A More General One Period Model

2019 ◽  
pp. 27-40
Author(s):  
Tomas Björk
Keyword(s):  
Fundamental Theorem ◽  
Financial Derivatives ◽  
Sample Space ◽  
Martingale Measure ◽  
Finite Sample ◽  
Equivalent Martingale Measure ◽  
No Arbitrage ◽  
Equivalent Martingale ◽  
Absence Of Arbitrage ◽  
Period Model

In this chapter we study a general one period model living on a finite sample space. The concepts of no arbitrage and completeness are introduced, as well as the concept of a martingale measure. We then prove the First Fundamental Theorem, stating that absence of arbitrage is equivalent to the existence of an equivalent martingale measure. We also prove the Second Fundamental Theorem which says that the market is complete if and only if the martingale measure is unique. Using this theory, we derive pricing and hedging formulas for financial derivatives.

scholarly journals The equivalent martingale measure conditions in a general model for interest rates

2005 ◽  
Vol 37 (2) ◽  
pp. 415-434 ◽  
Author(s):  
Kais Hamza ◽  
Saul Jacka ◽  
Fima Klebaner
Keyword(s):  
Interest Rates ◽  
General Model ◽  
Field Model ◽  
Sufficient Conditions ◽  
Martingale Measure ◽  
Equivalent Martingale Measure ◽  
Forward Rates ◽  
No Arbitrage ◽  
Equivalent Martingale ◽  
Integrated Processes

Assuming that the forward rates ftu are semimartingales, we give conditions on their components under which the discounted bond prices are martingales. To achieve this, we give sufficient conditions for the integrated processes ftu=∫0uftvdv to be semimartingales, and identify their various components. We recover the no-arbitrage conditions in models well known in the literature and, finally, we formulate a new random field model for interest rates and give its equivalent martingale measure (no-arbitrage) condition.

scholarly journals The equivalent martingale measure conditions in a general model for interest rates

2005 ◽  
Vol 37 (02) ◽  
pp. 415-434 ◽  
Author(s):  
Kais Hamza ◽  
Saul Jacka ◽  
Fima Klebaner
Keyword(s):  
Interest Rates ◽  
General Model ◽  
Field Model ◽  
Sufficient Conditions ◽  
Martingale Measure ◽  
Equivalent Martingale Measure ◽  
Forward Rates ◽  
No Arbitrage ◽  
Equivalent Martingale ◽  
Integrated Processes

Assuming that the forward rates f t u are semimartingales, we give conditions on their components under which the discounted bond prices are martingales. To achieve this, we give sufficient conditions for the integrated processes f t u =∫0 uf t v dv to be semimartingales, and identify their various components. We recover the no-arbitrage conditions in models well known in the literature and, finally, we formulate a new random field model for interest rates and give its equivalent martingale measure (no-arbitrage) condition.

Multidimensional Models: Martingale Approach

2019 ◽  
pp. 191-201
Author(s):  
Tomas Björk
Keyword(s):  
Market Price ◽  
Representation Formula ◽  
Explicit Representation ◽  
Martingale Measure ◽  
Market Price Of Risk ◽  
Martingale Approach ◽  
Price Of Risk ◽  
Equivalent Martingale ◽  
Absence Of Arbitrage ◽  
Multidimensional Models

In this chapter we study a very general multidimensional Wiener-driven model using the martingale approach. Using the Girsanov Theorem we derive the martingale equation which is used to find an equivalent martingale measure. We provide conditions for absence of arbitrage and completeness of the model, and we discuss hedging and pricing. For Markovian models we derive the relevant pricing PDE and we also provide an explicit representation formula for the stochastic discount factor. We discuss the relation between the market price of risk and the Girsanov kernel and finally we derive the Hansen–Jagannathan bounds for the Sharpe ratio.

scholarly journals Mean-Variance Hedging Under Additional Market Information

2003 ◽  
Vol 06 (06) ◽  
pp. 613-636 ◽  
Author(s):  
F. Thierbach
Keyword(s):  
Martingale Measure ◽  
Option Prices ◽  
Market Information ◽  
Hedging Strategy ◽  
No Arbitrage ◽  
Finite Set ◽  
The Mean ◽  
Hedging Problem ◽  
Equivalent Martingale ◽  
Mean Variance

In this paper we analyze the mean-variance hedging approach in an incomplete market under the assumption of additional market information, which is represented by a given, finite set of observed prices of non-attainable contingent claims. Due to no-arbitrage arguments, our set of investment opportunities increases and the set of possible equivalent martingale measures shrinks. Therefore, we obtain a modified mean-variance hedging problem, which takes into account the observed additional market information. Solving this we obtain an explicit description of the optimal hedging strategy and an admissible, constrained variance-optimal signed martingale measure, that generates both the approximation price and the observed option prices.

scholarly journals Regime-Switching Risk: To Price or Not to Price?

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Tak Kuen Siu
Keyword(s):  
Relative Entropy ◽  
Regime Switching ◽  
Martingale Measure ◽  
Martingale Measures ◽  
Equivalent Martingale Measure ◽  
Equivalent Martingale Measures ◽  
Black Scholes ◽  
Normative Issues ◽  
Equivalent Martingale ◽  
Markovian Regime Switching

Should the regime-switching risk be priced? This is perhaps one of the important “normative” issues to be addressed in pricing contingent claims under a Markovian, regime-switching, Black-Scholes-Merton model. We address this issue using a minimal relative entropy approach. Firstly, we apply a martingale representation for a double martingale to characterize the canonical space of equivalent martingale measures which may be viewed as the largest space of equivalent martingale measures to incorporate both the diffusion risk and the regime-switching risk. Then we show that an optimal equivalent martingale measure over the canonical space selected by minimizing the relative entropy between an equivalent martingale measure and the real-world probability measure does not price the regime-switching risk. The optimal measure also justifies the use of the Esscher transform for option valuation in the regime-switching market.

The Martingale Approach to Arbitrage Theory

2019 ◽  
pp. 150-170
Author(s):  
Tomas Björk
Keyword(s):  
Fundamental Theorem ◽  
Martingale Measure ◽  
Alternative Form ◽  
Martingale Theory ◽  
Martingale Approach ◽  
Financial Derivative ◽  
Pricing Theory ◽  
Equivalent Martingale ◽  
Guided Tour ◽  
Pricing Formula

In this chapter the theoretical level is substantially increased, and we discuss in detail the deep connection between financial pricing theory and martingale theory. The first main result of the chapter is the First Fundamental Theorem which says that the market is free of arbitrage if and only if there exists an equivalent martingale measure. We provide a guided tour through the Delbaen–Schachemayer proof and we then apply the theory to derive a general risk neutral pricing formula for an arbitrary financial derivative. We also discuss the Second Fundamental Theorem which says that the market is complete if and only if the martingale measure is unique. We define the stochastic discount factor and use it to provide an alternative form of the pricing formula. Finally, we provide a summary for the reader who wishes to go lighter on the (rather advanced) theory.

Multi-Asset Option Pricing Based on Exponential Lévy Process

2013 ◽  
Vol 380-384 ◽  
pp. 4537-4540
Author(s):  
Nan Liu ◽  
Mei Ling Wang ◽  
Xue Bin Lü
Keyword(s):  
Differential Equation ◽  
Fourier Transform ◽  
Option Pricing ◽  
Numerical Method ◽  
Martingale Measure ◽  
Esscher Transform ◽  
Equivalent Martingale Measure ◽  
Integro Differential Equation ◽  
Equivalent Martingale ◽  
The Fourier Transform

The multi-dimensional Esscher transform was used to find a locally equivalent martingale measure to price the options based on multi-asset. An integro-differential equation was driven for the prices of multi-asset options. The numerical method based on the Fourier transform was used to calculate some special multi-asset options in exponential Lévy models. As an example we give the calculation of extreme options.

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