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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.
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