Hybrid Lévy Models: Design and Computational Aspects

ABSTRACT

A hybrid model is a model, where two markets are studied jointly such that stochastic dependence can be taken into account. Such a dependence is well known for equity and interest rate markets on which we focus here. Other pairs can be considered in a similar way. Two different versions of a hybrid approach are developed. Independent time-inhomogeneous Lévy processes are used as the drivers of the dynamics of interest rates and equity. In both versions, the dynamics of the interest rate side is described by an equation for the instantaneous forward rate. Dependence between the markets is generated by introducing the driver of the interest rate market as an additional term into the dynamics of equity in the first version. The second version starts with the equity dynamics and uses a corresponding construction for the interest rate side. Dependence can be quantified in both cases by a single parameter. Numerically efficient valuation formulas for interest rate and equity derivatives are developed. Using market quotes for liquidly traded assets we show that the hybrid approach can be successfully calibrated.     

Valuation of Two-Factor Interest Rate Contingent Claims Using Green's Theorem

Abstract

Over the years a number of two-factor interest rate models have been proposed that have formed the basis for the valuation of interest rate contingent claims. This valuation equation often takes the form of a partial differential equation that is solved using the finite difference approach. In the case of two-factor models this has resulted in solving two second-order partial derivatives leading to boundary errors, as well as numerous first-order derivatives. In this article we demonstrate that using Green's theorem, second-order derivatives can be reduced to first-order derivatives that can be easily discretized; consequently, two-factor partial differential equations are easier to discretize than one-factor partial differential equations. We illustrate our approach by applying it to value contingent claims based on the two-factor CIR model. We provide numerical examples that illustrate that our approach shows excellent agreement with analytical prices and the popular Crank–Nicolson method.     

The Valuation of American Options with Stochastic Stopping Time Constraints

Abstract

This paper concerns the pricing of American options with stochastic stopping time constraints expressed in terms of the states of a Markov process. Following the ideas of Menaldi et al., we transform the constrained into an unconstrained optimal stopping problem. The transformation replaces the original payoff by the value of a generalized barrier option. We also provide a Monte Carlo method to numerically calculate the option value for multidimensional Markov processes. We adapt the Longstaff–Schwartz algorithm to solve the stochastic Cauchy–Dirichlet problem related to the valuation problem of the barrier option along a set of simulated trajectories of the underlying Markov process.     

Volatility Targeting Using Delayed Diffusions

ABSTRACT

A target volatility strategy (TVS) is a risky asset-riskless bond dynamic portfolio allocation which makes use of the risky asset historical volatility as an allocation rule with the aim of maintaining the instantaneous volatility of the investment constant at a target level. In a market with stochastic volatility, we consider a diffusion model for the value of a target volatility fund (TVF) which employs a system of stochastic delayed differential equations (SDDEs) involving the asset realized variance. First we prove that under some technical assumptions, contingent claim valuation on a TVF is approximately of Black-Scholes type, which is consistent with and supports the standing market practice. In second place, we develop a computational framework using recent results on Markovian approximations of SDDEs systems, which we then implement in the Heston variance model using an ad hoc Euler scheme. Our framework allows for efficient numerical valuation of derivatives on TVFs, whose typical purpose is the assessment of the guarantee costs of such funds for insurers.     

Vulnerable Derivatives and Good Deal Bounds: A Structural Model

Abstract

We price vulnerable derivatives – i.e. derivatives where the counterparty may default. These are basically the derivatives traded on the over-the-counter (OTC) markets. Default is modelled in a structural framework. The technique employed for pricing is good deal bounds (GDBs). The method imposes a new restriction in the arbitrage free model by setting upper bounds on the Sharpe ratios (SRs) of the assets. The potential prices that are eliminated represent unreasonably good deals. The constraint on the SR translates into a constraint on the stochastic discount factor. Thus, tight pricing bounds can be obtained. We provide a link between the objective probability measure and the range of potential risk-neutral measures, which has an intuitive economic meaning. We also provide tight pricing bounds for European calls and show how to extend the call formula to pricing other financial products in a consistent way. Finally, we numerically analyse the behaviour of the good deal pricing bounds.     

基于GC-MSV模型的沿海干散货运费衍生品套期保值研究

面对干散货航运运价波动,货主或者航运企业需要通过适当的方法进行风险管理,通过航运运费衍生品进行套期保值是一种主要的风险控制方法。本文采用GC-MSV、在最小方差准则下,研究了中国沿海煤炭运费衍生品的套期保值效果,估计了最优静态套期保值率和动态套期保值率,并与其他不同模型进行对比分析。从套期保值效果看,动态调整的GC-MSV模型优于其他模型,通过套期保值能降低20%~40%的波动率。尽管对资产方差降低的作用有限,沿海煤炭运费衍生品依然能够起到一定的对冲风险作用。     

Closed Formula for Options with Discrete Dividends and Its Derivatives

Abstract

We present a closed pricing formula for European options under the Black–Scholes model as well as formulas for its partial derivatives. The formulas are developed making use of Taylor series expansions and a proposition that relates expectations of partial derivatives with partial derivatives themselves. The closed formulas are attained assuming the dividends are paid in any state of the world. The results are readily extensible to time-dependent volatility models. For completeness, we reproduce the numerical results in Vellekoop and Nieuwenhuis, covering calls and puts, together with results on their partial derivatives. The closed formulas presented here allow a fast calculation of prices or implied volatilities when compared with other valuation procedures that rely on numerical methods.     

American Options in the Heston Model with Stochastic Interest Rate and Its Generalizations

Abstract

We consider the Heston model with the stochastic interest rate of Cox–Ingersoll–Ross (CIR) type and more general models with stochastic volatility and interest rates depending on two CIR-factors; the price, volatility and interest rate may correlate. Time-derivative and infinitesimal generator of the process for factors that determine the dynamics of the interest rate and/or volatility are discretized. The result is a sequence of embedded perpetual options arising in the time discretization of a Markov-modulated Lévy model. Options in this sequence are solved using an iteration method based on the Wiener–Hopf factorization. Typical shapes of the early exercise boundary are shown, and good agreement of option prices with prices calculated with the Longstaff–Schwartz method and Medvedev–Scaillet asymptotic method is demonstrated.     

Forward Variance Dynamics: Bergomi’s Model Revisited

Abstract

In this article, we propose an arbitrage-free modelling framework for the joint dynamics of forward variance along with the underlying index, which can be seen as a combination of the two approaches proposed by Bergomi. The difference between our modelling framework and the Bergomi (2008. Smile dynamics III. Risk, October, 90–96) models is mainly the ability to compute the prices of VIX futures and options by using semi-analytic formulas. Also, we can express the sensitivities of the prices of VIX futures and options with respect to the model parameters, which enables us to propose an efficient and easy calibration to the VIX futures and options. The calibrated model allows to Delta-hedge VIX options by trading in VIX futures, the corresponding hedge ratios can be computed analytically.     

Bonds and Options in Exponentially Affine Bond Models

Abstract

In this article we apply the Flesaker–Hughston approach to invert the yield curve and to price various options by letting the randomness in the economy be driven by a process closely related to the short rate, called the abstract short rate. This process is a pure deterministic translation of the short rate itself, and we use the deterministic shift to calibrate the models to the initial yield curve. We show that we can solve for the shift needed in closed form by transforming the problem to a new probability measure. Furthermore, when the abstract short rate follows a Cox–Ingersoll–Ross (CIR) process we compute bond option and swaption prices in closed form. We also propose a short-rate specification under the risk-neutral measure that allows the yield curve to be inverted and is consistent with the CIR dynamics for the abstract short rate, thus giving rise to closed form bond option and swaption prices.     

Real-World Scenarios With Negative Interest Rates Based on the LIBOR Market Model

ABSTRACT

In this article, we present a methodology to simulate the evolution of interest rates under real-world probability measure. More precisely, using the multidimensional Shifted Lognormal LIBOR market model and a specification of the market price of risk vector process, we explain how to perform simulations of the real-world forward rates in the future, using the Euler?Maruyama scheme with a predictor?corrector strategy. The proposed methodology allows for the presence of negative interest rates as currently observed in the markets.     

The freight allocation problem with lane cost balancing constraint

We consider a problem faced by a buying office for one of the largest retail distributors in the world. The buying office plans the distribution of goods from Asia to various destinations across Europe. The goods are transported along shipping lanes by shipping companies, many of which have collaborated to form strategic alliances; each lane must be serviced by a minimum number of companies belonging to a minimum number of alliances. The task involves purchasing freight capacity from shipping companies for each lane based on projected demand, and subject to minimum quantity requirements for each selected shipping company, such that the total transportation cost is minimized. In addition, the allocation must not assign an overly high proportion of freight to the more expensive shipping companies servicing any particular lane, which we call the lane cost balancing constraint.This study is the first to consider the lane cost balancing constraint in the context of freight allocation. We formulate the freight allocation problem with this lane cost balancing constraint as a mixed integer programming model, and show that even finding a feasible solution to this problem is computationally intractable. Hence, in order to produce high-quality solutions in practice, we devised a meta-heuristic approach based on tabu search. Experiments show that our approach significantly outperforms the branch-and-cut approach of CPLEX 11.0 when the problem increases to practical size and the lane cost balancing constraint is tight. Our approach was developed into an application that is currently employed by decision-makers at the buying office in question.     

Real options valuation of forest plantation investments in Brazil

In this paper, we consider investments in eucalyptus plantations in Brazil. For such projects, we discuss real options valuation in the place conventional methods such as IRR or NPV, possibly with CAPM. Traditionally, real options valuation assumes complete markets and neglects market imperfections. Yet, market frictions, such as transaction costs, interest rate spreads, and restricted short positions, can play an important role. We extend real options valuation to allow incomplete and imperfect markets. The value is obtained as a competitive price, given markets of competing investment opportunities, such as real and financial assets. Under perfect and complete markets, such valuation method is consistent with conventional real options theory. Stochastic programming and standard software is used for valuation of eucalyptus plantations. We estimate the underlying interdependent diffusion processes of stock market, interest rates, exchange rates and pulpwood price, and derive novel expressions of stochastic integrals to be employed in scenario generation for discrete time stochastic programming.     

Pricing and Hedging of Lookback Options in Hyper-exponential Jump Diffusion Models

ABSTRACT

In this article, we consider the problem of pricing lookback options in certain exponential Lévy market models. While in the classic Black-Scholes models the price of such options can be calculated in closed form, for more general asset price model, one typically has to rely on (rather time-intense) Monte-Carlo or partial (integro)-differential equation (P(I)DE) methods. However, for Lévy processes with double exponentially distributed jumps, the lookback option price can be expressed as one-dimensional Laplace transform (cf. Kou, S. G., Petrella, G., & Wang, H. (2005). Pricing path-dependent options with jump risk via Laplace transforms. The Kyoto Economic Review, 74(9), 1–23.). The key ingredient to derive this representation is the explicit availability of the first passage time distribution for this particular Lévy process, which is well-known also for the more general class of hyper-exponential jump diffusions (HEJDs). In fact, Jeannin and Pistorius (Jeannin, M., & Pistorius, M. (2010). A transform approach to calculate prices and Greeks of barrier options driven by a class of Lévy processes. Quntitative Finance, 10(6), 629–644.) were able to derive formulae for the Laplace transformed price of certain barrier options in market models described by HEJD processes. Here, we similarly derive the Laplace transforms of floating and fixed strike lookback option prices and propose a numerical inversion scheme, which allows, like Fourier inversion methods for European vanilla options, the calculation of lookback options with different strikes in one shot. Additionally, we give semi-analytical formulae for several Greeks of the option price and discuss a method of extending the proposed method to generalized hyper-exponential (as e.g. NIG or CGMY) models by fitting a suitable HEJD process. Finally, we illustrate the theoretical findings by some numerical experiments.     

干散货运价波动的杠杆效应特征研究——基于非对称随机波动模型

由信息冲击引起的干散货运价的剧烈波动给航运实体市场带来巨大风险,同等强度的利空消息通常要比利好消息引起更大的市场波动,本文对干散货航运市场运价波动存在的杠杆效应特征进行研究,为航运企业和租船人等把握市场态势、规避风险提供重要依据。考虑运价收益分布的厚尾特征,改变传统的非对称随机波动模型中随机误差项的正态分布假定,建立基于student-t分布的改进的非对称随机波动模型,在贝叶斯分析的基础上通过MCMC方法进行参数估计。通过实证研究发现,在考虑了极端风险情况后,改进的厚尾分布的非对称随机波动模型对干散货运价波动的杠杆效应特征刻画更加准确和优越。     

Closed-Form Pricing of Two-Asset Barrier Options with Stochastic Covariance

Abstract

Single and double barrier options on more than one underlying with stochastic volatility are usually priced via Monte Carlo simulation due to the non-existence of closed-form solutions for their value. In this paper, for a special dependence structure, the prices of some two-asset barrier derivatives, like double-digital options and correlation options can be derived analytically using generalized Fourier transforms and some conditions on the characteristic functions. We study the influence of the various parameters on these prices and show that these formulas can be easily and quickly computed. We also extend our approach to further allow for a random correlation structure.     

Accounting for risk aversion in derivatives purchase timing

We study the problem of optimal timing to buy/sell derivatives by a risk-averse agent in incomplete markets. Adopting the exponential utility indifference valuation, we investigate this timing flexibility and the associated delayed purchase premium. This leads to a stochastic control and optimal stopping problem that combines the observed market price dynamics and the agent??s risk preferences. Our results extend recent work on indifference valuation of American options, as well as the authors?? first paper (Leung and Ludkovski, SIAM J Finan Math 2(1) 768?C793, 2011). In the case of Markovian models of contracts on non-traded assets, we provide analytical characterizations and numerical studies of the optimal purchase strategies, with applications to both equity and credit derivatives.     

Rare Shock,Two-Factor Stochastic Volatility and Currency Option Pricing

Abstract

In this paper, we develop an option valuation model where the dynamics of the spot foreign exchange rate is governed by a two-factor Markov-modulated jump-diffusion process. The short-term fluctuation of stochastic volatility is driven by a Cox–Ingersoll–Ross (CIR) process and the long-term variation of stochastic volatility is driven by a continuous-time Markov chain which can be interpreted as economy states. Rare events are governed by a compound Poisson process with log-normal jump amplitude and stochastic jump intensity is modulated by a common continuous-time Markov chain. Since the market is incomplete under regime-switching assumptions, we determine a risk-neutral martingale measure via the Esscher transform and then give a pricing formula of currency options. Numerical results are presented for investigating the impact of the long-term volatility and the annual jump intensity on option prices.     

基于BEKK方差模型的干散货航运市场间波动溢出效应分析

针对灵便型、巴拿马型和海岬型干散货航运市场间的互动关系问题,选取波罗的海干散货运价指数,应用多元广义自回归条件异方差中的BEKK方差分析模型,研究了干散货航运市场间的波动溢出效应.发现海岬型干散货航运市场对灵便型和巴拿马型干散货航运市场存在波动溢出效应,而灵便型和巴拿马型干散货航运市场对海岬型干散货航运市场不存在波动溢出效应,灵便型干散货航运市场和巴拿马型干散货航运市场之间存在双向波动溢出效应,Wald检验验证了上述结论的正确性.从而可为航运经营者规避干散货航运市场波动风险提供决策参考.