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Register a new userExponential functionals of Levy processes and variable annuity guaranteed benefits
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Exponential functionals of Brownian motion have been extensively studied in financial and insurance mathematics due to their broad applications, for example, in the pricing of Asian options. The Black-Scholes model is appealing because of mathematical tractability, yet empirical evidence shows that geometric Brownian motion does not adequately capture features of market equity returns. One popular alternative for modeling equity returns consists in replacing the geometric Brownian motion by an exponential of a Levy process. In this paper we use this latter model to study variable annuity guaranteed benefits and to compute explicitly the distribution of certain exponential functionals.
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In this paper, we are concerned with the valuation of Guaranteed Annuity Options (GAOs) under the most generalised modelling framework where both interest and mortality rates are stochastic and correlated. Pricing these type of options in the correlated environment is a challenging task and no closed form solution exists in the literature. We employ the use of doubly stochastic stopping times to incorporate the randomness about the time of death and employ a suitable change of measure to facilitate the valuation of survival benefit, there by adapting the payoff of the GAO in terms of the payoff of a basket call option. We derive general price bounds for GAOs by utilizing a conditioning approach for the lower bound and arithmetic-geometric mean inequality for the upper bound. The theory is then applied to affine models to present some very interesting formulae for the bounds under the affine set up. Numerical examples are furnished and benchmarked against Monte Carlo simulations to estimate the price of a GAO for a variety of affine processes governing the evolution of mortality and the interest rate.
We consider a stochastic volatility model with Levy jumps for a log-return process $Z=(Z_{t})_{tgeq 0}$ of the form $Z=U+X$, where $U=(U_{t})_{tgeq 0}$ is a classical stochastic volatility process and $X=(X_{t})_{tgeq 0}$ is an independent Levy process with absolutely continuous Levy measure $ u$. Small-time expansions, of arbitrary polynomial order, in time-$t$, are obtained for the tails $bbp(Z_{t}geq z)$, $z>0$, and for the call-option prices $bbe(e^{z+Z_{t}}-1)_{+}$, $z eq 0$, assuming smoothness conditions on the {PaleGrey density of $ u$} away from the origin and a small-time large deviation principle on $U$. Our approach allows for a unified treatment of general payoff functions of the form $phi(x){bf 1}_{xgeq{}z}$ for smooth functions $phi$ and $z>0$. As a consequence of our tail expansions, the polynomial expansions in $t$ of the transition densities $f_{t}$ are also {Green obtained} under mild conditions.
We introduce a new class of processes for the evaluation of multivariate equity derivatives. The proposed setting is well suited for the application of the standard copula function theory to processes, rather than variables, and easily enables to enforce the martingale pricing requirement. The martingale condition is imposed in a general multidimensional Markov setting to which we only add the restriction of no-Granger-causality of the increments (Granger-independent increments). We call this class of processes GIMP (Granger Independent Martingale Processes). The approach can also be extended to the application of time change, under which the martingale restriction continues to hold. Moreover, we show that the class of GIMP processes is closed under time changing: if a Granger independent process is used as a multivariate stochastic clock for the change of time of a GIMP process, the new process is also GIMP.
A financial market model where agents trade using realistic combinations of buy-and-hold strategies is considered. Minimal assumptions are made on the discounted asset-price process - in particular, the semimartingale property is not assumed. Via a natural market viability assumption, namely, absence of arbitrages of the first kind, we establish that discounted asset-prices have to be semimartingales. In a slightly more specialized case, we extend the previous result in a weakened version of the Fundamental Theorem of Asset Pricing that involves strictly positive supermartingale deflators rather than Equivalent Martingale Measures.
This paper considers exponential utility indifference pricing for a multidimensional non-traded assets model subject to inter-temporal default risk, and provides a semigroup approximation for the utility indifference price. The key tool is the splitting method, whose convergence is proved based on the Barles-Souganidis monotone scheme, and the convergence rate is derived based on Krylovs shaking the coefficients technique. We apply our methodology to study the counterparty risk of derivatives in incomplete markets.
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