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.     

Hedging of game options in discrete markets with transaction costs

Game options introduced in [10] in 2000 were studied, by now, mostly in frictionless both complete and incomplete markets. In complete markets the fair price of a game option coincides with the value of an appropriate Dynkin's game, whereas in markets with friction and in incomplete ones there is a range of arbitrage free prices and superhedging comes into the picture. Here we consider game options in general discrete time markets with transaction costs and construct backward and forward induction algorithms for the computation of their prices and superhedging strategies from both seller's (upper arbitrage free price) and buyer's (lower arbitrage free price) points of view extending to the game options case most of the results from [12].     

Efficient L-stable method for parabolic problems with application to pricing American options under stochastic volatility

Efficient L-stable numerical method for semilinear parabolic problems with nonsmooth initial data is proposed and implemented to solve Heston’s stochastic volatility model based PDE for pricing American options under stochastic volatility. The proposed new method is also used to solve two asset American options pricing problem. Cox and Matthews [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics 176 (2002) 430-455] developed a class of exponential time differencing Runge-Kutta schemes (ETDRK) for nonlinear parabolic problems. Kassam and Trefethen [A.K. Kassam, L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM Journal on Scientific Computing 26 (4) (2005) 1214-1233] showed that while computing certain functions involved in the Cox-Matthews schemes, severe cancelation errors can occur which affect the accuracy and stability of the schemes. Kassam and Trefethen proposed complex contour integration technique to implement these schemes in a way that avoids these cancelation errors. But this approach creates new difficulties in choosing and evaluating the contour integrals for larger problems. We modify the ETDRK schemes using positivity preserving Padé approximations of the matrix exponential functions and construct computationally efficient parallel version using splitting technique. As a result of this approach it is required only to solve several backward Euler linear problems in serial or parallel.     

FINITE DIFFERENCE APPROXIMATION FOR PRICING THE AMERICAN LOOKBACK OPTION

In this paper we are concerned with the pricing of lookback options with American type constrains. Based on the differential linear complementary formula associated with the pricing problem, an implicit difference scheme is constructed and analyzed. We show that there exists a unique difference solution which is unconditionally stable. Using the notion of viscosity solutions, we also prove that the finite difference solution converges uniformly to the viscosity solution of the continuous problem. Furthermore, by means of the variational inequality analysis method, the O（△t ＋ △x^2）-order error estimate is derived in the discrete L2-norm provided that the continuous problem is sufficiently regular. In addition, a numerical example is provided to illustrate the theoretical results.     

Optimal static-dynamic hedges for exotic options under convex risk measures

We study the problem of optimally hedging exotic derivatives positions using a combination of dynamic trading strategies in underlying stocks and static positions in vanilla options when the performance is quantified by a convex risk measure. We establish conditions for the existence of an optimal static position for general convex risk measures, and then analyze in detail the case of shortfall risk with a power loss function. Here we find conditions for uniqueness of the static hedge. We illustrate the computational challenge of computing the market-adjusted risk measure in a simple diffusion model for an option on a non-traded asset.     

Dynamics of implied volatility surfaces from random matrix theory

We analyze the dynamics of the implied volatility surface of KOSPI 200 futures options from random matrix theory. To extract the informative data, we use random matrix criteria. Implied volatility data have a colossal eigenvalue, and the order of eigenvalues in a noisy regime is distinguishably smaller than a random matrix theory prediction. We discern the marketwide knowledge of the implied volatility surface movement such as the level, skew, and smile effect. These dynamics has the ergodic property and long range autocorrelation. We also study the relationship between the three implied volatility surface dynamics and the underlying asset dynamics, and confirm the existence of leverage effect even in the short time interval.     

随机利率下奇异期权的定价公式

在随机利率条件下,借助于测度变换获得了复合看涨期权的一般的定价公式,同时利用鞅理论和Girsanov定理,在利率服从于扩展的Vasicek利率模型时,得到了复合看涨期权精确的定价公式.用同样的方法,考虑了预设日期的重置看涨期权的定价问题,在利率服从同样的利率模型时,获得了重置看涨期权的定价公式.数值化的结果进一步说明了当利率遵循扩展的Vasicek利率模型时,B-S看涨期权的价格关于标的资产的价格是严格单调递增的,复合看涨期权的Geske公式是可以推广到随机利率的情况.     

基于VaR方法的我国外汇储备币种结构风险分析

基于VaR方法,把外汇储备看做是一个由不同币种组成的资产组合,通过实证分析测算各币种比例相对变化时的VaR值.结果表明,从控制风险的角度,目前我国应减少外汇储备中欧元的比重而增加美元的比重.     

Dynamic investment strategies with demand-side and cost-side risks

Investments in cost reductions are critical for the long run success of companies that operate in dynamic and stochastic market environments. This paper studies optimal investment in cost reductions as a real option under the assumption that a single firm faces two different sources of risk, stochastic demand and input prices. We derive optimal investment strategies for a monopoly as well as a firm in a perfectly competitive market and show that in case of high marginal costs, cost reductions take place earlier in competitive than in monopoly markets. While the existence of an option to invest in cost reductions increases firm value it also increases a firm’s systematic risk. Risk can be smaller in a monopolistic than in a competitive industry.