TIME‐CHANGED MARKOV PROCESSES IN UNIFIED CREDIT‐EQUITY MODELING

This paper develops a novel class of hybrid credit‐equity models with state‐dependent jumps, local‐stochastic volatility, and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time‐changed Markov diffusion process with state‐dependent local volatility and killing rate (default intensity). When the time change is a Lévy subordinator, the stock price process exhibits jumps with state‐dependent Lévy measure. When the time change is a time integral of an activity rate process, the stock price process has local‐stochastic volatility and default intensity. When the time change process is a Lévy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state‐dependent jumps, local‐stochastic volatility, and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the time‐changed process is expressed in closed form as a single integral in the complex plane. If the payoff is square integrable, the complex integral is further reduced to a spectral expansion. To illustrate our general framework, we time change the jump‐to‐default extended constant elasticity of variance model of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local‐stochastic volatility, and default intensity. These models can be used to jointly price equity and credit derivatives.     

RATING BASED LÉVY LIBOR MODEL

In this paper, we consider modeling of credit risk within the Libor market models. We extend the classical definition of the default‐free forward Libor rate and develop the rating based Libor market model to cover defaultable bonds with credit ratings. As driving processes for the dynamics of the default‐free and the predefault term structure of Libor rates, time‐inhomogeneous Lévy processes are used. Credit migration is modeled by a conditional Markov chain, whose properties are preserved under different forward Libor measures. Conditions for absence of arbitrage in the model are derived and valuation formulae for some common credit derivatives in this setup are presented.     

POWER UTILITY MAXIMIZATION IN CONSTRAINED EXPONENTIAL LÉVY MODELS

We study power utility maximization for exponential Lévy models with portfolio constraints, where utility is obtained from consumption and/or terminal wealth. For convex constraints, an explicit solution in terms of the Lévy triplet is constructed under minimal assumptions by solving the Bellman equation. We use a novel transformation of the model to avoid technical conditions. The consequences for q‐optimal martingale measures are discussed as well as extensions to nonconvex constraints.     

ROBUST UTILITY MAXIMIZATION WITH LÉVY PROCESSES

We study a robust portfolio optimization problem under model uncertainty for an investor with logarithmic or power utility. The uncertainty is specified by a set of possible Lévy triplets, that is, possible instantaneous drift, volatility, and jump characteristics of the price process. We show that an optimal investment strategy exists and compute it in semi‐closed form. Moreover, we provide a saddle point analysis describing a worst‐case model.     

Double continuation regions for American and Swing options with negative discount rate in Lévy models

In this paper, we study perpetual American call and put options in an exponential Lévy model. We consider a negative effective discount rate that arises in a number of financial applications including stock loans and real options, where the strike price can potentially grow at a higher rate than the original discount factor. We show that in this case a double continuation region arises and we identify the two critical prices. We also generalize this result to multiple stopping problems of Swing type, that is, when successive exercise opportunities are separated by i.i.d. random refraction times. We conduct an extensive numerical analysis for the Black–Scholes model and the jump‐diffusion model with exponentially distributed jumps.     

A NEW LOOK AT SHORT‐TERM IMPLIED VOLATILITY IN ASSET PRICE MODELS WITH JUMPS

We analyze the behavior of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalization of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal–Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short‐end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model‐independent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on short‐maturity option prices.     

LOCAL VARIANCE GAMMA AND EXPLICIT CALIBRATION TO OPTION PRICES

In some options markets (e.g., commodities), options are listed with only a single maturity for each underlying. In others (e.g., equities, currencies), options are listed with multiple maturities. In this paper, we analyze a special class of pure jump Markov martingale models and provide an algorithm for calibrating such models to match the market prices of European options with multiple strikes and maturities. This algorithm matches option prices exactly and only requires solving several one‐dimensional root‐search problems and applying elementary functions. We show how to construct a time‐homogeneous process which meets a single smile, and a piecewise time‐homogeneous process which can meet multiple smiles.     

When to sell an asset amid anxiety about drawdowns

We consider risk‐averse investors with different levels of anxiety about asset price drawdowns. The latter is defined as the distance of the current price away from its best performance since inception. These drawdowns can increase either continuously or by jumps, and will contribute toward the investor's overall impatience when breaching the investor's private tolerance level. We investigate the unusual reactions of investors when aiming to sell an asset under such adverse market conditions. Mathematically, we study the optimal stopping of the utility of an asset sale with a random discounting that captures the investor's overall impatience. The random discounting is given by the cumulative amount of time spent by the drawdowns in an undesirable high region, fine‐tuned by the investor's personal tolerance and anxiety about drawdowns. We prove that in addition to the traditional take‐profit sales, the real‐life employed stop‐loss orders and trailing stops may become part of the optimal selling strategy, depending on different personal characteristics. This paper thus provides insights on the effect of anxiety and its distinction with traditional risk aversion on decision making.     

The valuation of American options in a multidimensional exponential Lévy model

We consider the problem of valuation of American options written on dividend‐paying assets whose price dynamics follow a multidimensional exponential Lévy model. We carefully examine the relation between the option prices, related partial integro‐differential variational inequalities, and reflected backward stochastic differential equations. In particular, we prove regularity results for the value function and obtain the early exercise premium formula for a broad class of payoff functions.     

Static hedging and pricing of exotic options with payoff frames

We develop a general framework for statically hedging and pricing European‐style options with nonstandard terminal payoffs, which can be applied to mixed static–dynamic and semistatic hedges for many path‐dependent exotic options including variance swaps and barrier options. The goal is achieved by separating the hedging and pricing problems to obtain replicating strategies. Once prices have been obtained for a set of basis payoffs, the pricing and hedging of financial securities with arbitrary payoff functions is accomplished by computing a set of “hedge coefficients” for that security. This method is particularly well suited for pricing baskets of options simultaneously, and is robust to discontinuities of payoffs. In addition, the method enables a systematic comparison of the value of a payoff (or portfolio) across a set of competing model specifications with implications for security design.     

Multiple curve Lévy forward price model allowing for negative interest rates

In this paper, we develop a framework for discretely compounding interest rates that is based on the forward price process approach. This approach has a number of advantages, in particular in the current market environment. Compared to the classical as well as the Lévy Libor market model, it allows in a natural way for negative interest rates and has superb calibration properties even in the presence of extremely low rates. Moreover, the measure changes along the tenor structure are significantly simplified. These properties make it an excellent base for a postcrisis multiple curve setup. Two variants for multiple curve constructions based on the multiplicative spreads are discussed. Time‐inhomogeneous Lévy processes are used as driving processes. An explicit formula for the valuation of caps is derived using Fourier transform techniques. Relying on the valuation formula, we calibrate the two model variants to market data.     

VALUATION OF CONTINUOUSLY MONITORED DOUBLE BARRIER OPTIONS AND RELATED SECURITIES

In this paper, we apply Carr's randomization approximation and the operator form of the Wiener‐Hopf method to double barrier options in continuous time. Each step in the resulting backward induction algorithm is solved using a simple iterative procedure that reduces the problem of pricing options with two barriers to pricing a sequence of certain perpetual contingent claims with first‐touch single barrier features. This procedure admits a clear financial interpretation that can be formulated in the language of embedded options. Our approach results in a fast and accurate pricing method that can be used in a rather wide class of Lévy‐driven models including Variance Gamma processes, Normal Inverse Gaussian processes, KoBoL processes, CGMY model, and Kuznetsov's β ‐class. Our method can be applied to double barrier options with arbitrary bounded terminal payoff functions, which, in particular, allows us to price knock‐out double barrier put/call options as well as double‐no‐touch options.     

EFFICIENT PRICING OF BARRIER OPTIONS AND CREDIT DEFAULT SWAPS IN LÉVY MODELS WITH STOCHASTIC INTEREST RATE

Recently, advantages of conformal deformations of the contours of integration in pricing formulas for European options have been demonstrated in the context of wide classes of Lévy models, the Heston model, and other affine models. Similar deformations were used in one‐factor Lévy models to price options with barrier and lookback features and credit default swaps (CDSs). In the present paper, we generalize this approach to models, where the dynamics of the assets is modeled as , where X is a Lévy process, and the interest rate is stochastic. Assuming that X and r are independent, and , the infinitesimal generator of the pricing semigroup in the model for the short rate, satisfies weak regularity conditions, which hold for popular models of the short rate, we develop a variation of the pricing procedure for Lévy models which is almost as fast as in the case of the constant interest rate. Numerical examples show that about 0.15 second suffices to calculate prices of 8 options of same maturity in a two‐factor model with the error tolerance and less; in a three‐factor model, accuracy of order 0.001–0.005 is achieved in about 0.2 second. Similar results are obtained for quanto CDS, where an additional stochastic factor is the exchange rate. We suggest a class of Lévy models with the stochastic interest rate driven by 1–3 factors, which allows for fast calculations. This class can satisfy the current regulatory requirements for banks mandating sufficiently sophisticated credit risk models.     

LÉVY PROCESSES INDUCED BY DIRICHLET (B‐)SPLINES: MODELING MULTIVARIATE ASSET PRICE DYNAMICS

We consider a new class of processes, called LG processes, defined as linear combinations of independent gamma processes. Their distributional and path‐wise properties are explored by following their relation to polynomial and Dirichlet (B‐)splines. In particular, it is shown that the density of an LG process can be expressed in terms of Dirichlet (B‐)splines, introduced independently by Ignatov and Kaishev and Karlin, Micchelli, and Rinott. We further show that the well‐known variance gamma (VG) process, introduced by Madan and Seneta, and the bilateral gamma (BG) process, recently considered by Küchler and Tappe are special cases of an LG process. Following this LG interpretation, we derive new (alternative) expressions for the VG and BG densities and consider their numerical properties. The LG process has two sets of parameters, the B‐spline knots and their multiplicities, and offers further flexibility in controlling the shape of the Levy density, compared to the VG and the BG processes. Such flexibility is often desirable in practice, which makes LG processes interesting for financial and insurance applications. Multivariate LG processes are also introduced and their relation to multivariate Dirichlet and simplex splines is established. Expressions for their joint density, the underlying LG‐copula, the characteristic, moment and cumulant generating functions are given. A method for simulating LG sample paths is also proposed, based on the Dirichlet bridge sampling of gamma processes, due to Kaishev and Dimitriva. A method of moments for estimation of the LG parameters is also developed. Multivariate LG processes are shown to provide a competitive alternative in modeling dependence, compared to the various multivariate generalizations of the VG process, proposed in the literature. Application of multivariate LG processes in modeling the joint dynamics of multiple exchange rates is also considered.     

THE MULTIVARIATE supOU STOCHASTIC VOLATILITY MODEL

Using positive semidefinite supOU (superposition of Ornstein–Uhlenbeck type) processes to describe the volatility, we introduce a multivariate stochastic volatility model for financial data which is capable of modeling long range dependence effects. The finiteness of moments and the second‐order structure of the volatility, the log‐ returns, as well as their “squares” are discussed in detail. Moreover, we give several examples in which long memory effects occur and study how the model as well as the simple Ornstein–Uhlenbeck type stochastic volatility model behave under linear transformations. In particular, the models are shown to be preserved under invertible linear transformations. Finally, we discuss how (sup)OU stochastic volatility models can be combined with a factor modeling approach.     

HIGH‐ORDER SHORT‐TIME EXPANSIONS FOR ATM OPTION PRICES OF EXPONENTIAL LÉVY MODELS

The short‐time asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In this work, a novel second‐order approximation for at‐the‐money (ATM) option prices is derived for a large class of exponential Lévy models with or without Brownian component. The results hereafter shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In the presence of a Brownian component, the second‐order term, in time‐t, is of the form , with d₂ only depending on Y, the degree of jump activity, on σ, the volatility of the continuous component, and on an additional parameter controlling the intensity of the “small” jumps (regardless of their signs). This extends the well‐known result that the leading first‐order term is . In contrast, under a pure‐jump model, the dependence on Y and on the separate intensities of negative and positive small jumps are already reflected in the leading term, which is of the form . The second‐order term is shown to be of the form and, therefore, its order of decay turns out to be independent of Y. The asymptotic behavior of the corresponding Black–Scholes implied volatilities is also addressed. Our method of proof is based on an integral representation of the option price involving the tail probability of the log‐return process under the share measure and a suitable change of probability measure under which the pure‐jump component of the log‐return process becomes a Y‐stable process. Our approach is sufficiently general to cover a wide class of Lévy processes, which satisfy the latter property and whose Lévy density can be closely approximated by a stable density near the origin. Our numerical results show that the first‐order term typically exhibits rather poor performance and that the second‐order term can significantly improve the approximation's accuracy, particularly in the absence of a Brownian component.     

Static and semistatic hedging as contrarian or conformist bets

In this paper, we argue that, once the costs of maintaining the hedging portfolio are properly taken into account, semistatic portfolios should more properly be thought of as separate classes of derivatives, with nontrivial, model‐dependent payoff structures. We derive new integral representations for payoffs of exotic European options in terms of payoffs of vanillas, different from the Carr–Madan representation, and suggest approximations of the idealized static hedging/replicating portfolio using vanillas available in the market. We study the dependence of the hedging error on a model used for pricing and show that the variance of the hedging errors of static hedging portfolios can be sizably larger than the errors of variance‐minimizing portfolios. We explain why the exact semistatic hedging of barrier options is impossible for processes with jumps, and derive general formulas for variance‐minimizing semistatic portfolios. We show that hedging using vanillas only leads to larger errors than hedging using vanillas and first touch digitals. In all cases, efficient calculations of the weights of the hedging portfolios are in the dual space using new efficient numerical methods for calculation of the Wiener–Hopf factors and Laplace–Fourier inversion.     

Business Forecasting of Double‐trend Time Series: An Improved PLS‐based Time‐varying Weight Combination Approach

Business forecasting with double‐trend time series (long‐term trends and seasonal volatility) has been challenging due to its complexity. Neither a single time series model nor a fixed‐weight combination approach can fully capture the comprehensive information. We address this issue by proposing an improved partial least squares (PLS) based time‐varying weight combination approach. The proposed method can handle the relations both between the single models involved and between single models and time ordering with time‐varying weights. The test on 20 simulated datasets demonstrates the better and more robust performance of the method. We also apply it to three real datasets. The results show that our approach represents a significant improvement over the existing methods in terms of data fitness and prediction accuracy. Copyright © 2017 ASAC. Published by John Wiley & Sons, Ltd.     

HEDGING STRATEGIES AND MINIMAL VARIANCE PORTFOLIOS FOR EUROPEAN AND EXOTIC OPTIONS IN A LÉVY MARKET

This paper presents hedging strategies for European and exotic options in a Lévy market. By applying Taylor’s theorem, dynamic hedging portfolios are constructed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importance as such other derivatives might be readily available. Moment swaps or power jump assets are not typically liquidly traded. It is shown how minimal variance portfolios can be used to hedge the higher order terms in a Taylor expansion of the pricing function, investing only in a risk‐free bank account, the underlying asset, and potentially variance swaps. The numerical algorithms and performance of the hedging strategies are presented, showing the practical utility of the derived results.     

MCMC ESTIMATION OF LÉVY JUMP MODELS USING STOCK AND OPTION PRICES

We examine the performances of several popular Lévy jump models and some of the most sophisticated affine jump‐diffusion models in capturing the joint dynamics of stock and option prices. We develop efficient Markov chain Monte Carlo methods for estimating parameters and latent volatility/jump variables of the Lévy jump models using stock and option prices. We show that models with infinite‐activity Lévy jumps in returns significantly outperform affine jump‐diffusion models with compound Poisson jumps in returns and volatility in capturing both the physical and risk‐neutral dynamics of the S&P 500 index. We also find that the variance gamma model of Madan, Carr, and Chang with stochastic volatility has the best performance among all the models we consider.