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SIAM J. on Financial Mathematics

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2012

Volume 3 (partial)


Time-Consistent Portfolio Management

Ivar Ekeland, Oumar Mbodji, and Traian A. Pirvu

SIAM J. Finan. Math. 3, pp. 1-32 (32 pages)

Online Publication Date: January 03, 2012

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This paper considers the portfolio management problem for an investor with finite time horizon who is allowed to consume and take out life insurance. Natural assumptions, such as different discount rates for consumption and life insurance, lead to time inconsistency. This situation can also arise when the investor is in fact a group, the members of which have different utilities and/or different discount rates. As a consequence, the optimal strategies are not implementable. We focus on hyperbolic discounting, which has received much attention lately, especially in the area of behavioral finance. Following [I. Ekeland and T. A. Pirvu, Math. Financ. Econ., 2 (2008), pp. 57–86], we consider the resulting problem as a leader-follower game between successive selves, each of whom can commit for an infinitesimally small amount of time. We then define policies as subgame perfect equilibrium strategies. Policies are characterized by an integral equation which is shown to have a solution in the case of constant relative risk aversion utilities. Our results can be extended for more general preferences as long as the equations admit solutions. Numerical simulations reveal that for the Merton problem with hyperbolic discounting, the consumption increases up to a certain time, after which it decreases; this pattern does not occur in the case of exponential discounting and is therefore known in the literature as the “consumption puzzle.” Other numerical experiments explore the effect of time varying aggregation rate on the insurance premium.

The Small-Maturity Smile for Exponential Lévy Models

José E. Figueroa-López and Martin Forde

SIAM J. Finan. Math. 3, pp. 33-65 (33 pages)

Online Publication Date: January 17, 2012

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We derive a small-time expansion for out-of-the-money call options under an exponential Lévy model, using the small-time expansion for the distribution function given in [J. Figueroa-López and C. Houdré, Stochastic Process. Appl., 119 (2009), pp. 3862–3889], combined with a change of numéraire via the Esscher transform. In particular, we find that the effect of a nonzero volatility $\sigma$ of the Gaussian component of the driving Lévy process is to increase the call price by $\frac{1}{2}\sigma^2 t^2 e^{k}\nu(k)(1+o(1))$ as $t \to 0$, where $\nu$ is the Lévy density. Using the small-time expansion for call options, we then derive a small-time expansion for the implied volatility $\hat{\sigma}_{t}^{2}(k)$ at log-moneyness $k$, which sharpens the first order estimate $\hat{\sigma}_{t}^{2}(k)\sim \frac{\frac{1}{2}k^2}{t\log (1/t)}$ given in [P. Tankov, Pricing and hedging in exponential Lévy models: Review of recent results, in Paris-Princeton Lectures on Mathematical Finance, Springer, Berlin, 2011, pp. 319–359]. Our numerical results show that the second order approximation can significantly outperform the first order approximation. Our results are also extended to a class of time-changed Lévy models. We also consider a small-time, small log-moneyness regime for the CGMY model and apply this approach to the small-time pricing of at-the-money call options; we show that for $Y\in(1,2)$, $\lim_{t\to{}0}t^{-1/Y}\mathbb{E}(S_t-S_0)_{+}=S_{0}\mathbb{E}^{*}(Z_{+})$ and the corresponding at-the-money implied volatility $\hat{\sigma}_t(0)$ satisfies $\lim_{t \to 0}\hat{\sigma}_t(0)/t^{1/Y-1/2}=\sqrt{2\pi}\,\mathbb{E}^{*}(Z_{+})$, where $Z$ is a symmetric $Y$-stable random variable under $\mathbb{P}^*$ and $Y$ is the usual parameter for the CGMY model appearing in the Lévy density $\nu(x)=C x^{-1-Y}e^{-M x}{\bf 1}_{\{x>0\}}+C |x|^{-1-Y}e^{-G|x|}{\bf 1}_{\{x<0\}}$ of the process.

Option Pricing in Multivariate Stochastic Volatility Models of OU Type

Johannes Muhle-Karbe, Oliver Pfaffel, and Robert Stelzer

SIAM J. Finan. Math. 3, pp. 66-94 (29 pages)

Online Publication Date: January 17, 2012

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We present a multivariate stochastic volatility model with leverage, which is flexible enough to recapture the individual dynamics as well as the interdependencies between several assets, while still being highly analytically tractable. First, we derive the characteristic function and give conditions that ensure its analyticity and absolute integrability in some open complex strip around zero. Therefore we can use Fourier methods to compute the prices of multiasset options efficiently. To show the applicability of our results, we propose a concrete specification, the Ornstein–Uhlenbeck (OU)–Wishart model, where the dynamics of each individual asset coincide with the popular $\Gamma$-OU Barndorff-Nielsen–Shepard model. This model can be well calibrated to market prices, which we illustrate with an example using options on the exchange rates of some major currencies. Finally, we show that covariance swaps can also be priced in closed form.

An Asymptotic Expansion with Push-Down of Malliavin Weights

Akihiko Takahashi and Toshihiro Yamada

SIAM J. Finan. Math. 3, pp. 95-136 (42 pages)

Online Publication Date: January 24, 2012

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This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in multidimensional stochastic volatility models. In particular, the integration-by-parts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied. We provide an expansion formula for generalized Wiener functionals and closed-form approximation formulas in the stochastic volatility environment. In addition, we present applications of the general formula to expansions of option prices for the shifted log-normal model with stochastic volatility. Moreover, with some results of Malliavin calculus in jump-type models, we derive an approximation formula for the jump-diffusion model in the stochastic volatility environment. Some numerical examples are also shown.

Quadratic Transform Approximation for CDO Pricing in Multifactor Models

Paul Glasserman and Sira Suchintabandid

SIAM J. Finan. Math. 3, pp. 137-162 (26 pages)

Online Publication Date: January 26, 2012

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The multifactor version of copula models has the ability to generate complex correlation structure among defaults that is useful in fitting the base correlation skew. However, multifactor models have often been dismissed for their intractability. Even the semianalytical approach using Laplace transforms is computationally challenging, because although the model is tractable upon conditioning on the factors, unconditioning usually requires high efforts of integrating out the factors. To circumvent this problem, this paper develops a fast, closed-form approximation to the Laplace transform in multifactor models. The method, which approximates the conditional transform in a way that lends itself to closed-form unconditioning in arbitrarily high dimensions, is applicable to a range of models with Gaussian factors, including models that extend the standard Gaussian copula to allow stochastic recovery rates and factor loadings. We analyze the accuracy and convergence properties of the approximation. Numerical examples illustrate the speed and accuracy of the method.

Optimal Trading with Stochastic Liquidity and Volatility

Robert Almgren

SIAM J. Finan. Math. 3, pp. 163-181 (19 pages)

Online Publication Date: January 31, 2012

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We consider the problem of mean-variance optimal agency execution strategies, when the market liquidity and volatility vary randomly in time. Under specific assumptions for the stochastic processes satisfied by these parameters, we construct a Hamilton–Jacobi–Bellman equation for the optimal cost and strategy. We solve this equation numerically and illustrate optimal strategies for varying risk aversion. These strategies adapt optimally to the instantaneous variations of market quality.

Explicit Constructions of Martingales Calibrated to Given Implied Volatility Smiles

Peter Carr and Laurent Cousot

SIAM J. Finan. Math. 3, pp. 182-214 (33 pages)

Online Publication Date: January 31, 2012

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The construction of martingales with given marginal distributions at given times is a recurrent problem in financial mathematics. From a theoretical point of view, this problem is well known, as necessary and sufficient conditions for the existence of such martingales have been described. Moreover, several explicit constructions can even be derived from solutions to the Skorokhod embedding problem. However, these solutions have not been adopted by practitioners, who still prefer to construct the whole implied volatility surface and use the explicit constructions of calibrated (jump-) diffusions, available in the literature, when a continuum of marginal distributions is known. In this paper, we describe several new constructions of calibrated martingales, which do not rely on a potentially risky interpolation of the marginal distributions but only on the input marginal distributions. These calibrated martingales are intuitive since the continuous-time versions of our constructions can be interpreted as time-changed (jump-) diffusions. Moreover, we show that the valuation of claims, depending only on the values of the underlying process at maturities where the marginal distributions are known, can be extremely efficient in this setting. For example, path-independent claims of this type can be valued by solving a finite number of ordinary (integro-) differential equations. Finally, an example of calibration to the S&P 500 market is provided.

Asymptotic Approximations for Asian, European, and American Options with Discrete Averaging or Discrete Dividend/Coupon Payments

Sam Howison

SIAM J. Finan. Math. 3, pp. 215-241 (27 pages)

Online Publication Date: March 06, 2012

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We develop approximations to the pricing of options on an asset which makes discrete dividend payments, focusing on the case of frequent payments. The principal mathematical tool is the method of multiple timescales, allied to matched asymptotic expansions. We first analyze European style options, deriving the continuously paid dividend equation as a limiting case of the relevant discrete problem, and we analyze the range accrual note to compute the relevant “continuity correction.” We also carry out the same analysis for Asian options with discrete averaging. We then give a detailed description of the intricate exercise policies that arise for American put (and, to a lesser extent, call) options when dividends are paid discretely, for the cases of proportionate and fixed-amount dividends.

Weak Insider Trading and Behavioral Finance

L. Campi and M. Del Vigna

SIAM J. Finan. Math. 3, pp. 242-279 (38 pages)

Online Publication Date: March 22, 2012

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In this paper, we study the optimal portfolio selection problem for weakly informed traders in the sense of Baudoin [Stochastic Process. Appl., 100 (2002), pp. 109–145]. Apart from expected utility maximizers, we consider investors with other preference paradigms. In particular, we consider agents following cumulative prospect theory as developed by Tversky and Kahneman [J. Risk Uncertainty, 5 (1992), pp. 297–323] as well as Yaari's dual theory of choice [Econometrica, 55 (1987), pp. 95–115]. We solve the corresponding optimization problems, in both noninformed and informed case, i.e., when the agent has an additional weak information. Finally, comparison results among investors with different preferences and information sets are given, together with explicit examples. In particular, the insider's gain, i.e., the difference between the optimal values of an informed and a noninformed investor, is explicitly computed.

Processes of Class Sigma, Last Passage Times, and Drawdowns

Patrick Cheridito, Ashkan Nikeghbali, and Eckhard Platen

SIAM J. Finan. Math. 3, pp. 280-303 (24 pages)

Online Publication Date: April 03, 2012

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We propose a general framework for studying last passage times, suprema, and drawdowns of a large class of continuous-time stochastic processes. Our approach is based on processes of class Sigma and the more general concept of two processes, one of which moves only when the other is at the origin. After investigating certain transformations of such processes and their convergence properties, we provide three general representation results. The first allows the recovery of a process of class Sigma from its final value and the last time it visited the origin. In many situations this gives access to the distribution of the last time a stochastic process attains a certain level or is equal to its running maximum. It also leads to recently discovered formulas expressing option prices in terms of last passage times. Our second representation result is a stochastic integral representation that will allow us to price and hedge options on the running maximum of an underlying that are triggered when the underlying drops to a given level or, alternatively, when the drawdown or relative drawdown of the underlying attains a given height. The third representation gives conditional expectations of certain functionals of processes of class Sigma. It can be used to deduce the distributions of a variety of interesting random variables such as running maxima, drawdowns, and maximum drawdowns of suitably stopped processes.

The Relaxed Investor with Partial Information

Nicole Bäuerle, Sebastian P. Urban, and Luitgard A. M. Veraart

SIAM J. Finan. Math. 3, pp. 304-327 (24 pages)

Online Publication Date: April 10, 2012

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We consider an investor in a financial market consisting of a riskless bond and several risky assets. The price processes of the risky assets are geometric Brownian motions where either the drifts are modeled as random variables assuming a constant volatility matrix or the volatility matrix is considered random and drifts are assumed to be constant. The investor is only able to observe the asset prices but not all the model parameters and hence information is only partial. A Bayesian approach is used with known prior distributions for the random model parameters. We assume that the investor can only trade at discrete-time points which are multiples of $h>0$ and investigate the loss in expected utility of terminal wealth which is due to the fact that the investor cannot trade and observe continuously. It turns out that in general a discretization gap appears, i.e., for $h\to 0$ the expected utility of the $h$-investor does not converge to the expected utility of the continuous investor. This is in contrast to results under full information in [L.C.G. Rogers, Finance Stoch., 5(2001), pp. 131–154]. We also present simple asymptotically optimal portfolio strategies for the discrete-time problem. Our results are illustrated by some numerical examples.

Pricing and Hedging in Affine Models with Possibility of Default

Patrick Cheridito and Alexander Wugalter

SIAM J. Finan. Math. 3, pp. 328-350 (23 pages)

Online Publication Date: April 10, 2012

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We propose a general framework for the simultaneous modeling of equity, government bonds, corporate bonds, and derivatives. Uncertainty is generated by a general affine Markov process. The setting allows for stochastic volatility, jumps, the possibility of default, and correlation between different assets. We show how to calculate discounted complex moments by solving a coupled system of generalized Riccati equations. This yields an efficient method to compute prices of power payoffs. European calls and puts as well as binaries and asset-or-nothing options can be priced with the fast Fourier transform methods of Carr and Madan [J. Comput. Finance, 2 (1999), pp. 61–73] and Lee [J. Comput. Finance, 7 (2005), pp. 51–86]. Other European payoffs can be approximated with a linear combination of government bonds, power payoffs, and vanilla options. We show the results to be superior to using only government bonds and power payoffs or government bonds and vanilla options. We also give conditions for European continent claims in our framework to be replicable if enough financial instruments are liquidly tradable and study dynamic hedging strategies. As an example we discuss a Heston-type stochastic volatility model with possibility of default and stochastic interest rates.

Valuation Equations for Stochastic Volatility Models

Erhan Bayraktar, Constantinos Kardaras, and Hao Xing

SIAM J. Finan. Math. 3, pp. 351-373 (23 pages)

Online Publication Date: April 17, 2012

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We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the diffusion coefficients may grow faster than linearly and degenerate on the boundaries of the state space. We allow for various types of model behavior: the volatility process in our model can potentially reach zero and either stay there or instantaneously reflect, and the asset-price process may be a strict local martingale. Our main result is a necessary and sufficient condition on the uniqueness of classical solutions to the valuation equation: the value function is the unique nonnegative classical solution to the valuation equation among functions with at most linear growth if and only if the asset price is a martingale.

Estimation of Asset Distributions from Option Prices: Analysis and Regularization

J. Orozco Rodriguez and F. Santosa

SIAM J. Finan. Math. 3, pp. 374-401 (28 pages)

Online Publication Date: May 17, 2012

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We analyze the stability of a method for estimating the risk-neutral density (RND) for the price of an asset from option prices. The method first applies the principle of maximum entropy, where the maximum entropy solution (MES) corresponds to the estimated RND. Next, it provides an effective characterization of the constraint qualification (CQ) under which the MES can be computed by solving the dual problem, where an explicit function in finitely many variables is minimized. In our analysis, we show that the MES is stable under parameter perturbation, but the parameters are unstable under data perturbation. When noisy data are used, we show how to project the data so that the CQ is satisfied and the method can be used. To stabilize the method, we use Tikhonov regularization and choose the penalty parameter via the L-curve method. We demonstrate with numerical examples that the method then becomes much more stable to perturbation in data. Accordingly, we perform a convergence analysis of the regularized solution.
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