1,666 research outputs found
The Riccati Equation in Mathematical Finance
AbstractThis paper uses ideas from symbolic computation to classify solutions to an important class of problems in mathematical finance and thus provides a linkage between these two fields. We show that Kovacic’s concept of closed-form solutions to the Riccati ordinary differential equation can be used to provide a precise mathematical definition that is useful in certain financial models. We extend this definition to a broader class of problems and discuss how these ideas can be usefully applied to practical problems in the finance area. We provide a specific application by developing a new implementation of the Cox–Ingersoll–Ross interest-rate model that may be of practical interest
Geometric Asian Option Pricing in General Affine Stochastic Volatility Models with Jumps
In this paper we present some results on Geometric Asian option valuation for
affine stochastic volatility models with jumps. We shall provide a general
framework into which several different valuation problems based on some average
process can be cast, and we shall obtain close-form solutions for some relevant
affine model classes.Comment: 20 page
Nonlinear Parabolic Equations arising in Mathematical Finance
This survey paper is focused on qualitative and numerical analyses of fully
nonlinear partial differential equations of parabolic type arising in financial
mathematics. The main purpose is to review various non-linear extensions of the
classical Black-Scholes theory for pricing financial instruments, as well as
models of stochastic dynamic portfolio optimization leading to the
Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both
problems can be represented by solutions to nonlinear parabolic equations.
Qualitative analysis will be focused on issues concerning the existence and
uniqueness of solutions. In the numerical part we discuss a stable
finite-volume and finite difference schemes for solving fully nonlinear
parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387
Continuous time mean-variance portfolio selection with nonlinear wealth equations and random coefficients
This paper concerns the continuous time mean-variance portfolio selection
problem with a special nonlinear wealth equation. This nonlinear wealth
equation has nonsmooth random coefficients and the dual method developed in [7]
does not work. To apply the completion of squares technique, we introduce two
Riccati equations to cope with the positive and negative part of the wealth
process separately. We obtain the efficient portfolio strategy and efficient
frontier for this problem. Finally, we find the appropriate sub-derivative
claimed in [7] using convex duality method.Comment: arXiv admin note: text overlap with arXiv:1606.0548
Statistical analysis of the mixed fractional Ornstein--Uhlenbeck process
This paper addresses the problem of estimating drift parameter of the
Ornstein - Uhlenbeck type process, driven by the sum of independent standard
and fractional Brownian motions. The maximum likelihood estimator is shown to
be consistent and asymptotically normal in the large-sample limit, using some
recent results on the canonical representation and spectral structure of mixed
processes.Comment: to appear in Theory of Probability and its Application
- …