Change of Numeraire

2019 ◽  
pp. 202-218
Author(s):  
Tomas Björk
Keyword(s):  
General Theory ◽  
Option Pricing ◽  
General Formula ◽  
Financial Derivatives ◽  
Martingale Measure ◽  
Girsanov Transformation ◽  
Suitable Change ◽  
Pricing Formula ◽  
Exchange Option ◽  
Numeraire Portfolio

In this chapter we discuss how a suitable change of numeraire and the corresponding change of martingale measure, can simplify the computation of pricing formula for financial derivatives. We derive a general formula for the likelihood process related to an arbitrary numeraire, and we identify the corresponding Girsanov transformation. As an example, we compute the price of an exchange option. In particular we study the class of forward measures related to zero coupon bonds and we derive a general option pricing formula. As an application of the general theory we also study the so-called numeraire portfolio.

Arbitrage Theory in Continuous Time

2019 ◽  
Author(s):  
Tomas Björk
Keyword(s):  
Incomplete Markets ◽  
Continuous Time ◽  
Dynamic Equilibrium ◽  
Factor Model ◽  
Good Deal ◽  
Financial Derivatives ◽  
Equilibrium Theory ◽  
Martingale Measure ◽  
Fourth Edition ◽  
Optimal Stopping Theory

The fourth edition of this textbook on pricing and hedging of financial derivatives, now also including dynamic equilibrium theory, continues to combine sound mathematical principles with economic applications. Concentrating on the probabilistic theory of continuous time arbitrage pricing of financial derivatives, including stochastic optimal control theory and optimal stopping theory, the book is designed for graduate students in economics and mathematics, and combines the necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. All concepts and ideas are discussed, not only from a mathematics point of view, but the mathematical theory is also always supplemented with lots of intuitive economic arguments. In the substantially extended fourth edition Tomas Björk has added completely new chapters on incomplete markets, treating such topics as the Esscher transform, the minimal martingale measure, f-divergences, optimal investment theory for incomplete markets, and good deal bounds. There is also an entirely new part of the book presenting dynamic equilibrium theory. This includes several chapters on unit net supply endowments models, and the Cox–Ingersoll–Ross equilibrium factor model (including the CIR equilibrium interest rate model). Providing two full treatments of arbitrage theory—the classical delta hedging approach and the modern martingale approach—the book is written in such a way that these approaches can be studied independently of each other, thus providing the less mathematically oriented reader with a self-contained introduction to arbitrage theory and equilibrium theory, while at the same time allowing the more advanced student to see the full theory in action.

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.

scholarly journals Existence of Maximum Entropy Problem Solution in a General N-Dimensional Case

2018 ◽  
pp. 97-102
Author(s):  
Ruben Gevorgyan ◽  
Narek Margaryan
Keyword(s):  
Option Pricing ◽  
Maximum Entropy ◽  
Optimization Algorithms ◽  
Dimensional Case ◽  
Computational Time ◽  
Problem Solution ◽  
Call Options ◽  
Price Range ◽  
Risk Neutral ◽  
Pricing Formula

In the following paper, we will define conditions, which need to be satisfied in order for the maximum entropy problem applied in European call options to have a solution in a general n-dimensional case. We will also find a minimum right boundary for the price range in order to have at least one risk neutral measure satisfying the option pricing formula. The results significantly reduce the computational time of optimization algorithms used in maximum entropy problem.

scholarly journals Reverse Bonus Certificate Design and Valuation Using Pricing by Duplication Methods

2015 ◽  
Vol 62 (3) ◽  
pp. 277-289
Author(s):  
Martina Bobriková ◽  
Monika Harčariková
Keyword(s):  
Option Pricing ◽  
Exotic Options ◽  
Pricing Models ◽  
Underlying Asset ◽  
Replication Method ◽  
Maturity Date ◽  
Derivative Market ◽  
Replicating Portfolio ◽  
Pricing Formula

Abstract In this paper we perform an analysis of a capped reverse bonus certificate, the value of which is derived from the value of an underlying asset. A pricing formula for the portfolio replication method is applied to price the capped reverse bonus certificate. A replicating portfolio has profit that is identical to profit from a combination of positions in spot and derivative market, i.e. vanilla and exotic options. Based upon the theoretical option pricing models, the replicating portfolio for capped reverse bonus certificate on the Euro Stoxx 50 index is engineered. We design the capped reverse bonus certificate with various parameters and calculate the issue prices in the primary market. The profitability for the potential investor at the maturity date is provided. The relation between the profit change of the investor and parameters’ change is detected. The best capped reverse bonus certificate for every estimated development of the index is identified.

scholarly journals Option Pricing under Risk-Minimization Criterion in an Incomplete Market with the Finite Difference Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Xinfeng Ruan ◽  
Wenli Zhu ◽  
Shuang Li ◽  
Jiexiang Huang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Option Pricing ◽  
Difference Method ◽  
Incomplete Market ◽  
Martingale Measure ◽  
Risk Minimization ◽  
European Option ◽  
The Finite Difference Method ◽  
Nikodym Derivative

We study option pricing with risk-minimization criterion in an incomplete market where the dynamics of the risky underlying asset is governed by a jump diffusion equation with stochastic volatility. We obtain the Radon-Nikodym derivative for the minimal martingale measure and a partial integro-differential equation (PIDE) of European option. The finite difference method is employed to compute the European option valuation of PIDE.

The Condition of the Conditionality

2021 ◽  
Author(s):  
Tumellano Sebehela
Keyword(s):  
Probability Theory ◽  
Option Pricing ◽  
Simultaneous Occurrence ◽  
Option Value ◽  
Hedging Strategy ◽  
Compound Options ◽  
Exotic Option ◽  
Exchange Options ◽  
Nikodym Derivative ◽  
Exchange Option

The interdependence of options is common among compound options. Moreover, this interconnectedness is synonymous with probability theory-how a set of axioms are treated. The conditionality, where one option value is dependent on another option, has spilled over to option pricing, especially exchange options. However, it seems that no study has explored whether that simultaneous occurrence of two options is conditional or not. This study uses conditional approaches (Radon–Nikodým derivative and probability theory) to illustrate conditionality in an exchange option. Furthermore, hedging strategy is derived based on straddles. The results show that due to conditionality another exotic option, tri-conditional option (also known as triple option) is derived. The hedging of a triple option encompasses both dynamic and static techniques.

Sign in / Sign up

Export Citation Format

Share Document