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ABSTRACT: We solve the problem of valuing and optimal exercise of American call-type options in markets which do not necessarily admit
an equivalent local martingale measure. This resolves an open question proposed by Karatzas and Fernholz (Handbook of Numerical
Analysis, vol.15, pp.89–167, Elsevier, Amsterdam, 2009).
KeywordsStrict local martingales–Deflators–American call options
Finance and Stochastics 04/2012; 16(2):275-291. · 1.20 Impact Factor
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ABSTRACT: We approximate the price of the American put for jump diffusions by a sequence of functions, which are computed iteratively.
This sequence converges to the price function uniformly and exponentially fast. Each element of the approximating sequence
solves an optimal stopping problem for geometric Brownian motion, and can be numerically computed using the classical finite
difference methods. We prove the convergence of this numerical scheme and present examples to illustrate its performance.
Mathematical Methods of Operational Research 04/2012; 70(3):505-525. · 0.48 Impact Factor
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ABSTRACT: We construct a sequence of functions that uniformly converge (on compact sets) to the price of an Asian option, which is written on a stock whose dynamics follow a jump diffusion. The convergence is exponentially fast. We show that each element in this sequence is the unique classical solution of a parabolic partial differential equation (not an integro-differential equation). As a result we obtain a fast numerical approximation scheme whose accuracy versus speed characteristics can be controlled. We analyze the performance of our numerical algorithm on several examples.
Mathematical Finance 12/2010; 21(1):117 - 143. · 1.25 Impact Factor
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ABSTRACT: 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.
04/2010;
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ABSTRACT: Calibrating option pricing models to market prices often leads to optimisation problems to which standard methods (like such based on gradients) cannot be applied. We investigate two models: Heston’s stochastic volatility model, and Bates’s model which also includes jumps. We discuss how to price options under these models, and how to calibrate the parameters of the models with heuristic techniques.
arXiv.org, Quantitative Finance Papers. 01/2010;
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ABSTRACT: We solve the problem of pricing and optimal exercise of American call-type options in markets which do not necessarily admit an equivalent local martingale measure. This resolves an open question proposed by Fernholz and Karatzas [Stochastic Portfolio Theory: A Survey, Handbook of Numerical Analysis, 15:89-168, 2009]. Comment: Key Words: Strict local martingales, deflators, American call options
08/2009;
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ABSTRACT: The value function of an optimal stopping problem for a process with L\'{e}vy jumps is known to be a generalized solution of a variational inequality. Assuming the diffusion component of the process is nondegenerate and a mild assumption on the singularity of the L\'{e}vy measure, this paper shows that the value function of obstacle problems on an unbounded domain with finite/infinite variation jumps is in $W^{2,1}_{p, loc}$. As a consequence, the smooth-fit property holds.
arXiv.org, Quantitative Finance Papers. 01/2009;
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ABSTRACT: Given that the terminal condition is of at most linear growth, it is well known that a Cauchy problem admits a unique classical solution when the coefficient multiplying the second derivative (i.e., the volatility) is also a function of at most linear growth. In this note, we give a condition on the volatility that is necessary and sufficient for a Cauchy problem to admit a unique solution.
arXiv.org, Quantitative Finance Papers. 01/2009;
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ABSTRACT: In this paper we show that the optimal exercise boundary / free boundary of the American put option pricing problem for jump diffusions is continuously differentiable (except at the maturity). This differentiability result has been established by Yang et al. (European Journal of Applied Mathematics, 17(1):95-127, 2006) in the case where the condition $r\geq q+ \lambda \int_{\R_+} \left(e^z-1\right) \nu(dz)$ is satisfied. We extend the result to the case where the condition fails using a unified approach that treats both cases simultaneously. We also show that the boundary is infinitely differentiable under a regularity assumption on the jump distribution.
01/2008;
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ABSTRACT: We construct a sequence of functions that uniformly converge (on compact sets) to the price of Asian option, which is written on a stock whose dynamics follows a jump diffusion, exponentially fast. Each of the element in this sequence solves a parabolic partial differen- tial equation (not an integro-differential equation). As a result we obtain a fast numerical approximation scheme whose accuracy versus speed characteristics can be controlled. We analyze the performance of our numerical algorithm on several examples.
08/2007;
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CoRR. 01/2007; abs/0706.2331.
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ABSTRACT: We develop an efficient method for pricing American options for jump diffusion models. The price function is approximated by a sequence of functions, each of which is the solution of an optimal stopping problem for diffusion. The convergence of this sequence is uniform and exponentially fast. We present the numerical performance of our algorithm and compare it with the performance of other methods in the literature.
