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SIAM J. on Financial Mathematics

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2010

Volume 1, pp. 1-961


Message From the Editors-in-Chief

René Carmona and Ronnie Sircar, Editors-in-Chief

SIAM J. Finan. Math. 1, pp. 1-1 (1 page)

Online Publication Date: January 21, 2010

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Dear Reader,
It is a great pleasure and privilege to welcome you to the first volume of the SIAM Journal on Financial Mathematics. This journal, with the help of its editorial board, contributors, and readers and the SIAM staff, follows in the outstanding tradition of high-quality SIAM journals and, we hope, will become a leading journal in mathematical modeling and analysis of financial problems.
The journal began accepting submissions in October 2008. You will find the first papers accepted for this all-electronic publication. These papers provide a representative glance at the diversity of topics covered by the journal, something clearly observable from the outstanding editorial board itself. A number of papers currently in the pipeline further stress the broad vision of the journal.
The SIAM Journal on Financial Mathematics came to exist thanks to the tremendous effort of numerous people. The impetus came from the SIAM Activity Group on Financial Mathematics and Engineering (SIAG/FME) at its inaugural meeting in Boston in July 2006. While the idea of a new journal focused on computational finance had been discussed for a number of years, the enthusiasm at this meeting persuaded us, and other officers of the SIAG, to work toward this end. We received enthusiastic support from the publications committee and SIAM's board and council, and the journal was approved in July 2008. We are grateful to Tim Kelley and David Marshall for their hard work and advice in establishing the journal.
Preparing for the creation of the journal was arduous but rewarding. We aimed high, and at times, we wondered whether our goals were realistic. After agreeing on a charter for the journal, we made a wish list of associate editors, and it was no surprise to be asked by members of the council, “What makes you think that these distinguished scholars will accept your invitation to work for a new journal?" Except for two special cases, we were delighted that all of them accepted. In hindsight, the warm reception by the applied mathematics community at large has made the experience rewarding.
The SIAM staff is a wonderful asset. In particular, Mitch Chernoff, Brian Fauth, and Heather Blythe have been a critical source of support.
The current editorial board is working very hard to guarantee timely and high-quality reviews. The whole cycle of reviewing and production has been fast on average. Please enjoy your reading, and we look forward to receiving your high-quality submissions in the future.

Local Volatility Enhanced by a Jump to Default

Peter Carr and Dilip B. Madan

SIAM J. Finan. Math. 1, pp. 2-15 (14 pages)

Online Publication Date: January 21, 2010

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A local volatility model is enhanced by the possibility of a single jump to default. The jump has a hazard rate that is the product of the stock price raised to a prespecified negative power and a deterministic function of time. The empirical work uses a power of $-1.5$. It is shown how one may simultaneously recover from the prices of credit default swap contracts and equity option prices both the deterministic component of the hazard rate function and revised local volatility. The procedure is implemented on prices of credit default swaps and equity options for General Motors and the Ford Motor Company over the period October 2004 to September 2007.

Minimizing the Expected Market Time to Reach a Certain Wealth Level

Constantinos Kardaras and Eckhard Platen

SIAM J. Finan. Math. 1, pp. 16-29 (14 pages)

Online Publication Date: January 21, 2010

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In a financial market model, we consider variations of the problem of minimizing the expected time to upcross a certain wealth level. For exponential Lévy markets, we show the asymptotic optimality of the growth-optimal portfolio for the above problem and obtain tight bounds for the value function for any wealth level. In an Itô market, we employ the concept of market time, which is a clock that runs according to the underlying market growth. We show the optimality of the growth-optimal portfolio for minimizing the expected market time to reach any wealth level. This reveals a general definition of market time which can be useful from an investor's point of view. We utilize this last definition to extend the previous results in a general semimartingale setting.

Optimal Convergence Rate of the Binomial Tree Scheme for American Options with Jump Diffusion and Their Free Boundaries

Jin Liang, Bei Hu, and Lishang Jiang

SIAM J. Finan. Math. 1, pp. 30-65 (36 pages)

Online Publication Date: January 21, 2010

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An American put option with jump diffusion can be modeled as an integro-variational inequality. With a penalization approximation and under the stability condition $\frac{\sigma^2 \Delta t}{\Delta x^2}\le 1$, where $\Delta x ={\rm ln}\,\frac {S_{n+1}}{S_n}$ ($S_t$-underlying asset price), we obtain the optimal convergence rate $O((\Delta x)+(\Delta t)^{1/2})$ of the binomial tree scheme for this variational inequality. Moreover, we define an approximate optimal exercise boundary within the framework of the binomial tree scheme and derive the convergence rate estimate $O((\Delta t)^{1/4})$ to the actual free boundary.

Duality for Set-Valued Measures of Risk

Andreas H. Hamel and Frank Heyde

SIAM J. Finan. Math. 1, pp. 66-95 (30 pages)

Online Publication Date: January 21, 2010

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Extending the approach of Jouini, Meddeb, and Touzi [Finance Stoch., 8 (2004), pp. 531–552] we define set-valued (convex) measures of risk and their acceptance sets, and we give dual representation theorems. A scalarization concept is introduced that has a meaning in terms of internal prices of portfolios of reference instruments. Using primal and dual descriptions, we introduce new examples for set-valued measures of risk, e.g., set-valued upper expectations, value at risk, average value at risk, and entropic risk measure.

Continuous-Time Markowitz's Model with Transaction Costs

Min Dai, Zuo Quan Xu, and Xun Yu Zhou

SIAM J. Finan. Math. 1, pp. 96-125 (30 pages)

Online Publication Date: January 21, 2010

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A continuous-time Markowitz's mean-variance portfolio selection problem is studied in a market with one stock, one bond, and proportional transaction costs. This is a singular stochastic control problem, inherently with a finite time horizon. Via a series of transformations, the problem is turned into a so-called double obstacle problem, a well-studied problem in physics and PDE literature, featuring two time-varying free boundaries. The two boundaries, which define the buy, sell, and no-trade regions, are proved to be smooth in time. This in turn characterizes the optimal strategy, via a Skorokhod problem, as one that tries to keep a certain adjusted bond-stock position within the no-trade region. Several features of the optimal strategy are revealed that are remarkably different from its no-transaction-cost counterpart. It is shown that there exists a critical length in time, which is dependent on the stock excess return as well as the transaction fees but independent of the investment target and the stock volatility, so that an expected terminal return may not be achievable if the planning horizon is shorter than that critical length (while in the absence of transaction costs any expected return can be reached in an arbitrary period of time). It is further demonstrated that anyone following the optimal strategy should not buy the stock beyond the point when the time to maturity is shorter than the aforementioned critical length. Moreover, the investor would be less likely to buy the stock and more likely to sell the stock when the maturity date is getting closer. These features, while consistent with the widely accepted investment wisdom, suggest that the planning horizon is an integral part of the investment opportunities.

Short-Maturity Asymptotics for a Fast Mean-Reverting Heston Stochastic Volatility Model

Jin Feng, Martin Forde, and Jean-Pierre Fouque

SIAM J. Finan. Math. 1, pp. 126-141 (16 pages)

Online Publication Date: February 03, 2010

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In this paper, we study the Heston stochastic volatility model in a regime where the maturity is small but large compared to the mean-reversion time of the stochastic volatility factor. We derive a large deviation principle and compute the rate function by a precise study of the moment generating function and its asymptotic. We then obtain asymptotic prices for out-of-the-money call and put options and their corresponding implied volatilities.

A Fourier Transform Method for Spread Option Pricing

T. R. Hurd and Zhuowei Zhou

SIAM J. Finan. Math. 1, pp. 142-157 (16 pages)

Online Publication Date: February 03, 2010

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Spread options are a fundamental class of derivative contracts written on multiple assets and are widely traded in a range of financial markets. There is a long history of approximation methods for computing such products, but as yet there is no preferred approach that is accurate, efficient, and flexible enough to apply in general asset models. The present paper introduces a new formula for general spread option pricing based on Fourier analysis of the payoff function. Our detailed investigation, including a flexible and general error analysis, proves the effectiveness of a fast Fourier transform implementation of this formula for the computation of spread option prices. It is found to be easy to implement, stable, efficient, and applicable in a wide variety of asset pricing models.

Hedging of Claims with Physical Delivery under Convex Transaction Costs

Teemu Pennanen and Irina Penner

SIAM J. Finan. Math. 1, pp. 158-178 (21 pages)

Online Publication Date: February 17, 2010

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We study superhedging of contingent claims with physical delivery in a discrete-time market model with convex transaction costs. Our model extends Kabanov's currency market model by allowing for nonlinear illiquidity effects. We show that an appropriate generalization of Schachermayer's robust no-arbitrage condition implies that the set of claims hedgeable with zero cost is closed in probability. Combined with classical techniques of convex analysis, the closedness yields a dual characterization of premium processes that are sufficient to superhedge a given claim process. We also extend the fundamental theorem of asset pricing for general conical models.

Weak Kyle–Back Equilibrium Models for Max and ArgMax

A. Kohatsu-Higa and S. Ortiz-Latorre

SIAM J. Finan. Math. 1, pp. 179-211 (33 pages)

Online Publication Date: February 17, 2010

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The goal of this article is to introduce a new approach to model equilibrium in financial markets with an insider. We prove the existence and uniqueness in law of equilibrium for these markets. Our setting is weaker than that of Back, and it can be interpreted as a first theoretical step towards developing statistical test procedures. Additionally, it allows various forms of insider information to be considered under the same framework and compared. As major examples, we consider the cases of the maximum of the demand and the time at which this maximum is taken, which have not previously been treated in the literature of equilibrium in financial markets with inside information. Simulations indicate that the expected wealth for the maximum is greater than the expected wealth for its argument.

Common Forward Rate Volatility

Victor Goodman and Kyounghee Kim

SIAM J. Finan. Math. 1, pp. 212-229 (18 pages)

Online Publication Date: February 19, 2010

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Statistical analyses of forward interest rate behavior provide evidence that these rates share a common volatility. We develop a risk-neutral term structure model based on this assumption. The main feature of this model is that each discounted bond price is both an explicit local martingale and a diffusion. The Markov property of discounted bonds is convenient for pricing interest rate derivatives. We give price formulas for caps and swaptions and compare caplet prices to the market-standard Black formula. The two formulas have nearly identical numerical values.

Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility

Martino Bardi, Annalisa Cesaroni, and Luigi Manca

SIAM J. Finan. Math. 1, pp. 230-265 (36 pages)

Online Publication Date: February 19, 2010

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We study singular perturbations of a class of stochastic control problems under assumptions motivated by models of financial markets with stochastic volatilities evolving on a fast time scale. We prove the convergence of the value function to the solution of a limit (effective) Cauchy problem for a parabolic equation of HJB type. We use methods of the theory of viscosity solutions and of the homogenization of fully nonlinear PDEs. We test the result on some financial examples, such as Merton portfolio optimization problem.

Maturity-Independent Risk Measures

Thaleia Zariphopoulou and Gordan Žitković

SIAM J. Finan. Math. 1, pp. 266-288 (23 pages)

Online Publication Date: March 24, 2010

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The new notion of maturity-independent risk measures is introduced and contrasted with the existing risk measurement concepts. It is shown, by means of two examples, one set on a finite probability space and the other in a diffusion framework, that, surprisingly, some of the widely utilized risk measures cannot be used to build maturity-independent counterparts. We construct a large class of maturity-independent risk measures and give representative examples in both continuous- and discrete-time financial models.

Time Dependent Heston Model

E. Benhamou, E. Gobet, and M. Miri

SIAM J. Finan. Math. 1, pp. 289-325 (37 pages)

Online Publication Date: April 21, 2010

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The use of the Heston model is still challenging because it has a closed formula only when the parameters are constant [S. Heston, Rev. Financ. Stud., 6 (1993), pp. 327–343] or piecewise constant [S. Mikhailov and U. Nogel, Wilmott Magazine, July (2003), pp. 74–79]. Hence, using a small volatility of volatility expansion and Malliavin calculus techniques, we derive an accurate analytical formula for the price of vanilla options for any time dependent Heston model (the accuracy is less than a few bps for various strikes and maturities). In addition, we establish tight error estimates. The advantage of this approach over Fourier-based methods is its rapidity (gain by a factor 100 or more) while maintaining a competitive accuracy. From the approximative formula, we also derive some corollaries related first to equivalent Heston models (extending some work of Piterbarg on stochastic volatility models [V. Piterbarg, Risk Magazine, 18 (2005), pp. 71–75]) and second, to the calibration procedure in terms of ill-posed problems.

Portfolio Choice under Space-Time Monotone Performance Criteria

M. Musiela and T. Zariphopoulou

SIAM J. Finan. Math. 1, pp. 326-365 (40 pages)

Online Publication Date: May 26, 2010

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The class of time-decreasing forward performance processes is analyzed in a portfolio choice model of Itô-type asset dynamics. The associated optimal wealth and portfolio processes are explicitly constructed and their probabilistic properties are discussed. These formulae are, in turn, used in analyzing how the investor's preferences can be calibrated to the market, given his desired investment targets.

Merton Problem with Taxes: Characterization, Computation, and Approximation

Imen Ben Tahar, H. Mete Soner, and Nizar Touzi

SIAM J. Finan. Math. 1, pp. 366-395 (30 pages)

Online Publication Date: May 26, 2010

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We formulate a computationally tractable extension of the classical Merton optimal consumption-investment problem to include the capital gains taxes. This is the continuous-time version of the model introduced by Dammon, Spatt, and Zhang [Rev. Financ. Stud., 14 (2001), pp. 583–616]. In this model the tax basis is computed as the average cost of the stocks in the investor's portfolio. This average rule introduces only one additional state variable, namely the tax basis. Since the other tax rules such as the first in first out rule require the knowledge of all past transactions, the average model is computationally much easier. We emphasize the linear taxation rule, which allows for tax credits when capital gains losses are experienced. In this context wash sales are optimal, and we prove it rigorously. Our main contributions are a first order explicit approximation of the value function of the problem and a unique characterization by means of the corresponding dynamic programming equation. The latter characterization builds on technical results isolated in the accompanying paper [I. Ben Tahar, H. M. Soner, and N. Touzi, SIAM J. Control Optim., 46 (2007), pp. 1779–1801]. We also suggest a numerical computation technique based on a combination of finite differences and the Howard iteration algorithm. Finally, we provide some numerical results on the welfare consequences of taxes and the quality of the first order approximation.

Multivariate Extension of Put-Call Symmetry

Ilya Molchanov and Michael Schmutz

SIAM J. Finan. Math. 1, pp. 396-426 (31 pages)

Online Publication Date: May 26, 2010

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Multivariate analogues of the put-call symmetry can be expressed as certain symmetry properties of basket options and options on the maximum of several assets with respect to some (or all) permutations of the weights and the strike. The so-called self-dual distributions satisfying these symmetry conditions are completely characterized and their properties explored. It is also shown how to relate some multivariate asymmetric distributions to symmetric ones by a power transformation that is useful to adjust for carrying costs. Particular attention is devoted to the case of asset prices driven by Lévy processes. Based on this, semistatic hedging techniques for multiasset barrier options are suggested.

On the Microstructural Hedging Error

Christian Y. Robert and Mathieu Rosenbaum

SIAM J. Finan. Math. 1, pp. 427-453 (27 pages) | Cited 1 time

Online Publication Date: June 03, 2010

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We consider the issue of hedging a European derivative security in the presence of microstructure noise. In a market where the efficient price of the asset is driven by a stochastic volatility process, we assume an agent wants to use a (possibly misspecified) local volatility-type replication strategy. Focusing on microstructure noise effects, our goal is to evaluate the error between the theoretical, but practically unfeasible, strategy and its market adapted versions. The microstructural hedging error is in particular due to transaction price discreteness and endogenous trading times. Thus, we consider a transaction price model that accommodates such inherent properties of ultrahigh frequency data with the assumption of a continuous semimartingale efficient price. In this framework, we study two hedging strategies derived from the local volatility-type hedging strategy: (i) the hedging portfolio is rebalanced every time that the transaction price moves; (ii) the hedging portfolio is rebalanced only once the transaction price has varied by more than a selected value. To assess these strategies, we use an asymptotic approach where the number of rebalancing transactions goes to infinity. For the first strategy, we show that, because of microstructure noise effects, the hedging error does not vanish. However, an optimal strategy of the second type enables us to reduce it significantly.

Option Pricing in Hilbert Space-Valued Jump-Diffusion Models Using Partial Integro-Differential Equations

Peter Hepperger

SIAM J. Finan. Math. 1, pp. 454-489 (36 pages)

Online Publication Date: July 01, 2010

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Hilbert space-valued jump-diffusion models are employed for various markets and derivatives. Examples include swaptions, which depend on continuous forward curves, and basket options on stocks. Usually, no analytical pricing formulas are available for such products. Numerical methods, on the other hand, suffer from exponentially increasing computational effort with increasing dimension of the problem, the “curse of dimension.” In this paper, we present an efficient approach using partial integro-differential equations. The key to this method is a dimension reduction technique based on a Karhunen–Loève expansion, which is also known as proper orthogonal decomposition. Using the eigenvectors of a covariance operator, the differential equation is projected to a low-dimensional problem. Convergence results for the projection are given, and the numerical aspects of the implementation are discussed. An approximate solution is computed using a sparse grid combination technique and discontinuous Galerkin discretization. The main goal of this article is to combine the different analytical and numerical techniques needed, presenting a computationally feasible method for pricing European options. Numerical experiments show the effectiveness of the algorithm.

Optimal Trade Execution and Absence of Price Manipulations in Limit Order Book Models

Aurélien Alfonsi and Alexander Schied

SIAM J. Finan. Math. 1, pp. 490-522 (33 pages)

Online Publication Date: July 01, 2010

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We analyze the existence of price manipulation and optimal trade execution strategies in a model for an electronic limit order book with nonlinear price impact and exponential resilience. Our main results show that, under general conditions on the shape function of the limit order book, placing deterministic trade sizes at trading dates that are homogeneously spaced is optimal within a large class of adaptive strategies with arbitrary trading dates. This extends results from our earlier work with A. Fruth. Perhaps even more importantly, our analysis yields as a corollary that our model does not admit price manipulation strategies. This latter result contrasts the recent findings of Gatheral [Quant. Finance, to appear], where, in a related but different model, exponential resilience was found to give rise to price manipulation strategies when price impact is nonlinear.

Term Structure Models Driven by Wiener Processes and Poisson Measures: Existence and Positivity

Damir Filipović, Stefan Tappe, and Josef Teichmann

SIAM J. Finan. Math. 1, pp. 523-554 (32 pages)

Online Publication Date: July 01, 2010

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In the spirit of [T. Björk et al., Finance Stoch., 1 (1997), pp. 141–174], we investigate term structure models driven by Wiener processes and Poisson measures with forward curve dependent volatilities. This includes a full existence and uniqueness proof for the corresponding Heath–Jarrow–Morton-type term structure equation. Furthermore, we characterize positivity preserving models by means of the characteristic coefficients. A key role in our investigation is played by the method of the moving frame, which allows us to transform term structure equations to time-dependent SDEs.

Default Intensities Implied by CDO Spreads: Inversion Formula and Model Calibration

Rama Cont, Romain Deguest, and Yu Hang Kan

SIAM J. Finan. Math. 1, pp. 555-585 (31 pages) | Cited 1 time

Online Publication Date: July 08, 2010

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We propose a simple computational method for constructing an arbitrage-free collateralized debt obligation (CDO) pricing model which matches a prespecified set of CDO tranche spreads. The key ingredient of the method is an inversion formula for computing the aggregate default rate in a portfolio, as a function of the number of defaults, from its expected tranche notionals. This formula can be seen as an analogue of the Dupire formula for portfolio credit derivatives. Together with a quadratic programming method for recovering expected tranche notionals from CDO spreads, our inversion formula leads to an efficient nonparametric method for calibrating CDO pricing models. Contrarily to the base correlation method, our method yields an arbitrage-free model. Comparing this approach to other calibration methods, we find that model-dependent quantities such as the forward starting tranche spreads and jump-to-default ratios are quite sensitive to the calibration method used, even within the same model class. On the other hand, comparing the local intensity functions implied by different credit portfolio models reveals that apparently different models, such as the static Student-t copula models and the reduced-form affine jump-diffusion models, lead to similar marginal loss distributions and tranche spreads.

Path-Dependence of Leveraged ETF Returns

Marco Avellaneda and Stanley Zhang

SIAM J. Finan. Math. 1, pp. 586-603 (18 pages)

Online Publication Date: July 08, 2010

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It is well known that leveraged exchange-traded funds (LETFs) do not reproduce the corresponding multiple of index returns over extended (quarterly or annual) investment horizons. For instance, in 2008 and early 2009, most LETFs underperformed the corresponding static strategies. In this paper, we study this phenomenon in detail. We give an exact formula linking the return of a leveraged fund with the corresponding multiple of the return of the unleveraged fund and its realized variance. This formula is tested empirically over quarterly horizons for 56 leveraged funds (44 double-leveraged and 12 triple-leveraged) using daily prices since January 2008 or since inception, according to the fund considered. The results indicate excellent agreement between the formula and the empirical data. The study also shows that leveraged funds can be used to replicate the returns of the underlying index, provided we use a dynamic rebalancing strategy. Empirically, we find that rebalancing frequencies required to achieve this goal are moderate—on the order of one week between rebalancings. Nevertheless, this need for dynamic rebalancing leads to the conclusion that LETFs as currently designed may be unsuitable for buy-and-hold investors.

Dual Valuation and Hedging of Bermudan Options

L. C. G. Rogers

SIAM J. Finan. Math. 1, pp. 604-608 (5 pages)

Online Publication Date: July 15, 2010

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Some years ago, a different characterization of the value of a Bermudan option was discovered which can be thought of as the viewpoint of the seller of the option, in contrast to the conventional characterization which took the viewpoint of the buyer. Since then, there has been a lot of interest in finding numerical methods which exploit this dual characterization. This paper presents a pure dual algorithm for pricing and hedging Bermudan options.

Asymptotic Formulas with Error Estimates for Call Pricing Functions and the Implied Volatility at Extreme Strikes

Archil Gulisashvili

SIAM J. Finan. Math. 1, pp. 609-641 (33 pages)

Online Publication Date: August 17, 2010

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In this paper, we obtain asymptotic formulas with error estimates for the implied volatility associated with a European call pricing function. We show that these formulas imply Lee's moment formulas for the implied volatility and the tail-wing formulas due to Benaim and Friz. In addition, we analyze Pareto-type tails of stock price distributions in uncorrelated Hull–White, Stein–Stein, and Heston models and find asymptotic formulas with error estimates for call pricing functions in these models.

Affine Point Processes and Portfolio Credit Risk

Eymen Errais, Kay Giesecke, and Lisa R. Goldberg

SIAM J. Finan. Math. 1, pp. 642-665 (24 pages) | Cited 1 time

Online Publication Date: September 16, 2010

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This paper analyzes a family of multivariate point process models of correlated event timing whose arrival intensity is driven by an affine jump diffusion. The components of an affine point process are self- and cross-exciting and facilitate the description of complex event dependence structures. ODEs characterize the transform of an affine point process and the probability distribution of an integer-valued affine point process. The moments of an affine point process take a closed form. This guarantees a high degree of computational tractability in applications. We illustrate this in the context of portfolio credit risk, where the correlation of corporate defaults is the main issue. We consider the valuation of securities exposed to correlated default risk and demonstrate the significance of our results through market calibration experiments. We show that a simple model variant can capture the default clustering implied by index and tranche market prices during September 2008, a month that witnessed significant volatility.

Real Options Games in Complete and Incomplete Markets with Several Decision Makers

Alain Bensoussan, J. David Diltz, and SingRu Hoe

SIAM J. Finan. Math. 1, pp. 666-728 (63 pages)

Online Publication Date: September 29, 2010

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We consider optimal investment policies for irreversible capital investment projects under uncertainty in a monopoly situation and in a Stackelberg leader-follower game. We consider two types of payoffs: lump-sum and cash flows. The decisions are the times to enter into the market. The problems belong to the class of optimal stopping times, for which the right approach is that of variational inequalities (V.I.s). In the case of complete markets, payoffs are expected values with respect to the risk-neutral probability. In the case of incomplete markets, the risk-neutral probability is not defined. We consider an investor maximizing his/her utility function, and we consider the investment in the project as an additional decision, besides portfolio investment and consumption decisions. This decision remains a stopping time, conversely to the portfolio investment and consumption decisions (continuous controls). The game problem raises new difficulties. The leader's V.I. has a nondifferentiable obstacle. The weak formulation of the V.I. handles this difficulty. In some cases, the solution of the V.I. may be continuously differentiable although the obstacle is not. An additional difficulty occurs for lump-sum payoffs in the case of incomplete markets. We cannot compare gains and losses at different times. We propose an alternative approach, using equivalence (indifference) considerations. In the case of payoffs characterized by cash flows, this difficulty does not exist, but an intermediary problem arises which has a nice interpretation as a differential game. The solutions thus obtained for the Stackelberg game are not intuitive. Therefore, competition has important consequences on investment decisions.

Storage Costs in Commodity Option Pricing

Juri Hinz and Max Fehr

SIAM J. Finan. Math. 1, pp. 729-751 (23 pages)

Online Publication Date: October 12, 2010

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Unlike derivatives of financial contracts, commodity options exhibit distinct particularities owing to physical aspects of the underlying. An adaptation of no-arbitrage pricing to this kind of derivative turns out to be a stress test, challenging the martingale-based models with diverse technical and technological constraints, with storability and short selling restrictions, and sometimes with the lack of an efficient dynamic hedging. In this work, we study the effect of storability on risk neutral commodity price modeling and suggest a model class where arbitrage is excluded for both commodity futures trading and simultaneous dynamical management of the commodity stock. The proposed framework is based on key results from interest rate theory.

Optimal Allocation of a Futures Portfolio Utilizing Numerical Market Phase Detection

L. Putzig, D. Becherer, and I. Horenko

SIAM J. Finan. Math. 1, pp. 752-779 (28 pages)

Online Publication Date: October 21, 2010

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This paper presents an application of the recently developed method for simultaneous dimension reduction and metastability analysis of high-dimensional time series in the context of computational finance. Further extensions are included to combine state-specific principal component analysis (PCA) and state-specific regressive trend models to handle the high-dimensional, nonstationary data. The identification of market phases allows one to control the involved phase-specific risk for futures portfolios. The numerical optimization strategy for futures portfolios based on Tikhonov-type regularization is presented. The application of proposed strategies to online detection of the market phases is exemplified first on the simulated data and then on historical futures prices for oil and wheat from 2005–2008. Numerical tests demonstrate the comparison of the presented methods with existing approaches.

Trend Following Trading under a Regime Switching Model

M. Dai, Q. Zhang, and Q. J. Zhu

SIAM J. Finan. Math. 1, pp. 780-810 (31 pages)

Online Publication Date: October 21, 2010

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This paper is concerned with the optimality of a trend following trading rule. The idea is to catch a bull market at its early stage, ride the trend, and liquidate the position at the first evidence of the subsequent bear market. We characterize the bull and bear phases of the markets mathematically using the conditional probabilities of the bull market given the up to date stock prices. The optimal buying and selling times are given in terms of a sequence of stopping times determined by two threshold curves. Numerical experiments are conducted to validate the theoretical results and demonstrate how they perform in a marketplace.

Representations for Optimal Stopping under Dynamic Monetary Utility Functionals

Volker Krätschmer and John Schoenmakers

SIAM J. Finan. Math. 1, pp. 811-832 (22 pages)

Online Publication Date: October 21, 2010

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In this paper we consider the optimal stopping problem for general dynamic monetary utility functionals. Sufficient conditions for the Bellman principle and the existence of optimal stopping times are provided. Particular attention is paid to representations which allow for a numerical treatment in real situations. To this aim, generalizations of standard evaluation methods like policy iteration and dual and consumption based approaches are developed in the context of general dynamic monetary utility functionals. As a result, it turns out that the possibility of a particular generalization depends on specific properties of the utility functional under consideration.

Parametrix Approximation of Diffusion Transition Densities

Francesco Corielli, Paolo Foschi, and Andrea Pascucci

SIAM J. Finan. Math. 1, pp. 833-867 (35 pages)

Online Publication Date: November 11, 2010

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A new analytical approximation tool, derived from the classical PDE theory, is introduced in order to build approximate transition densities of diffusions. The tool is useful for approximate pricing and hedging of financial derivatives and for maximum likelihood and method of moments estimates of diffusion parameters. The approximation is uniform with respect to time and space variables. Moreover, easily computable error bounds are available in any dimension.

Exact and Efficient Simulation of Correlated Defaults

K. Giesecke, H. Kakavand, M. Mousavi, and H. Takada

SIAM J. Finan. Math. 1, pp. 868-896 (29 pages)

Online Publication Date: November 30, 2010

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Correlated default risk plays a significant role in financial markets. Dynamic intensity-based models, in which a firm default is governed by a stochastic intensity process, are widely used to model correlated default risk. The computations in these models can be performed by Monte Carlo simulation. The standard simulation method, which requires the discretization of the intensity process, leads to biased simulation estimators. The magnitude of the bias is often hard to quantify. This paper develops an exact simulation method for intensity-based models that leads to unbiased estimators of credit portfolio loss distributions, risk measures, and derivatives prices. In a first step, we construct a Markov chain that matches the marginal distribution of the point process describing the binary default state of each firm. This construction reduces the original estimation problem to one involving a Markov chain expectation. In a second step, we estimate the Markov chain expectation using a simple acceptance/rejection scheme that facilitates exact sampling. To address rare event situations, the acceptance/rejection scheme is embedded in an overarching selection/mutation scheme, in which a selection mechanism adaptively forces the chain into the regime of interest. Numerical experiments demonstrate the effectiveness of the method for a self-exciting model of correlated default risk.

Optimal Portfolio Liquidation with Execution Cost and Risk

Idris Kharroubi and Huyên Pham

SIAM J. Finan. Math. 1, pp. 897-931 (35 pages)

Online Publication Date: December 07, 2010

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We study the optimal portfolio liquidation problem over a finite horizon in a limit order book with bid-ask spread and temporary market price impact penalizing speedy execution trades. We use a continuous-time modeling framework, but in contrast with previous related papers (see, e.g., [L. C. G. Rogers and S. Singh, Math. Finance, 20 (2010), pp. 597–615] and [A. Schied and T. Schöneborn, Finance Stoch., 13 (2009), pp. 181–204]), we do not assume continuous-time trading strategies. We consider instead real trading that occur in discrete time, and this is formulated as an impulse control problem under a solvency constraint, including the lag variable tracking the time interval between trades. A first important result of our paper is to prove rigorously that nearly optimal execution strategies in this context actually lead to a finite number of trades with strictly increasing trading times, and this holds true without assuming ad hoc any fixed transaction fee. Next, we derive the dynamic programming quasi-variational inequality satisfied by the value function in the sense of constrained viscosity solutions. We also introduce a family of value functions which converges to our value function and is characterized as the unique constrained viscosity solutions of an approximation of our dynamic programming equation. This convergence result is useful for numerical purpose but is postponed until a companion paper [F. Guilbaud, M. Mnif, and H. Pham, Numerical Methods for an Optimal Order Execution Problem, preprint, 2010].

Portfolio Selection Using Tikhonov Filtering to Estimate the Covariance Matrix

Sungwoo Park and Dianne P. O'Leary

SIAM J. Finan. Math. 1, pp. 932-961 (30 pages)

Online Publication Date: December 14, 2010

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Markowitz's portfolio selection problem chooses weights for stocks in a portfolio based on an estimated covariance matrix of stock returns. Our study proposes reducing noise in the estimation by using a Tikhonov filter function. In addition, we prevent rank deficiency of the estimated covariance matrix and propose a method for effectively choosing the Tikhonov parameter, which determines the filtering intensity. We put previous estimators into a common framework and compare their filtering functions for eigenvalues of the correlation matrix. We demonstrate the effectiveness of our estimator using stock return data from 1958 through 2007.
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