models with double-gamma jump size distributions are found to outperform benchmark models with normally distributed jump sizes.     

Homogeneous semi-Markov reliability models for credit risk management*

Abstract The credit risk problem is one of the most important issues of modern financial mathematics. Fundamentally it consists in computing the default probability of a company going into debt. The problem can be studied by means of Markov transition models. The generalization of the transition models by means of homogeneous semi-Markov models is presented in this paper. The idea is to consider the credit risk problem as a reliability problem. In a semi-Markov environment it is possible to consider transition probabilities that change as a function of waiting time inside a state. The paper also shows how to apply semi-Markov reliability models in a credit risk environment. In the last section an example of the model is provided. Mathematics Subject Classification (2000): 60K15, 60K20, 90B25, 91B28 Journal of Economic Literature Classification: G21, G33     

Approximate arbitrage-free option pricing under the SABR model

The stochastic-alpha-beta-rho (SABR) model introduced by Hagan et al. (2002) provides a popular vehicle to model the implied volatilities in the interest rate and foreign exchange markets. To exclude arbitrage opportunities, we need to specify an absorbing boundary at zero for this model, which the existing analytical approaches to pricing derivatives under the SABR model typically ignore. This paper develops closed-form approximations to the prices of vanilla options to incorporate the effect of such a boundary condition. Different from the traditional normal distribution-based approximations, our method stems from an expansion around a one-dimensional Bessel process. Extensive numerical experiments demonstrate its accuracy and efficiency. Furthermore, the explicit expression yielded from our method is appealing from the practical perspective because it can lead to fast calibration, pricing, and hedging.     

Generalized affine transform on pricing quanto range accrual note

This paper was to price and hedge a quanto floating range accrual note (QFRAN) by an affine term structure model with affine-jump processes. We first generalized the affine transform proposed by Duffie et al. (2000) under both the domestic and foreign risk-neutral measures with a change of measure, which provides a flexible structure to value quanto derivatives. Then, we provided semi-analytic pricing and hedging solutions for QFRAN under a four-factor affine-jump model with the stochastic mean, stochastic volatility, and jumps. The numerical results demonstrated that both the common and local factors significantly affect the value and hedging strategy of QFRAN. Notably,  the factor of stochastic mean plays the most important role in either valuation or hedging. This study suggested that ignorance of these factors in a term-structure model will result in significant pricing and hedging errors in QFRAN. In summary, this study provided flexible and easily implementable solutions in valuing quanto derivatives.     

Equal risk pricing under convex trading constraints

In an incomplete market model where convex trading constraints are imposed upon the underlying assets, it is no longer possible to obtain unique arbitrage-free prices for derivatives using standard replication arguments. Most existing derivative pricing approaches involve the selection of a suitable martingale measure or the optimisation of utility functions as well as risk measures from the perspective of a single trader.We propose a new and effective derivative pricing method, referred to as the equal risk pricing approach, for markets with convex trading constraints. The approach analyses the risk exposure of both the buyer and seller of the derivative, and seeks an equal risk price which evenly distributes the expected loss for both parties under optimal hedging. The existence and uniqueness of the equal risk price are established for both European and American options. Furthermore, if the trading constraints are removed, the equal risk price agrees with the standard arbitrage-free price.Finally, the equal risk pricing approach is applied to a constrained Black–Scholes market model where short-selling is banned. In particular, simple pricing formulas are derived for European calls, European puts and American puts.     

Pure jump models for pricing and hedging VIX derivatives

Recent non-parametric statistical analysis of high-frequency VIX data (Todorov and Tauchen, 2011) reveals that VIX dynamics is a pure jump semimartingale with infinite jump activity and infinite variation. To our best knowledge, existing models in the literature for pricing and hedging VIX derivatives do not have these features. This paper fills this gap by developing a novel class of parsimonious pure jump models with such features for VIX based on the additive time change technique proposed in Li et al., 2016a, Li et al., 2016b. We time change the 3/2 diffusion by a class of additive subordinators with infinite activity, yielding pure jump Markov semimartingales with infinite activity and infinite variation. These processes have time and state dependent jumps that are mean reverting and are able to capture stylized features of VIX. Our models take the initial term structure of VIX futures as input and are analytically tractable for pricing VIX futures and European options via eigenfunction expansions. Through calibration exercises, we show that our model is able to achieve excellent fit for the VIX implied volatility surface which typically exhibits very steep skews. Comparison to two other models in terms of calibration reveals that our model performs better both in-sample and out-of-sample. We explain the ability of our model to fit the volatility surface by evaluating the matching of moments implied from market VIX option prices. To hedge VIX options, we develop a dynamic strategy which minimizes instantaneous jump risk at each rebalancing time while controlling transaction cost. Its effectiveness is demonstrated through a simulation study on hedging Bermudan style VIX options.     

APPROACHES TO PRICE FORMATION IN FINANCIALIZED COMMODITY MARKETS

A recent debate about the financialization of commodity markets has stimulated the development of new approaches to price formation which incorporate index traders as a new trader category. I survey these new approaches by retracing their emergence to traditional price formation models and show that they arise from a synthesis between commodity arbitrage pricing and behavioural pricing theories in the tradition of Keynesian inspired hedging pressure models. Based on these insights, I derive testable hypotheses and provide guidance for a growing literature that seeks to empirically evaluate the effects of index traders on price discovery in commodity futures markets.     

Option Pricing under the Double Exponential Jump-Diffusion Model with Stochastic Volatility and Interest Rate

This paper proposes an efficient option pricing model that incorporates stochastic interest rate (SIR), stochastic volatility (SV), and double exponential jump into the jump-diffusion settings. The model comprehensively considers the leptokurtosis and heteroscedasticity of the underlying asset’s returns, rare events, and an SIR. Using the model, we deduce the pricing characteristic function and pricing formula of a European option. Then, we develop the Markov chain Monte Carlo method with latent variable to solve the problem of parameter estimation under the double exponential jump-diffusion model with SIR and SV. For verification purposes, we conduct time efficiency analysis, goodness of fit analysis, and jump/drift term analysis of the proposed model. In addition, we compare the pricing accuracy of the proposed model with those of the Black–Scholes and the Kou (2002) models. The empirical results show that the proposed option pricing model has high time efficiency, and the goodness of fit and pricing accuracy are significantly higher than those of the other two models.     

A two-step simulation procedure to analyze the exercise features of American options

Abstract In this contribution we propose a two-step simulation procedure that enables to compute the exercise features of American options and analyze the properties of the optimal exercise times and exercise probabilities. The first step of the procedure is based on the calculation of an accurate approximation of the optimal exercise boundary. In particular, we use a smoothed binomial method which effectively reduces the fluctuating behavior of a discrete boundary. In the second step the boundary is used to define a stopping rule which is embodied in a Monte Carlo simulation method. A broad experimental analysis is carried out in order to test the procedure and study the behavior of the exercise features. Mathematics Subject Classification (2000): 60G40, 60J60, 65C20 Journal of Economic Literature Classification: G13     

The valuation of American barrier options using the decomposition technique

In this paper, we propose an alternative approach for pricing and hedging American barrier options. Specifically, we obtain an analytic representation for the value and hedge parameters of barrier options, using the decomposition technique of separating the European option value from the early exercise premium. This allows us to identify some new put-call ‘symmetry’ relations and the homogeneity in price parameters of the optimal exercise boundary. These properties can be utilized to increase the computational efficiency of our method in pricing and hedging American options. Our implementation of the obtained solution indicates that the proposed approach is both efficient and accurate in computing option values and option hedge parameters. Our numerical results also demonstrate that the approach dominates the existing lattice methods in both accuracy and efficiency. In particular, the method is free of the difficulty that existing numerical methods have in dealing with spot prices in the proximity of the barrier, the case where the barrier options are most problematic.     

Arbitrage pricing theory and risk-neutral measures

Abstract We consider a market with countably many risky assets and finite factor structure, as in the “arbitrage pricing theory” of Ross (1976). We prove necessary and sufficient conditions in terms of parameters for the existence of an equivalent risk-neutral measure, i.e., a measure under which each asset return has zero expected value. We relate these conditions to a certain absence of arbitrage property of the model. Mathematics Subject Classification (2000): 91B24, 91B28 Journal of Economic Literature Classification: G10, G12     

Option pricing with discrete time jump processes

In this paper we propose new option pricing models based on class of models with jumps contained in the Lévy-type based models (NIG-Lévy, Schoutens, 2003, Merton-jump, Merton, 1976 and Duan based model, Duan et al., 2007). By combining these different classes of models with several volatility dynamics of the GARCH type, we aim at taking into account the dynamics of financial returns in a realistic way. The associated risk neutral dynamics of the time series models is obtained through two different specifications for the pricing kernel: we provide a characterization of the change in the probability measure using the Esscher transform and the Minimal Entropy Martingale Measure. We finally assess empirically the performance of this modelling approach, using a dataset of European options based on the S&P 500 and on the CAC 40 indices. Our results show that models involving jumps and a time varying volatility provide realistic pricing and hedging results for options with different kinds of time to maturities and moneyness. These results are supportive of the idea that a realistic time series model can provide realistic option prices making the approach developed here interesting to price options when option markets are illiquid or when such markets simply do not exist.     

Hedging gas bills with weather derivatives

Natural gas company managers concerned with customer satisfaction attempt to minimize the occurrence of extreme bills. Previously, only price fluctuations were addressed with derivative instruments; exchange-traded weather derivatives present a means of hedging exposure to increases in quantity of gas demanded during colder than expected winter months. We model a natural gas company’s ability to adjust for consumer sensitivity and exposure to extreme bills with the use of an optimal mix of weather derivatives and gas pricing derivatives. We find consumer exposure to extreme bills is minimized when the utility uses pricing and weather derivatives.(JEL G11, L51)