We extend the approach of Carr, Itkin and Muravey, 2021 for getting semi-analytical prices of barrier options for the time-dependent Heston model with time-dependent barriers by applying it to the so-called $lambda$-SABR stochastic volatility model.
In doing so we modify the general integral transform method (see Itkin, Lipton, Muravey, Generalized integral transforms in mathematical finance, World Scientific, 2021) and deliver solution of this problem in the form of Fourier-Bessel series. The weights of this series solve a linear mixed Volterra-Fredholm equation (LMVF) of the second kind also derived in the paper. Numerical examples illustrate speed and accuracy of our method which are comparable with those of the finite-difference approach at small maturities and outperform them at high maturities even by using a simplistic implementation of the RBF method for solving the LMVF.
We continue a series of papers devoted to construction of semi-analytic solutions for barrier options. These options are written on underlying following some simple one-factor diffusion model, but all the parameters of the model as well as the barrie
rs are time-dependent. We managed to show that these solutions are systematically more efficient for pricing and calibration than, eg., the corresponding finite-difference solvers. In this paper we extend this technique to pricing double barrier options and present two approaches to solving it: the General Integral transform method and the Heat Potential method. Our results confirm that for double barrier options these semi-analytic techniques are also more efficient than the traditional numerical methods used to solve this type of problems.
We continue a series of papers where prices of the barrier options written on the underlying, which dynamics follows some one factor stochastic model with time-dependent coefficients and the barrier, are obtained in semi-closed form, see (Carr and It
kin, 2020, Itkin and Muravey, 2020). This paper extends this methodology to the CIR model for zero-coupon bonds, and to the CEV model for stocks which are used as the corresponding underlying for the barrier options. We describe two approaches. One is generalization of the method of heat potentials for the heat equation to the Bessel process, so we call it the method of Bessel potentials. We also propose a general scheme how to construct the potential method for any linear differential operator with time-independent coefficients. The second one is the method of generalized integral transform, which is also extended to the Bessel process. In all cases, a semi-closed solution means that first, we need to solve numerically a linear Volterra equation of the second kind, and then the option price is represented as a one-dimensional integral. We demonstrate that computationally our method is more efficient than both the backward and forward finite difference methods while providing better accuracy and stability. Also, it is shown that both method dont duplicate but rather compliment each other, as one provides very accurate results at small maturities, and the other one - at high maturities.
In this paper we derive semi-closed form prices of barrier (perhaps, time-dependent) options for the Hull-White model, ie., where the underlying follows a time-dependent OU process with a mean-reverting drift. Our approach is similar to that in (Carr
and Itkin, 2020) where the method of generalized integral transform is applied to pricing barrier options in the time-dependent OU model, but extends it to an infinite domain (which is an unsolved problem yet). Alternatively, we use the method of heat potentials for solving the same problems. By semi-closed solution we mean that first, we need to solve numerically a linear Volterra equation of the first kind, and then the option price is represented as a one-dimensional integral. Our analysis shows that computationally our method is more efficient than the backward and even forward finite difference methods (if one uses them to solve those problems), while providing better accuracy and stability.
We present an alternative to the well-known Andersons formula for the probability that a first exit time from the planar region between two slopping lines -a_1 t -b_1 and a_2 t + b_2 by a standard Brownian motion is greater than T. As the Andersons f
ormula, our representation is an infinite series from special functions. We show that convergence rate of both formulas depends only on terms (a_1 + a_2)(b_1 + b_2) and (b_1 + b_2)^2 /T and deduce simple rules of appropriate representations choose. We prove that for any given set of parameters a_1, b_1, a_2, b_2, T the sum of first 6 terms ensures precision 10^{-16}.
This paper uses Lie symmetry methods to analyze boundary crossing probabilities for a large class of diffusion processes. We show that if Fokker--Planck--Kolmogorov equation has non-trivial Lie symmetry, then boundary crossing identity exists and dep
ends only on parameters of process and symmetry. For time-homogeneous diffusion processes we found necessary and sufficient conditions of symmetries existence. This paper shows that if drift function satisfy one of a family of Ricatti equations, then the problem has nontrivial Lie symmetries. For each case we present symmetries in explicit form. Based on obtained results, we derive two-parametric boundary crossing identities and prove its uniqueness. Further, we present boundary crossing identities between different process. We show, that if the problem has 6 or 4 group of symmetries then the first passage time density to any boundary can be explicitly represented in terms of the first passage time by Brownian motion or Bessel process. Many examples are presented to illustrate the method.
This paper studies an optimal investment problem under M-CEV with power utility function. Using Laplace transform we obtain explicit expression for optimal strategy in terms of confluent hypergeometric functions. For obtained representations we deriv
e asymptotic and approximation formulas contains only elementary functions and continued fractions. These formulas allow to make analysis of impact of models parameters and effects of parameters misspecification. In addition we propose some extensions of obtained results that can be applicable for algorithmic strategies.
In this paper we consider a variation of the Mertons problem with added stochastic volatility and finite time horizon. It is known that the corresponding optimal control problem may be reduced to a linear parabolic boundary problem under some assumpt
ions on the underlying process and the utility function. The resulting parabolic PDE is often quite difficult to solve, even when it is linear. The present paper contributes to the pool of explicit solutions for stochastic optimal control problems. Our main result is the exact solution for optimal investment in Heston model.
We provide explicit formulas for the Green function of an elliptic PDE in the infinite strip and the half-plane. They are expressed in elementary and special functions. Proofs of uniqueness and existence are also given.
I present the technique which can analyse some interest rate models: Constantinides-Ingersoll, CIR-model, geometric CIR and Geometric Brownian Motion. All these models have the unified structure of Whittaker function. The main focus of this text is c
losed-form solutions of the zero-coupon bond value in these models. In text I emphasize the specific details of mathematical methods of their determination such as Laplace transform and hypergeometric functions.
