scholarly journals A Note on the Distribution of Multivariate Brownian Extrema

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Marcos Escobar ◽  
Julio Hernandez
Keyword(s):  
Financial Markets ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Joint Probability ◽  
Form Solution ◽  
Financial Mathematics ◽  
Method Of Images ◽  
Correlation Matrices ◽  
Correlation Structures ◽  
Multidimensional Wiener Process

This paper presents a closed-form solution for the joint probability of the endpoints and minimums of a multidimensional Wiener process for some correlation matrices. This is the only explicit expressions found in the literature for this joint probability. The analysis can only be carried out for special correlation structures as it is related to the fundamentals regions of irreducible spherical simplexes generated by reflections and the link to the method of images. This joint distribution can be used in financial mathematics to obtain prices of credit or market related products in high dimension. The solution could be generalized to account for stochastic volatility and other stylized features of the financial markets.

A CLOSED-FORM PRICING FORMULA FOR EUROPEAN EXCHANGE OPTIONS WITH STOCHASTIC VOLATILITY

2021 ◽  
pp. 1-10
Author(s):  
Puneet Pasricha ◽  
Anubha Goel
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Form Solution ◽  
Stochastic Volatility Model ◽  
Strike Price ◽  
Volatility Model ◽  
Exchange Options ◽  
Pricing Formula ◽  
Exchange Option

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations.

MULTI-ASSET PORTFOLIO OPTIMIZATION WITH STOCHASTIC SHARPE RATIO UNDER DRAWDOWN CONSTRAINT

2020 ◽  
Vol 15 (01) ◽  
pp. 2080001
Author(s):  
SUBHOJIT BISWAS ◽  
SAIF JAWAID ◽  
DIGANTA MUKHERJEE
Keyword(s):  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Fixed Time ◽  
Form Solution ◽  
Risk Tolerance ◽  
Finite Difference Schemes ◽  
Value Functions ◽  
Pairs Trading ◽  
Risky Assets ◽  
Drawdown Constraint

We consider an investor who seeks to maximize his expected utility of the portfolio, consisting of multiple risky assets and one risk-free asset, derived from the terminal wealth relative to the maximum wealth achieved over a fixed time horizon. This is achieved under a portfolio draw down constraint, in a market with local stochastic volatility. In empirical application, considering two risky assets, the assets have been identified with the help of pairs trading. In the absence of closed form solution of the value function and the optimal strategy, we obtain the approximates of these quantities using coefficient series expansion techniques and finite difference schemes. We utilize the risk tolerance factor function to ease our approximations of this value functions and the strategies. All the parameters were estimated from the triplets and used to illustrate and compare the stochastic volatility with the constant volatility situation, and how an investor can deploy different portfolio plans.

scholarly journals The method of images in the pricing of barrier derivatives in three dimensions

2021 ◽  
Author(s):  
Xianzhang Wen
Keyword(s):  
Wiener Process ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Financial Mathematics ◽  
Method Of Images ◽  
Linear Combinations ◽  
Correlation Matrices ◽  
Joint Distributions ◽  
Barrier Type ◽  
Spherical Triangles

The thesis describes the joint distributions of minima, maxima and endpoint values for a three dimensional Wiener process. In particular, the results provide the point cumulative distributions for the maxima and/or minima of the components of the process. The densities are obtained explicitly for special type of correlations by the method of images; the analysis requires a detailed study of partitions of the sphere by means of spherical triangles. The joint densities obtained can be used to obtain explicit expressions for price of options in financial mathematics. We provide closed-form expressions for the price of several barrier type derivatives with a three dimensional geometric Wiener process as underlying. These solutions are found for special correlation matrices and are given by linear combinations of three dimensional Gaussian cumulative distributions. In order to extend the results to a wider set of correlation matrices the method of random correlations is outlined.

EXACT PRICING WITH STOCHASTIC VOLATILITY AND JUMPS

2010 ◽  
Vol 13 (06) ◽  
pp. 901-929 ◽  
Author(s):  
FERNANDA D'IPPOLITI ◽  
ENRICO MORETTO ◽  
SARA PASQUALI ◽  
BARBARA TRIVELLATO
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Exact Algorithm ◽  
Form Solution ◽  
Closed Form Expression ◽  
Barrier Options ◽  
Form Expression ◽  
Variance Swaps ◽  
Jump Diffusion Model

A stochastic volatility jump-diffusion model for pricing derivatives with jumps in both spot return and volatility underlying dynamics is presented. This model admits, in the spirit of Heston, a closed-form solution for European-style options. The structure of the model is also suitable to explicitly obtain the fair delivery price for variance swaps. To evaluate derivatives whose value does not admit a closed-form expression, a methodology based on an "exact algorithm", in the sense that no discretization of equations is required, is developed and applied to barrier options. Goodness of pricing algorithm is tested using DJ Euro Stoxx 50 market data for European options. Finally, the algorithm is applied to compute prices and Greeks for barrier options and to determine the fair delivery prices for variance swaps.

scholarly journals A PAIR OF PERFECTLY CONDUCTING DISKS IN AN EXTERNAL FIELD

2015 ◽  
Vol 20 (2) ◽  
pp. 273-288 ◽  
Author(s):  
Natalia Rylko
Keyword(s):  
Closed Form Solution ◽  
Elliptic Functions ◽  
Convergent Series ◽  
Fiber Composites ◽  
Form Solution ◽  
Local Fields ◽  
Trigonometric Functions ◽  
Method Of Images ◽  
Asymptotic Investigation ◽  
Asymptotic Formulae

A pair of non-overlapping perfectly conducting equal disks embedded in a two-dimensional background was investigated by the classic method of images, by Poincar´e series, by use of the bipolar coordinates and by the elliptic functions in the previous works. In particular, successive application of the inversions with respect to circles were applied to obtain the field in the form of a series. For closely placed disks, the previous methods yield slowly convergent series. In this paper, we study the local fields around closely placed disks by the elliptic functions. The problem of small gap is completely investigated since the obtained closed form solution admits a precise asymptotic investigation in terms of the trigonometric functions when the gap between the disks tends to zero. The exact and asymptotic formulae are extended to the case when a prescribed singularity is located in the gap. This extends applications of structural approximations to estimations of the local fields in densely packed fiber composites in various external fields.

scholarly journals DECOMPOSITION FORMULA FOR JUMP DIFFUSION MODELS

2018 ◽  
Vol 21 (08) ◽  
pp. 1850052
Author(s):  
R. MERINO ◽  
J. POSPÍŠIL ◽  
T. SOBOTKA ◽  
J. VIVES
Keyword(s):  
Option Pricing ◽  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Form Solution ◽  
Heston Model ◽  
Approximation Formula ◽  
Option Prices ◽  
Currency Options ◽  
Decomposition Formula

In this paper, we derive a generic decomposition of the option pricing formula for models with finite activity jumps in the underlying asset price process (SVJ models). This is an extension of the well-known result by Alòs [(2012) A decomposition formula for option prices in the Heston model and applications to option pricing approximation, Finance and Stochastics 16 (3), 403–422, doi: https://doi.org/10.1007/s00780-012-0177-0 ] for Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ] SV model. Moreover, explicit approximation formulas for option prices are introduced for a popular class of SVJ models — models utilizing a variance process postulated by Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ]. In particular, we inspect in detail the approximation formula for the Bates [(1996), Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options, The Review of Financial Studies 9 (1), 69–107, doi: https://doi.org/10.1093/rfs/9.1.69 ] model with log-normal jump sizes and we provide a numerical comparison with the industry standard — Fourier transform pricing methodology. For this model, we also reformulate the approximation formula in terms of implied volatilities. The main advantages of the introduced pricing approximations are twofold. Firstly, we are able to significantly improve computation efficiency (while preserving reasonable approximation errors) and secondly, the formula can provide an intuition on the volatility smile behavior under a specific SVJ model.

scholarly journals Network of interacting synthetic molecules in steady state

2008 ◽  
Vol 464 (2096) ◽  
pp. 2219-2228 ◽  
Author(s):  
Erol Gelenbe
Keyword(s):  
Steady State ◽  
Closed Form Solution ◽  
Joint Probability ◽  
Reaction Network ◽  
Mass Conservation ◽  
Reaction Rates ◽  
Form Solution ◽  
Joint Probability Distribution ◽  
State Behaviour ◽  
Steady State Behaviour

In this paper, we study the steady-state behaviour of a reaction network of interacting molecules using the chemical master equation (CME). The model considers a set of base species from which further compounds are created via binary reactions, as well as by monomolecular and dissipation reactions. The model includes external arrivals of molecules into the reaction volume and assumes that the reaction rates are proportional to the number of molecules of the reactants that are present. We obtain an explicit expression for the solution of the CME in equilibrium under the assumption that the system obeys a mass conservation law for the overall rate of incoming and outgoing molecules. This closed-form solution shows that the joint probability distribution of the number of molecules of each species is in ‘product form’, i.e. it is the product of the marginal distributions for the number of molecules of each species. We also show that the steady-state distribution of the number of molecules of each base and synthesized species follows a Poisson distribution.

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