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2011

Volume 2, pp. 1-1076


Good Deal Bounds Induced by Shortfall Risk

Takuji Arai

SIAM J. Finan. Math. 2, pp. 1-21 (21 pages)

Online Publication Date: January 11, 2011

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We consider, throughout this paper, an incomplete financial market which is governed by a possibly nonlocally bounded right-continuous with left-limits (RCLL) special semimartingale. We shall provide good deal bounds for contingent claims induced by shortfall risk in the framework of the Orlicz heart setting. We prove that the upper and lower bounds of such a good deal bound are expressed by a convex risk measure on an Orlicz heart. In addition, we obtain representation results for three types of model, which are an unconstrained portfolio model, a $W$-admissible model, and a predictably convex model.

Jump-Diffusion Risk-Sensitive Asset Management I: Diffusion Factor Model

Mark Davis and Sébastien Lleo

SIAM J. Finan. Math. 2, pp. 22-54 (33 pages)

Online Publication Date: January 13, 2011

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This paper considers a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure, with drifts that are functions of an auxiliary diffusion factor process. The criterion, following earlier work by Bielecki, Pliska, Nagai, and others, is risk-sensitive optimization (equivalent to maximizing the expected growth rate subject to a constraint on variance). By using a change of measure technique introduced by Kuroda and Nagai we show that the problem reduces to solving a certain stochastic control problem in the factor process, which has no jumps. The main result of this paper is to show that the risk-sensitive jump-diffusion problem can be fully characterized in terms of a parabolic Hamilton–Jacobi–Bellman PDE rather than a partial integro-differential equation, and that this PDE admits a classical $(C^{1,2})$ solution.

On the Convergence of Higher Order Hedging Schemes: The Delta-Gamma Case

Mats Brodén and Magnus Wiktorsson

SIAM J. Finan. Math. 2, pp. 55-78 (24 pages)

Online Publication Date: January 20, 2011

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Hedging errors induced by discrete rebalancing of the hedge portfolio of a delta-gamma hedging strategy are investigated. The rate of convergence of the expected squared hedging error as the number of adjustments of the hedge portfolio goes to infinity is analyzed. It is found that the delta-gamma strategy produces higher convergence rates than the usual delta strategy.

Convergence of Price and Sensitivities in Carr's Randomization Approximation Globally and Near Barrier

Sergei Levendorskiĭ

SIAM J. Finan. Math. 2, pp. 79-111 (33 pages)

Online Publication Date: January 20, 2011

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Barrier options under wide classes of Lévy processes with exponentially decaying jump densities, including the variance gamma model, KoBoL and CGMY models, normal inverse Gaussian processes, and $\beta$-class, are studied. The leading term of asymptotics of the option price and the leading term of asymptotics in Carr's randomization approximation to the price are calculated as the price of the underlying approaches the barrier. We prove that the order of asymptotics is the same in both cases and that the asymptotic coefficient in the asymptotic formula for Carr's randomization approximation converges to the asymptotic coefficient for the price as the number of time steps $N\to+\infty$. Also, we justify Richardson extrapolation of arbitrary order. Similar results are derived for sensitivities and the leading terms of their asymptotics in Carr's randomization approximation. The convergence of prices and sensitivities is proved in appropriate weighted Hölder spaces.

Dynamic Hedging of Portfolio Credit Derivatives

Rama Cont and Yu Hang Kan

SIAM J. Finan. Math. 2, pp. 112-140 (29 pages)

Online Publication Date: February 01, 2011

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We compare the performance of various hedging strategies for index collateralized debt obligation (CDO) tranches across a variety of models and hedging methods during the recent credit crisis. Our empirical analysis shows evidence for market incompleteness: a large proportion of risk in the CDO tranches appears to be unhedgeable. We also show that, unlike what is commonly assumed, dynamic models do not necessarily perform better than static models, nor do high-dimensional bottom-up models perform better than simpler top-down models. When it comes to hedging, top-down and regression-based hedging with the index provide significantly better results during the credit crisis than bottom-up hedging with single-name credit default swap (CDS) contracts. Our empirical study also reveals that while significantly large moves—“jumps”—do occur in CDS, index, and tranche spreads, these jumps do not necessarily occur on the default dates of index constituents, an observation which shows the insufficiency of some recently proposed portfolio credit risk models.

Robust Hedging of Double Touch Barrier Options

A. M. G. Cox and Jan Obloj

SIAM J. Finan. Math. 2, pp. 141-182 (42 pages)

Online Publication Date: February 03, 2011

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We consider robust pricing of digital options, which pay out if the underlying asset has crossed both upper and lower barriers. We make only weak assumptions about the underlying process (typically continuity), but assume that the initial prices of call options with the same maturity and all strikes are known. In such circumstances, we are able to give upper and lower bounds on the arbitrage-free prices of the relevant options and show that these bounds are tight. Moreover, pathwise inequalities are derived, which provide the trading strategies with which we are able to realize any potential arbitrages. These super- and subhedging strategies have a simple quasi-static structure, their associated hedging error is bounded below, and in practice they carry low transaction costs. We show that, depending on the risk aversion of the investor, they can outperform significantly the standard delta/vega-hedging in presence of market frictions and/or model misspecification. We make use of embeddings techniques; in particular, we develop two new solutions to the (optimal) Skorokhod embedding problem.

Optimal Execution in a General One-Sided Limit-Order Book

Silviu Predoiu, Gennady Shaikhet, and Steven Shreve

SIAM J. Finan. Math. 2, pp. 183-212 (30 pages)

Online Publication Date: March 09, 2011

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We construct an optimal execution strategy for the purchase of a large number of shares of a financial asset over a fixed interval of time. Purchases of the asset have a nonlinear impact on price, and this is moderated over time by resilience in the limit-order book that determines the price. The limit-order book is permitted to have arbitrary shape. The form of the optimal execution strategy is to make an initial lump purchase and then purchase continuously for some period of time during which the rate of purchase is set to match the order book resiliency. At the end of this period, another lump purchase is made, and following that there is again a period of purchasing continuously at a rate set to match the order book resiliency. At the end of this second period, there is a final lump purchase. Any of the lump purchases could be of size zero. A simple condition is provided that guarantees that the intermediate lump purchase is of size zero.

Is the Minimum Value of an Option on Variance Generated by Local Volatility?

Mathias Beiglböck, Peter Friz, and Stephan Sturm

SIAM J. Finan. Math. 2, pp. 213-220 (8 pages)

Online Publication Date: March 09, 2011

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We discuss the possibility of obtaining model-free bounds on volatility derivatives, given present market data in the form of a calibrated local volatility model. A counterexample to a widespread conjecture is given.

A Fast Mean-Reverting Correction to Heston's Stochastic Volatility Model

Jean-Pierre Fouque and Matthew J. Lorig

SIAM J. Finan. Math. 2, pp. 221-254 (34 pages)

Online Publication Date: March 09, 2011

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We propose a multiscale stochastic volatility model in which a fast mean-reverting factor of volatility is built on top of the Heston stochastic volatility model. A singular perturbative expansion is then used to obtain an approximation for European option prices. The resulting pricing formulas are semianalytic, in the sense that they can be expressed as integrals. Difficulties associated with the numerical evaluation of these integrals are discussed, and techniques for avoiding these difficulties are provided. Overall, it is shown that computational complexity for our model is comparable to the case of a pure Heston model, but our correction brings significant flexibility in terms of fitting to the implied volatility surface. This is illustrated numerically and with option data.

On the Heston Model with Stochastic Interest Rates

Lech A. Grzelak and Cornelis W. Oosterlee

SIAM J. Finan. Math. 2, pp. 255-286 (32 pages)

Online Publication Date: March 15, 2011

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We discuss the Heston model [Rev. Financ. Stud., 6 (1993), pp. 327–343] with stochastic interest rates driven by Hull–White (HW) [J. Derivatives, 4 (1996), pp. 26–36] or Cox–Ingersoll–Ross (CIR) [Econometrica, 53 (1985), pp. 385–407] processes. Two projection techniques to derive affine approximations of the original hybrid models are presented. In these approximations we can prescribe a nonzero correlation structure between all underlying processes. The affine approximate models admit pricing basic derivative products by Fourier techniques [P. P. Carr and D. B. Madan, J. Comput. Finance, 2 (1999), pp. 61–73, F. Fang and C. W. Oosterlee, SIAM J. Sci. Comput., 31 (2008), pp. 826–848] and can therefore be used for fast calibration of the hybrid model.

A Reduced Basis for Option Pricing

Rama Cont, Nicolas Lantos, and Olivier Pironneau

SIAM J. Finan. Math. 2, pp. 287-316 (30 pages)

Online Publication Date: March 29, 2011

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We introduce a reduced basis method for the efficient numerical solution of partial integro-differential equations (PIDEs) which arise in option pricing theory. Our method constructs the solution as a linear combination of basis functions constructed from a sequence of Black–Scholes solutions with different volatilities. We show that this a priori choice of basis leads to a sparse representation of option pricing functions, yielding an approximation error which decays exponentially in the number of basis functions. A Galerkin method using this basis for solving the pricing PDE is shown to have better numerical performance relative to commonly used finite-difference and finite-element methods for the CEV diffusion model and the Merton jump diffusion model. We also compare our method with a numerical proper orthogonal decomposition (POD). Finally, we show that this approach may be used advantageously for the calibration of local volatility functions.

Arbitrage Opportunities in Misspecified Stochastic Volatility Models

Rudra P. Jena and Peter Tankov

SIAM J. Finan. Math. 2, pp. 317-341 (25 pages)

Online Publication Date: May 24, 2011

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There is vast empirical evidence that given a set of assumptions on the real-world dynamics of an asset, the European options on this asset are not efficiently priced in options markets, giving rise to arbitrage opportunities. We study these opportunities in a generic stochastic volatility model and exhibit the strategies which maximize the arbitrage profit. In the case when the misspecified dynamics are classical Black–Scholes ones, we give a new interpretation of the butterfly and risk reversal contracts in terms of their performance for volatility arbitrage. Our results are illustrated by a numerical example including transaction costs.

Nonquadratic Local Risk-Minimization for Hedging Contingent Claims in Incomplete Markets

Frédéric Abergel and Nicolas Millot

SIAM J. Finan. Math. 2, pp. 342-356 (15 pages)

Online Publication Date: May 24, 2011

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We introduce a new criterion to perform hedging of contingent claims in incomplete markets. Our approach is close to the one proposed by Schweizer [Stochastic Process. Appl., 37 (1991), pp. 339–363] in that it uses the concept of locally risk-minimizing strategies. But we aim at being more general by defining the local risk as a general, nonnecessarily quadratic, convex function of the local cost process. We derive the corresponding optimal strategies and value function in both discrete and continuous time settings. Finally we give an application of our hedging method in the stochastic volatility case as well as in the jump diffusion case. We work with a single traded asset, but our approach may be generalized to deal with claims depending on multiple assets.

Dual Representation of Quasi-convex Conditional Maps

Marco Frittelli and Marco Maggis

SIAM J. Finan. Math. 2, pp. 357-382 (26 pages)

Online Publication Date: May 24, 2011

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We provide a dual representation of quasi-convex maps $\pi :L_{\mathcal{F}% }\rightarrow L_{\mathcal{G}}$, between two locally convex lattices of random variables, in terms of conditional expectations. This generalizes the dual representation of quasi-convex real valued functions $\pi :L_{\mathcal{F}% }\rightarrow \mathbb{R}$ and the dual representation of conditional convex maps $\pi :L_{\mathcal{F}}\rightarrow L_{\mathcal{G}}.$ These results were inspired by the theory of dynamic measurements of risk and are applied in this context.

Pricing Discretely Monitored Asian Options by Maturity Randomization

Gianluca Fusai, Daniele Marazzina, and Marina Marena

SIAM J. Finan. Math. 2, pp. 383-403 (21 pages)

Online Publication Date: June 07, 2011

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We present a new methodology based on maturity randomization to price discretely monitored arithmetic Asian options when the underlying asset evolves according to a generic Lévy process. Our randomization technique considers the option expiry to be a random variable distributed according to a geometric distribution of a parameter independent of the underlying process. This allows one to transform the pricing backward procedure into a set of independent integral equations. Numerical procedures for a fast and accurate solution of the pricing problem are provided.

Optimal Control of Trading Algorithms: A General Impulse Control Approach

Bruno Bouchard, Ngoc-Minh Dang, and Charles-Albert Lehalle

SIAM J. Finan. Math. 2, pp. 404-438 (35 pages)

Online Publication Date: June 07, 2011

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We propose a general framework for intraday trading based on the control of trading algorithms. Given a set of generic parameterized algorithms (which have to be specified by the controller ex-ante), our aim is to optimize the dates $(\tau_i)_i$ at which they are launched, the length $(\delta_i)_i$ of the trading period, and the value of the parameters $({\cal E}_i)_i$ kept during the time interval $[\tau_i,\tau_i + \delta_i)$. This provides the financial agent a decision tool for selecting which algorithm (and for which set of parameters and time length) should be used in the different phases of the trading period. From the mathematical point of view, this gives rise to a nonclassical impulse control problem where not only the regime ${\cal E}_i$ but also the period $[\tau_i,\tau_i+ \delta_i)$ have to be determined by the controller at the impulse time $\tau_i$. We adapt the weak dynamic programming principle of Bouchard and Touzi [SIAM J. Control Optim., 49 (2011), pp. 948–962] to our context to provide a characterization of the associated value function as a discontinuous viscosity solution of a system of partial differential equations with appropriate boundary conditions, for which we prove a comparison principle. We also propose a numerical scheme for the resolution of the above system and show that it is convergent. We finally provide a simple example of application to a problem of optimal stock trading with a nonlinear market impact function. This shows how parameters adapt to the market.

A Fourier-Based Valuation Method for Bermudan and Barrier Options under Heston's Model

Fang Fang and Cornelis W. Oosterlee

SIAM J. Finan. Math. 2, pp. 439-463 (25 pages)

Online Publication Date: July 19, 2011

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We develop an efficient Fourier-based numerical method for pricing Bermudan and discretely monitored barrier options under the Heston stochastic volatility model. The two-dimensional pricing problem is dealt with by a combination of a Fourier cosine series expansion, as in [F. Fang and C. W. Oosterlee, SIAM J. Sci. Comput., 31 (2008), pp. 826–848, F. Fang and C. W. Oosterlee, Numer. Math., 114 (2009), pp. 27–62], and high-order quadrature rules in the other dimension. Error analysis and experiments confirm a fast error convergence.

Lévy-Based Cross-Commodity Models and Derivative Valuation

Sebastian Jaimungal and Vladimir Surkov

SIAM J. Finan. Math. 2, pp. 464-487 (24 pages)

Online Publication Date: July 21, 2011

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Energy commodities, such as oil, gas, and electricity, lack the liquidity of equity markets, have large costs associated with storage, exhibit high volatilities, and can have significant spikes in prices. Furthermore, and possibly more importantly, commodities tend to revert to long run equilibrium prices. Many complex commodity contingent claims exist in the markets, such as swing and interruptible options; however, the current method of valuation relies heavily on Monte Carlo simulations and tree-based methods. In this article, we develop a new cross-commodity modeling framework which accounts for jumps and cointegration in prices and introduce a new derivative valuation methodology by working in Fourier space. The method is based on the Fourier space time-stepping algorithm of Jackson, Jaimungal, and Surkov [J. Comput. Finance, 12 (2008), pp. 1–28] but is tailored for mean-reverting models. We demonstrate the utility of the method by applying it to European, American, and barrier options on a single commodity, to European and Bermudan spread options on two commodities, and to a particular class of swing options.

Stochastic Switching Games and Duopolistic Competition in Emissions Markets

Michael Ludkovski

SIAM J. Finan. Math. 2, pp. 488-511 (24 pages)

Online Publication Date: August 09, 2011

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We study optimal behavior of energy producers under a CO$_2$ emission abatement program. We focus on a two-player discrete-time model where each producer is sequentially optimizing her emission and production schedules. The game-theoretic aspect is captured through a reduced-form price-impact model for the CO$_2$ allowance price. Such duopolistic competition results in a new type of non–zero-sum stochastic switching game with finite horizon. Existence of game Nash equilibria is established through generalization to randomized switching strategies. No uniqueness is possible, and we therefore consider a variety of correlated equilibrium mechanisms. We prove existence of correlated equilibrium points in switching games and give a recursive description of equilibrium game values. A simulation-based algorithm to solve for the game values is constructed, and a numerical example is presented.

An Option to Reduce Transaction Costs

Jonathan Goodman and Daniel N. Ostrov

SIAM J. Finan. Math. 2, pp. 512-537 (26 pages)

Online Publication Date: August 09, 2011

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For small transaction costs, we determine the leading order optimal dynamic trading strategy of a portfolio of stock, cash, and options. Except for the transaction costs, our market assumptions are those of Black, Scholes, and Merton. Without transaction costs, the option is redundant in the portfolio. With transaction costs, however, we show that adding the option to the portfolio can significantly reduce overall trading costs compared to optimal strategies that use only stock and cash. The analysis is based on an asymptotic expansion with three scales: macroscopic, mesoscopic, and microscopic. The macroscopic analysis is Merton's optimal investment problem. Within a plane defined by the amount of stock and options held, the macroscopic analysis yields a Merton line of optimal portfolios. We show that there is a particular magic point on the Merton line that minimizes expensive stochastic movement away from the Merton line. The mesoscopic scale governs less expensive deviations of the portfolio away from the magic point but along the Merton line. The microscopic scale governs the more expensive deviations of the portfolio away from the magic point, transverse to the Merton line. The resulting strategy is related to commonly used Delta and Gamma hedging strategies, but our scale analysis implies that some rebalancings are much more effective than others. We do not give rigorous mathematical proofs, only arguments of formal applied mathematics.

Regularity of the Exercise Boundary for American Put Options on Assets with Discrete Dividends

B. Jourdain and M. H. Vellekoop

SIAM J. Finan. Math. 2, pp. 538-561 (24 pages)

Online Publication Date: August 16, 2011

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We analyze the regularity of the optimal exercise boundary for the American Put option when the underlying asset pays a discrete dividend at a known time $t_d$ during the lifetime of the option. The ex-dividend asset price process is assumed to follow Black–Scholes dynamics, and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. The solution to the associated optimal stopping problem can be characterized in terms of an optimal exercise boundary which, in contrast to the case when there are no dividends, may no longer be monotone. In this paper we prove that when the dividend function is positive and concave, then the boundary is nonincreasing in a left-hand neighborhood of $t_d$ and tends to 0 as time tends to $t_d^-$ with a speed that we can characterize. When the dividend function is linear in a neighborhood of zero, then we show continuity of the exercise boundary and a high contact principle in the left-hand neighborhood of $t_d$. When it is globally linear, then the right-continuity of the boundary and the high contact principle are proved to hold globally. Finally, we show how all the previous results can be extended to multiple dividend payment dates in that case.

Primal and Dual Pricing of Multiple Exercise Options in Continuous Time

Christian Bender

SIAM J. Finan. Math. 2, pp. 562-586 (25 pages)

Online Publication Date: August 18, 2011

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In this paper we study the pricing problem of multiple exercise options in continuous time on a finite time horizon. For the corresponding multiple stopping problem, we prove, under quite general assumptions, the existence of the Snell envelope, a reduction principle as nested single stopping problems, and a Doob–Meyer-type decomposition for the Snell envelope. The main technical difficulty arises from the fact that the price process of a multiple exercise option typically exhibits discontinuities from the right-hand side, even if the payoff process of the option is right-continuous. We also derive a dual minimization problem for the price of the multiple exercise option in terms of martingales and processes of bounded variation. Moreover, we explain how the primal and dual pricing formulas can be applied to compute confidence intervals on the option price via Monte Carlo methods, and we present a numerical example.

On the Robustness of the Snell Envelope

Pierre Del Moral, Peng Hu, Nadia Oudjane, and Bruno Rémillard

SIAM J. Finan. Math. 2, pp. 587-626 (40 pages)

Online Publication Date: August 25, 2011

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We analyze the robustness properties of the Snell envelope backward evolution equation for the discrete time optimal stopping problem. We consider a series of approximation schemes, including cut–off-type approximations, Euler discretization schemes, interpolation models, quantization tree models, and the stochastic mesh method of Broadie and Glasserman. In each situation, we provide nonasymptotic convergence estimates, including $\mathbb{L}_p$-mean error bounds and exponential concentration inequalities. We deduce these estimates from a single and general robustness property of Snell envelope semigroups. In particular, this analysis allows us to recover existing convergence results for the quantization tree method and to improve significantly the rates of convergence obtained for the stochastic mesh estimator of Broadie and Glasserman. In the second part of the article, we propose a new approach based on a genealogical tree approximation model of the reference Markov process in terms of a neutral-type genetic model. In contrast to Broadie–Glasserman Monte Carlo models, the computational cost of this new stochastic approximation is linear in the number of particles. Some simulation results are provided and confirm the interest of this new algorithm.

Stochastic Evolution Equations in Portfolio Credit Modelling

N. Bush, B. M. Hambly, H. Haworth, L. Jin, and C. Reisinger

SIAM J. Finan. Math. 2, pp. 627-664 (38 pages)

Online Publication Date: September 15, 2011

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We consider a structural credit model for a large portfolio of credit risky assets where the correlation is due to a market factor. By considering the large portfolio limit of this system we show the existence of a density process for the asset values. This density evolves according to a stochastic partial differential equation, and we establish existence and uniqueness for the solution taking values in a suitable function space. The loss function of the portfolio is then a function of the evolution of this density at the default boundary. We develop numerical methods for pricing and calibration of the model to credit indices and consider its performance before and after the credit crunch.

Spectral Decomposition of Option Prices in Fast Mean-Reverting Stochastic Volatility Models

Jean-Pierre Fouque, Sebastian Jaimungal, and Matthew J. Lorig

SIAM J. Finan. Math. 2, pp. 665-691 (27 pages)

Online Publication Date: September 15, 2011

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Using spectral decomposition techniques and singular perturbation theory, we develop a systematic method to approximate the prices of a variety of European and path-dependent options in a fast mean-reverting stochastic volatility setting. Our method is shown to be equivalent to those developed in [J.-P. Fouque, G. Papanicolaou, and R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000] but has the advantage of being able to price options for which the methods of [J.-P. Fouque, G. Papanicolaou, and R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000] are unsuitable. In particular, we are able to price double-barrier options. To our knowledge, this is the first time that double-barrier options have been priced in a stochastic volatility setting in which the Brownian motions driving the stock and volatility are correlated.

Saddlepoint Approximations for Expectations and an Application to CDO Pricing

Xinzheng Huang and Cornelis W. Oosterlee

SIAM J. Finan. Math. 2, pp. 692-714 (23 pages)

Online Publication Date: September 22, 2011

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We derive two types of saddlepoint approximations for expectations in the form of $\mathbb{E}[(X-K)^+]$, where $X$ is the sum of $n$ independent random variables and $K$ is a known constant. We establish error convergence rates for both types of approximations in the independently and identically distributed case. The approximations are further extended to cover the case of lattice variables. An application of the saddlepoint approximations to CDO pricing is presented.

Managing Risk with Short-Term Futures Contracts

Zhijian Wu, Chunhui Yu, and Xiaohua Zheng

SIAM J. Finan. Math. 2, pp. 715-726 (12 pages)

Online Publication Date: September 22, 2011

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Under the constraint of terminal risk, we search for an optimal deterministic strategy to reduce the running risk in hedging a long-term commitment with short-term futures contracts. An explicit solution is given if the underlying stock follows the simple stochastic differential equation $dS_t=\mu dt+\sigma dB_t$, where $B_t$ is the standard Brownian motion. Our result generalizes the result of Larcher and Leobacher in [Math. Finance, 13 (2003), pp. 331–344]. As an application, we provide a solution to the utility optimization problem posed in that paper.

Smooth Value Functions for a Class of Nonsmooth Utility Maximization Problems

Baojun Bian, Sheng Miao, and Harry Zheng

SIAM J. Finan. Math. 2, pp. 727-747 (21 pages)

Online Publication Date: September 22, 2011

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In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if the optimal control satisfies an exponential moment condition or if the value function is continuous on the closure of its domain. The key idea is to work on the dual control problem and the dual HJB equation. We construct a smooth, strictly convex solution to the dual HJB equation and show that its conjugate function is a smooth, strictly concave solution to the primal HJB equation satisfying the terminal and boundary conditions.

Optimal Portfolios of Mean-Reverting Instruments

Gordana Dmitrašinović-Vidović and Antony Ware

SIAM J. Finan. Math. 2, pp. 748-767 (20 pages)

Online Publication Date: September 29, 2011

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In this paper we investigate portfolios consisting of instruments whose logarithms are mean-reverting. Under the assumption that portfolios are constant, we derive analytic expressions for the expected wealth and the quantile-based risk measure capital at risk. Assuming that short-selling and borrowing are allowed, we then solve the problems of global minimum capital at risk and the problem of finding maximal wealth subject to constrained capital at risk. We illustrate these results with some numerical examples that show the strong effect of the mean-reversion rates on the portfolio choice.

Optimal Timing to Purchase Options

Tim Leung and Mike Ludkovski

SIAM J. Finan. Math. 2, pp. 768-793 (26 pages)

Online Publication Date: October 11, 2011

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We study the optimal timing of derivative purchases in incomplete markets. In our model, an investor attempts to maximize the spread between her model price and the offered market price through optimally timing her purchase. Both the investor and the market value the options by risk-neutral expectations but under different equivalent martingale measures representing different market views. The structure of the resulting optimal stopping problem depends on the interaction between the respective market price of risk and the option payoff. In particular, a crucial role is played by the delayed purchase premium that is related to the stochastic bracket between the market price and the buyer's risk premia. Explicit characterization of the purchase timing is given for two representative classes of Markovian models: (i) defaultable equity models with local intensity; (ii) diffusion stochastic volatility models. Several numerical examples are presented to illustrate the results. Our model is also applicable to the optimal rolling of long-dated options and sequential buying and selling of options.

Static Hedging under Time-Homogeneous Diffusions

Peter Carr and Sergey Nadtochiy

SIAM J. Finan. Math. 2, pp. 794-838 (45 pages)

Online Publication Date: October 11, 2011

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We consider the problem of semistatic hedging of a single barrier option in a model where the underlying is a time-homogeneous diffusion, possibly running on an independent stochastic clock. The main result of the paper is an analytic expression for the payoff of a European-type contingent claim, which has the same price as the barrier option up to hitting the barrier. We then consider some examples, such as the Black–Scholes, constant elasticity of variance, and zero-correlation SABR models. Finally, we investigate an approximation of the static hedge with options of at most two different strikes.

How to Detect an Asset Bubble

Robert Jarrow, Younes Kchia, and Philip Protter

SIAM J. Finan. Math. 2, pp. 839-865 (27 pages)

Online Publication Date: October 12, 2011

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After the 2007 credit crisis, financial bubbles have once again emerged as a topic of current concern. An open problem is to determine in real time whether or not a given asset's price process exhibits a bubble. Due to recent progress in the characterization of asset price bubbles using the arbitrage-free martingale pricing technology, we are able to propose a new methodology for answering this question based on the asset's price volatility. We limit ourselves to the special case of a risky asset's price being modeled by a Brownian driven stochastic differential equation. Such models are ubiquitous both in theory and in practice. Our methods use sophisticated volatility estimation techniques combined with the method of reproducing kernel Hilbert spaces. We illustrate these techniques using several stocks from the alleged Internet dot-com episode of 1998–2001, where price bubbles were widely thought to have existed. Our results confirm the suspicions of the presence of bubbles in many of the dot-com stocks of 1998–2001.

Continuity Correction for Barrier Options in Jump-Diffusion Models

El Hadj Aly Dia and Damien Lamberton

SIAM J. Finan. Math. 2, pp. 866-900 (35 pages)

Online Publication Date: October 18, 2011

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The aim of this paper is to study the continuity correction for barrier options in jump-diffusion models. For this purpose, we express the payoff of a barrier option in terms of the maximum of the underlying process. We then condition with respect to the jump times and to the values of the underlying at the jump times and rely on the connection between the maximum of the Brownian motion and Bessel processes.

Closed-Form Asymptotics and Numerical Approximations of 1D Parabolic Equations with Applications to Option Pricing

Wen Cheng, Nick Costanzino, John Liechty, Anna Mazzucato, and Victor Nistor

SIAM J. Finan. Math. 2, pp. 901-934 (34 pages)

Online Publication Date: October 25, 2011

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We construct closed-form asymptotic formulas for the Green's function of parabolic equations (e.g., Fokker–Planck equations) with variable coefficients in one space dimension. More precisely, let $u(t,x)=\int\mathcal{G}_t(x,y)f(y)dy$ be the solution of $\partial_tu-(au”+bu'+cu)=0$ for $t>0$, $u(0,x)=f(x)$. Then we find computable approximations $\mathcal{G}_t^{[n]}$ of $\mathcal{G}_t$. The approximate kernels are derived by applying the Dyson–Taylor commutator method that we have recently developed for short-time expansions of heat kernels on arbitrary dimension Euclidean spaces. We then utilize these kernels to obtain closed-form pricing formulas for European call options. The validity of such approximations to large time is extended using a bootstrap scheme. We prove explicit error estimates in weighted Sobolev spaces, which we test numerically and compare to other methods.

Asymptotic Approximations to Deterministic and Stochastic Volatility Models

Richard Jordan and Charles Tier

SIAM J. Finan. Math. 2, pp. 935-964 (30 pages)

Online Publication Date: October 27, 2011

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The problem of pricing, hedging, and calibrating equity derivatives in a fast and consistent fashion is considered when the underlying asset does not follow the standard Black–Scholes model but instead the stochastic CEV (constant elasticity of variance) or SABR (stochastic alpha beta rho) model. The underlying process in the SABR model has the volatility as a stochastic function of the asset price. In such situations, trading desks often resort to numerical methods to solve the pricing and hedging problem. This can be problematic for complex models if real-time valuations, hedging, and calibration are required. A more efficient and practical alternative is to use a formula even if it is only an approximation. A systematic approach is presented, based on the WKB or ray method, to derive asymptotic approximations yielding simple formulas for the pricing problem. For the SABR model, default may be possible, and the original ray approximation is not valid near the default boundary, so a modified asymptotic approximation or boundary layer correction is derived. New results are also derived for the standard CEV model, which has deterministic volatility, as a special case of the SABR results. The applicability of the results is illustrated by deriving new analytical approximations for vanilla options based on the CEV and SABR models. The accuracy of the results is demonstrated numerically.

A Bridge between American and European Options: The “Ameripean” Delayed-Exercise Model

Paul V. Johnson, Nicholas J. Sharp, Peter W. Duck, and David P. Newton

SIAM J. Finan. Math. 2, pp. 965-988 (24 pages)

Online Publication Date: November 03, 2011

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A new class of option-pricing model is discussed, motivated initially by the practical observation that contracts with embedded options are not always exercised immediately when an implicit barrier is breached; this may occur for a number of reasons, for example linked to behavior of the investor. Rather than using a conventional barrier, this option class model takes the first touching of the payoff function by the option value as the start of a waiting period before exercise. This presents itself as a free-boundary problem, similar to, but somewhat more complicated than, that found with the usual American option. It turns out that this gives insight into the dynamics of the American option itself, as the “Ameripean” delayed-exercise option model provides a fluid link between a European and an American option. It also prompts the development of an improved numerical technique, based on boundary-fitted coordinates, together with some useful asymptotic analyses (which give further insights into valuations).

A Finite-Dimensional Approximation for Pricing Moving Average Options

Marie Bernhart, Peter Tankov, and Xavier Warin

SIAM J. Finan. Math. 2, pp. 989-1013 (25 pages)

Online Publication Date: November 15, 2011

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We propose a method for pricing American options whose payoff depends on the moving average of the underlying asset price. The method uses a finite-dimensional approximation of the infinite-dimensional dynamics of the moving average process based on a truncated Laguerre series expansion. The resulting problem is a finite-dimensional optimal stopping problem, which we propose solving with a least squares Monte Carlo approach. We analyze the theoretical convergence rate of our method and present numerical results in the Black–Scholes framework.

Maximization of Recursive Utilities: A Dynamic Maximum Principle Approach

Wahid Faidi, Anis Matoussi, and Mohamed Mnif

SIAM J. Finan. Math. 2, pp. 1014-1041 (28 pages)

Online Publication Date: November 29, 2011

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We study a maximization problem from terminal wealth and consumption for a class of robust utility functions introduced in Bordigoni, Matoussi, and Schweizer [A stochastic control approach to a robust utility maximization problem, in Stochastic Analysis and Applications, Abel Symp. 2, F. E. Benth, G. Di Nunno, T. Lindstrøm, B. Øksendal, and T. Zhang, eds., Springer, Berlin, 2007, pp. 125–151]. Our method is based on backward stochastic differential equation theory techniques. We prove a dynamic maximum principle for the optimal control. We study the existence and the uniqueness of the consumption-investment strategy which is characterized as the unique solution of a forward-backward system.

Optimal Split of Orders Across Liquidity Pools: A Stochastic Algorithm Approach

Sophie Laruelle, Charles-Albert Lehalle, and Gilles Pagès

SIAM J. Finan. Math. 2, pp. 1042-1076 (35 pages)

Online Publication Date: December 20, 2011

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Evolutions of the trading landscape lead to the capability to exchange the same financial instrument on different venues. Because of liquidity issues, the trading firms split large orders across several trading destinations to optimize their execution. To solve this problem we devised two stochastic recursive learning procedures which adjust the proportions of the order to be sent to the different venues, one based on an optimization principle and the other on some reinforcement ideas. Both procedures are investigated from a theoretical point of view: we prove a.s. convergence of the optimization algorithm under some light ergodic (or “averaging") assumption on the input data process. No Markov property is needed. When the inputs are independent and identically distributed we show that the convergence rate is ruled by a central limit theorem. A variant including some market impact effect is also proposed. Finally, the mutual performances of both algorithms are compared on simulated and real data with respect to an “oracle" strategy devised by an “insider" who a priori knows the executed quantities by all venues.
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