The density process of the minimal entropy martingale measure in a stochastic volatility market. A PDE Approach

Kufakunesu, Rodwell

doi:10.2989/16073606.2011.594229

Cited by 3 publications

(11 citation statements)

References 16 publications

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“…The former option is not really feasible in the case of an evolution equation about which nothing is known apart from the equation itself. 4 It tends to be rather problematic even when one is dealing with an ordinary differential equation [25,26].…”

Section: )mentioning

confidence: 99%

“…In [3] α, β, ρ and ξ are taken as constants whereas Kufakunesu [4] takes them to have an explicit dependence upon the time.…”

Section: Introductionmentioning

confidence: 99%

“…The solution was provided in two instances. The first was the autonomous problem presented in [3] and the second was a nonautonomous version of the same problem introduced in [4].…”

Section: Introductionmentioning

confidence: 99%

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Symmetry analysis of a model of stochastic volatility with time-dependent parameters

Sophocleous

O’Hara

2011

Journal of Computational and Applied Mathematics

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a b s t r a c tWe provide the solutions for the Heston model of stochastic volatility when the parameters of the model are constant and when they are functions of time. In the former case, the solution follows immediately from the determination of the Lie point symmetries of the governing 1 + 1 evolution partial differential equation. This is not the situation in the latter case, but we are able to infer the essential structure of the required nonlocal symmetry from that of the autonomous problem and hence can present the solution to the nonautonomous problem. As in the case of the standard Black-Scholes problem the presence of timedependent parameters is not a hindrance to the demonstration of a solution.

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Section: )mentioning

confidence: 99%

“…In [3] α, β, ρ and ξ are taken as constants whereas Kufakunesu [4] takes them to have an explicit dependence upon the time.…”

Section: Introductionmentioning

confidence: 99%

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Symmetry analysis of a model of stochastic volatility with time-dependent parameters

Sophocleous

O’Hara

2011

Journal of Computational and Applied Mathematics

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“…The set of admissible controls is donated by A(K). Considering Markov controls, means that the investor will allocate the amount π ≡ π(t, x, y) ∈ A(K) into the risky asset when the wealth X t = x and volatility Y t = y (see e.g., [11], [9], [1]). We consider an investor with a time-dependent relative risk aversion utility function U :…”

Section: The Modelmentioning

confidence: 99%

“…Solving the resultant PDE when finding solution for stochastic volatility models has been a major focus of many researchers (see Benth and Kalsen [1], Kufakunesu [9], Hobson [6], and Heston [5] among others). The HJB equation leads to the characterization of the value function which then gives birth to highly nonlinear PDE [11].…”

Section: Introductionmentioning

confidence: 99%

Optimal investment models with stochastic volatility: the time inhomogeneous case

Kufakunesu

2015

Quaestiones Mathematicae

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Abstract. In a recent paper by Pham [11] a multidimensional model with stochastic volatility and portfolio constraints has been proposed, solving a class of investment problems. One feature which is common with these problems is that the resultant HamiltonJacobi-Bellman (HJB) partial differential equation (PDE) is highly nonlinear. Therefore, a transform is primordial to express the value function in terms of a semilinear PDE with quadratic growth on the derivative term. Some proofs for the existence of smooth solution to this equation have been provided for this equation by Pham [11]. In that paper they illustrated some common stochastic volatility examples in which most of the parameters are time-homogeneous. However, there are cases where time-dependent parameters are needed, such as in the calibrating financial models. Therefore, in this paper we extend the work of Pham[11] to the time-inhomogeneous case.Mathematics Subject Classification (2000): 93E20, 35K55, 60H30, 91B10.

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On the multi-dimensional portfolio optimization with stochastic volatility

Kufakunesu

2017

Quaestiones Mathematicae

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In a recent paper by Mnif [17], a solution to the portfolio optimization with stochastic volatility and constraints problem has been proposed, in which most of the model parameters are time-homogeneous. However, there are cases where time-dependent parameters are needed, such as in the calibration of financial models. Therefore, the purpose of this paper is to generalize the work of Mnif [17] to the time-inhomogeneous case. We consider a time-dependent exponential utility function of which the objective is to maximize the expected utility from the investor's terminal wealth. The derived Hamilton-Jacobi-Bellman(HJB) equation, is highly nonlinear and is reduced to a semilinear partial differential equation (PDE) by a suitable transformation. The existence of a smooth solution is proved and a verification theorem presented. A multi-asset stochastic volatility model with jumps and endowed with time-dependent parameters is illustrated.

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The density process of the minimal entropy martingale measure in a stochastic volatility market. A PDE Approach

Cited by 3 publications

References 16 publications

Symmetry analysis of a model of stochastic volatility with time-dependent parameters

Symmetry analysis of a model of stochastic volatility with time-dependent parameters

Optimal investment models with stochastic volatility: the time inhomogeneous case

On the multi-dimensional portfolio optimization with stochastic volatility

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