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.

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 Pricing European options with stochastic volatility under the minimal entropy martingale measure

2015 ◽  
Vol 27 (2) ◽  
pp. 233-247 ◽  
Author(s):  
XIN-JIANG HE ◽  
SONG-PING ZHU
Keyword(s):  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Martingale Measure ◽  
Minimal Entropy Martingale Measure ◽  
Volatility Model ◽  
In Series ◽  
Expected Utility Maximization ◽  
Equivalent Martingale ◽  
The One ◽  
Pricing Formula

In this paper, a closed-form pricing formula in the form of an infinite series for European call options is derived for the Heston stochastic volatility model under a chosen martingale measure. Given that markets with the stochastic volatility are incomplete, there exists a number of equivalent martingale measures and consequently investors face a problem of making a choice of appropriate measure when they price options. The one we adopt here is the so-called minimal entropy martingale measure shown to be related to the expected utility maximization theory (Frittelli 2000 Math. Finance10(1), 39–52) and the financial rationality for choosing this measure will be further illustrated in this paper. A great advantage of our newly-derived pricing formula is that the convergence of the solution in series form can be proved theoretically; such a proof of the convergence is also complemented by some numerical examples to demonstrate the speed of convergence. To further show the validity of our formula, a comparison of prices calculated through the newly derived formula is made with those obtained directly from the Monte Carlo simulation as well as those from solving the PDE (partial differential equation) with the finite difference method.

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.

Black–Scholes from a Martingale Point of View

2019 ◽  
pp. 185-190
Author(s):  
Tomas Björk
Keyword(s):  
Point Of View ◽  
Martingale Measure ◽  
Martingale Approach ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Delta Hedging ◽  
Pricing Formula

In this chapter we return to the Black–Scholes model. This model was discussed in Chapter 7, using the classical delta hedging approach but we now analyze the model in much more detail by using the martingale approach. We prove that the model is free of arbitrage and complete, we find the unique martingale measure, and we easily rederive the pricing formula from Chapter 7.

scholarly journals Pricing Participating Products under a Generalized Jump-Diffusion Model

2008 ◽  
Vol 2008 ◽  
pp. 1-30 ◽  
Author(s):  
Tak Kuen Siu ◽  
John W. Lau ◽  
Hailiang Yang
Keyword(s):  
Diffusion Model ◽  
Life Insurance ◽  
Regime Switching ◽  
Jump Diffusion ◽  
Martingale Measure ◽  
Practical Implementation ◽  
Switching Models ◽  
Jump Diffusion Model ◽  
Equivalent Martingale ◽  
Markovian Regime Switching

We propose a model for valuing participating life insurance products under a generalized jump-diffusion model with a Markov-switching compensator. It also nests a number of important and popular models in finance, including the classes of jump-diffusion models and Markovian regime-switching models. The Esscher transform is employed to determine an equivalent martingale measure. Simulation experiments are conducted to illustrate the practical implementation of the model and to highlight some features that can be obtained from our model.

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.

Forward and Futures Contracts

2019 ◽  
pp. 236-243
Author(s):  
Tomas Björk
Keyword(s):  
Option Pricing ◽  
Elementary Theory ◽  
Futures Contracts ◽  
Forward Contracts ◽  
Martingale Approach ◽  
Futures Contract ◽  
Pricing Formula

This chapter contains a substantial extension of the more elementary theory of forwards and futures developed in Chapter 7. We derive a general pricing formula for forward contracts. Futures contracts are discussed in some detail and it is shown that a futures contract can be viewed as a certain price dividend pair. Using the dividend theory of Chapter 16 we derive formulas for futures contracts using the martingale approach. As an application we derive the Black-76 futures option pricing formula.

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