Extension of Black-Scholes Equation for Derivatives with Time Dependent Payoff Specifications

2001 ◽  
Author(s):  
Shahram Alavian
Keyword(s):  
Time Dependent ◽  
Black Scholes

scholarly journals Continuous Time Portfolio Selection under Conditional Capital at Risk

2010 ◽  
Vol 2010 ◽  
pp. 1-26 ◽  
Author(s):  
Gordana Dmitrasinovic-Vidovic ◽  
Ali Lari-Lavassani ◽  
Xun Li ◽  
Antony Ware
Keyword(s):  
At Risk ◽  
Portfolio Selection ◽  
Continuous Time ◽  
Risk Measures ◽  
Risk Measure ◽  
Optimal Strategies ◽  
Time Dependent ◽  
Risky Assets ◽  
Black Scholes ◽  
Capital At Risk

Portfolio optimization with respect to different risk measures is of interest to both practitioners and academics. For there to be a well-defined optimal portfolio, it is important that the risk measure be coherent and quasiconvex with respect to the proportion invested in risky assets. In this paper we investigate one such measure—conditional capital at risk—and find the optimal strategies under this measure, in the Black-Scholes continuous time setting, with time dependent coefficients.

AN ANALYTICAL OPTION PRICING FORMULA FOR MEAN-REVERTING ASSET WITH TIME-DEPENDENT PARAMETER

2021 ◽  
Vol 63 (2) ◽  
pp. 178-202
Author(s):  
P. NONSOONG ◽  
K. MEKCHAY ◽  
S. RUJIVAN
Keyword(s):  
Option Pricing ◽  
Option Price ◽  
Time Dependent ◽  
Initial Time ◽  
Agricultural Commodity ◽  
Time Integral ◽  
Long Run ◽  
Black Scholes ◽  
Dependent Parameter ◽  
Pricing Formula

AbstractWe present an analytical option pricing formula for the European options, in which the price dynamics of a risky asset follows a mean-reverting process with a time-dependent parameter. The process can be adapted to describe a seasonal variation in price such as in agricultural commodity markets. An analytical solution is derived based on the solution of a partial differential equation, which shows that a European option price can be decomposed into two terms: the payoff of the option at the initial time and the time-integral over the lifetime of the option driven by a time-dependent parameter. Finally, results obtained from the formula have been compared with Monte Carlo simulations and a Black–Scholes-type formula under various kinds of long-run mean functions, and some examples of option price behaviours have been provided.

scholarly journals SWITCHING TO NONAFFINE STOCHASTIC VOLATILITY: A CLOSED-FORM EXPANSION FOR THE INVERSE GAMMA MODEL

2016 ◽  
Vol 19 (05) ◽  
pp. 1650031 ◽  
Author(s):  
NICOLAS LANGRENÉ ◽  
GEOFFREY LEE ◽  
ZILI ZHU
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Time Dependent ◽  
Market Data ◽  
Affine Models ◽  
Inverse Gamma ◽  
Volatility Model ◽  
Black Scholes

We examine the inverse gamma (IGa) stochastic volatility model with time-dependent parameters. This nonaffine model compares favorably in terms of volatility distribution and volatility paths to classical affine models such as the Heston model, while being as parsimonious (only four stochastic parameters). In practice, this means more robust calibration and better hedging, explaining its popularity among practitioners. Closed-form volatility-of-volatility expansions are obtained for the price of vanilla options, which allow for very fast pricing and calibration to market data. Specifically, the price of a European put option with IGa volatility is approximated by a Black–Scholes price plus a weighted combination of Black–Scholes Greeks, with weights depending only on the four time-dependent parameters of the model. The accuracy of the expansion is illustrated on several calibration tests on foreign exchange market data. This paper shows that the IGa model is as simple, more realistic, easier to implement and faster to calibrate than classical transform-based affine models. We therefore hope that the present work will foster further research on nonaffine models favored by practitioners such as the IGa model.

scholarly journals Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Yuzi Jin ◽  
Jian Wang ◽  
Sangkwon Kim ◽  
Youngjin Heo ◽  
Changwoo Yoo ◽  
...  
Keyword(s):  
Cost Function ◽  
Piecewise Linear ◽  
Reconstruction Algorithm ◽  
Descent Method ◽  
Time Dependent ◽  
Steepest Descent Method ◽  
Market Values ◽  
Black Scholes ◽  
Volatility Function ◽  
The Cost

We propose a simple and robust numerical algorithm to estimate a time-dependent volatility function from a set of market observations, using the Black–Scholes (BS) model. We employ a fully implicit finite difference method to solve the BS equation numerically. To define the time-dependent volatility function, we define a cost function that is the sum of the squared errors between the market values and the theoretical values obtained by the BS model using the time-dependent volatility function. To minimize the cost function, we employ the steepest descent method. However, in general, volatility functions for minimizing the cost function are nonunique. To resolve this problem, we propose a predictor-corrector technique. As the first step, we construct the volatility function as a constant. Then, in the next step, our algorithm follows the prediction step and correction step at half-backward time level. The constructed volatility function is continuous and piecewise linear with respect to the time variable. We demonstrate the ability of the proposed algorithm to reconstruct time-dependent volatility functions using manufactured volatility functions. We also present some numerical results for real market data using the proposed volatility function reconstruction algorithm.

AN APPLICATION OF MELLIN TRANSFORM TECHNIQUES TO A BLACK–SCHOLES EQUATION PROBLEM

2007 ◽  
Vol 05 (01) ◽  
pp. 51-66 ◽  
Author(s):  
MARIANITO R. RODRIGO ◽  
ROGEMAR S. MAMON
Keyword(s):  
Existence And Uniqueness ◽  
Mellin Transform ◽  
Formal Solution ◽  
Option Price ◽  
Contingent Claims ◽  
Time Dependent ◽  
European Option ◽  
Equation Problem ◽  
Black Scholes Model ◽  
Black Scholes

In this article, we use a Mellin transform approach to prove the existence and uniqueness of the price of a European option under the framework of a Black–Scholes model with time-dependent coefficients. The formal solution is rigorously shown to be a classical solution under quite general European contingent claims. Specifically, these include claims that are bounded and continuous, and claims whose difference with some given but arbitrary polynomial is bounded and continuous. We derive a maximum principle and use it to prove uniqueness of the option price. An extension of the put-call parity which relates the aforementioned two classes of claims is also given.

OPTIMAL SUPERHEDGING UNDER NON-CONVEX CONSTRAINTS — A BSDE APPROACH

2008 ◽  
Vol 11 (04) ◽  
pp. 363-380 ◽  
Author(s):  
CHRISTIAN BENDER ◽  
MICHAEL KOHLMANN
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Numerical Methods ◽  
Borrowing Constraints ◽  
Time Dependent ◽  
Convex Constraints ◽  
Numerical Examples ◽  
Black Scholes ◽  
Backward Sdes ◽  
Theoretical Results

We apply theoretical results by Peng on supersolutions for Backward SDEs (BSDEs) to the problem of finding optimal superhedging strategies in a generalized Black–Scholes market under constraints. Constraints may be imposed simultaneously on wealth process and portfolio. They may be non-convex, time-dependent, and random. The BSDE method turns out to be an extremely useful tool for modeling realistic markets: in this paper, it is shown how more realistic constraints on the portfolio may be formulated via BSDE theory in terms of the amount of money invested, the portfolio proportion, or the number of shares held. Based on recent advances on numerical methods for BSDEs (in particular, the forward scheme by Bender and Denk [1]), a Monte Carlo method for approximating the superhedging price is given, which demonstrates the practical applicability of the BSDE method. Some numerical examples concerning European and American options under non-convex borrowing constraints are presented.

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