PERFECT HEDGING OF INDEX DERIVATIVES UNDER A MINIMAL MARKET MODEL

The paper presents a financial market model that generates stochastic volatility using a minimal set of factors. These factors, formed by transformations of square root processes, model the dynamics of different denominations of a benchmark portfolio. Benchmarked prices are assumed to be local martingales. Numerical results for the pricing and hedging of basic derivatives on indices are described for the minimal market model. This includes cases where the standard risk neutral pricing methodology fails because of the presence of a strict local martingale measure. However, payoffs can be perfectly hedged using self-financing strategies and a form of arbitrage exists. This is illustrated by hedge simulations. The different term structure of implied volatilities is documented for calls and puts on an index.

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Non-Arbitrage Models of Financial Markets

Global Journal of Science Frontier Research ◽

10.34257/gjsfravol21is4pg67 ◽

2021 ◽

pp. 67-112

Author(s):

N. S. Gonchar

Keyword(s):

Financial Markets ◽

Financial Market ◽

Wide Class ◽

Market Model ◽

Martingale Measure ◽

Fair Price ◽

Risky Assets ◽

Contingent Liabilities ◽

Initial Measure

In the first part of the paper, we construct the models of the complete non-arbitrage financial markets for a wide class of evolutions of risky assets.This construction is based on the observation that for a certain class of risky as set evolutions the martingale measure is invariant with respect to these evolutions. For such a financial market model the only martingale measure being equivalent to an initial measure is built. On such a financial market,formulas for the fair price of contingent liabilities are presented. A multi-parameter model of the financial market is proposed, the martingale measure of which does not depend on the parameters of the model of the evolution of risky assets and is the only one.

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Asset price bubbles: Invariance theorems

Frontiers of Mathematical Finance ◽

10.3934/fmf.2021006 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Robert Jarrow ◽

Philip Protter ◽

Jaime San Martin

Keyword(s):

Asset Price ◽

Martingale Measure ◽

Local Martingale ◽

Statistical Probability ◽

Asset Price Bubbles ◽

Price Bubbles ◽

New Family ◽

Price Bubble ◽

Markov Diffusion ◽

Strict Local Martingale

<p style='text-indent:20px;'>This paper provides invariance theorems that facilitate testing for the existence of an asset price bubble in a market where the price evolves as a Markov diffusion process. The test involves only the properties of the price process' quadratic variation under the statistical probability. It does not require an estimate of either the equivalent local martingale measure or the asset's drift. To augment its use, a new family of stochastic volatility price processes is also provided where the processes' strict local martingale behavior can be characterized.</p>

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STRONG BUBBLES AND STRICT LOCAL MARTINGALES

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024916500229 ◽

2016 ◽

Vol 19 (04) ◽

pp. 1650022 ◽

Cited By ~ 11

Author(s):

MARTIN HERDEGEN ◽

MARTIN SCHWEIZER

Keyword(s):

Financial Market ◽

Asset Prices ◽

Martingale Measure ◽

Local Martingale ◽

Strict Local Martingales ◽

Fundamental Value ◽

Absence Of Arbitrage

In a numéraire-independent framework, we study a financial market with [Formula: see text] assets which are all treated in a symmetric way. We define the fundamental value ∗S of an asset [Formula: see text] as its super-replication price and say that the market has a strong bubble if ∗S and [Formula: see text] deviate from each other. None of these concepts needs any mention of martingales. Our main result then shows that under a weak absence-of-arbitrage assumption (basically NUPBR), a market has a strong bubble if and only if in all numéraire s for which there is an equivalent local martingale measure (ELMM), asset prices are strict local martingales under all possible ELMMs. We show by an example that our bubble concept lies strictly between the existing notions from the literature. We also give an example where asset prices are strict local martingales under one ELMM, but true martingales under another, and we show how our approach can lead naturally to endogenous bubble birth.

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DEFAULTABLE TERM STRUCTURES DRIVEN BY SEMIMARTINGALES

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024921500321 ◽

2021 ◽

Author(s):

SANDRINE GÜMBEL ◽

THORSTEN SCHMIDT

Keyword(s):

Term Structure ◽

General Setting ◽

Martingale Measure ◽

Local Martingale ◽

Current Time ◽

Forward Rates ◽

Large Financial Market ◽

Risky Bonds ◽

Drift Conditions ◽

The Right

In this paper, we consider a market with a term structure of credit risky bonds in the single-name case. We aim at minimal assumptions extending existing results in this direction: first, the random field of forward rates is driven by a general semimartingale. Second, the Heath–Jarrow–Morton (HJM) approach is extended with an additional component capturing those future jumps in the term structure which are visible from the current time. Third, the associated recovery scheme is as general as possible, it is only assumed to be nonincreasing. In this general setting, we derive generalized drift conditions which characterize when a given measure is a local martingale measure, thus yielding no asymptotic free lunch with vanishing risk (NAFLVR), the right notion for this large financial market to be free of arbitrage.

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