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.

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 Detection of arbitrage opportunities in multi-asset derivatives markets

2021 ◽  
Vol 9 (1) ◽  
pp. 439-459
Author(s):  
Antonis Papapantoleon ◽  
Paulo Yanez Sarmiento
Keyword(s):  
Financial Market ◽  
Sufficient Conditions ◽  
Martingale Measure ◽  
No Arbitrage ◽  
Additional Information ◽  
Arbitrage Opportunities ◽  
Equivalent Martingale ◽  
Arbitrage Opportunity ◽  
Derivatives Markets ◽  
Hoeffding Bounds

Abstract We are interested in the existence of equivalent martingale measures and the detection of arbitrage opportunities in markets where several multi-asset derivatives are traded simultaneously. More specifically, we consider a financial market with multiple traded assets whose marginal risk-neutral distributions are known, and assume that several derivatives written on these assets are traded simultaneously. In this setting, there is a bijection between the existence of an equivalent martingale measure and the existence of a copula that couples these marginals. Using this bijection and recent results on improved Fréchet–Hoeffding bounds in the presence of additional information on functionals of a copula by [18], we can extend the results of [33] on the detection of arbitrage opportunities to the general multi-dimensional case. More specifically, we derive sufficient conditions for the absence of arbitrage and formulate an optimization problem for the detection of a possible arbitrage opportunity. This problem can be solved efficiently using numerical optimization routines. The most interesting practical outcome is the following: we can construct a financial market where each multi-asset derivative is traded within its own no-arbitrage interval, and yet when considered together an arbitrage opportunity may arise.

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.

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.

scholarly journals No-Arbitrage Principle in Conic Finance

2020 ◽  
Author(s):  
Mehdi Vazifedan ◽  
Qiji Jim Zhu
Keyword(s):  
Cash Flow ◽  
Financial Models ◽  
Unit Vector ◽  
Martingale Measure ◽  
Financial Model ◽  
No Arbitrage ◽  
Arbitrage Opportunities ◽  
Equivalent Martingale ◽  
Arbitrage Opportunity ◽  
Price Spread

In a one price economy, the Fundamental Theorem of Asset Pricing (FTAP) establishes that no-arbitrage is equivalent to the existence of an equivalent martingale measure. Such an equivalent measure can be derived as the normal unit vector of the hyperplane that separates the attainable gain subspace and the convex cone representing arbitrage opportunities. However, in two-price financial models (where there is a bid-ask price spread), the set of attainable gains is not a subspace anymore. We use convex optimization, and the conic property of this region to characterize the “No-Arbitrage” principle in financial models with the bid-ask price spread present. This characterization will lead us to the generation of a set of price factor random variables. Under such a set, we can find the lower and upper bounds (supper-hedging and sub-hedging bounds) for the price of any future cash flow. We will show that for any given cash flow, for which the price is outside the above range, we can build a trading strategy that provides one with an arbitrage opportunity. We will generalize this structure to any two-price finite-period financial model.

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