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31
Hedging under arbitrage
, 2010
"... It is shown that delta hedging provides the optimal trading strategy in terms of minimal required initial capital to replicate a given terminal payoff in a continuoustime Markovian context. This holds true in market models where no equivalent local martingale measure exists but only a squareintegr ..."
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Cited by 24 (6 self)
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It is shown that delta hedging provides the optimal trading strategy in terms of minimal required initial capital to replicate a given terminal payoff in a continuoustime Markovian context. This holds true in market models where no equivalent local martingale measure exists but only a squareintegrable market price of risk. A new probability measure is constructed, which takes the place of an equivalent local martingale measure. In order to ensure the existence of the delta hedge, sufficient conditions are derived for the necessary differentiability of expectations indexed over the initial market configuration. The recently often discussed phenomenon of “bubbles” is a special case of the setting in this paper. Several examples at the end illustrate the techniques described in this work.
BUBBLES, CONVEXITY AND THE BLACK–SCHOLES EQUATION
, 908
"... A bubble is characterized by the presence of an underlying asset whose discounted price process is a strict local martingale under the pricing measure. In such markets, many standard results from option pricing theory do not hold, and in this paper we address some of these issues. In particular, we ..."
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Cited by 18 (0 self)
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A bubble is characterized by the presence of an underlying asset whose discounted price process is a strict local martingale under the pricing measure. In such markets, many standard results from option pricing theory do not hold, and in this paper we address some of these issues. In particular, we derive existence and uniqueness results for the Black–Scholes equation, and we provide convexity theory for option pricing and derive related ordering results with respect to volatility. We show that American options are convexity preserving, whereas European options preserve concavity for general payoffs and convexity only for bounded contracts. 1. Introduction. Recently
Valuation equations for stochastic volatility models
 SIAM J. Financial Math
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Discretely sampled variance and volatility swaps versus their continuous approximations
, 2011
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Strict local martingales and bubbles
"... This paper deals with asset price bubbles modeled by strict local martingales. With any strict local martingale, one can associate a new measure, which is studied in detail in the first part of the paper. In the second part, we determine the “default term ” apparent in riskneutral option prices if ..."
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Cited by 8 (0 self)
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This paper deals with asset price bubbles modeled by strict local martingales. With any strict local martingale, one can associate a new measure, which is studied in detail in the first part of the paper. In the second part, we determine the “default term ” apparent in riskneutral option prices if the underlying stock exhibits a bubble modeled by a strict local martingale. Results for certain path dependent options and last passage time formulas are given. 1. Introduction. The
The BlackScholes equation in stochastic volatility models
, 2010
"... Abstract. We study the BlackScholes equation in stochastic volatility models. In particular, we show that the option price is the unique classical solution to a parabolic differential equation with a certain boundary behaviour for vanishing values of the volatility. If the boundary is attainable, ..."
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Cited by 7 (1 self)
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Abstract. We study the BlackScholes equation in stochastic volatility models. In particular, we show that the option price is the unique classical solution to a parabolic differential equation with a certain boundary behaviour for vanishing values of the volatility. If the boundary is attainable, then this boundary behaviour serves as a boundary condition and guarantees uniqueness in appropriate function spaces. On the other hand, if the boundary is nonattainable, then the boundary behaviour is not needed to guarantee uniqueness, but is nevertheless very useful for instance from a numerical perspective. 1.
Strict local martingales, bubbles, and no early exercise
, 2007
"... We show pathological behavior of asset price processes modeled by continuous strict local martingales under a riskneutral measure. The inspiration comes from recent results on financial bubbles. We analyze, in particular, the effect of the strict nature of the local martingale on the usual formula ..."
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We show pathological behavior of asset price processes modeled by continuous strict local martingales under a riskneutral measure. The inspiration comes from recent results on financial bubbles. We analyze, in particular, the effect of the strict nature of the local martingale on the usual formula for the price of a European call option, especially a strong anomaly when call prices decay monotonically with maturity. A complete and detailed analysis for the archetypical strict local martingale, the reciprocal of a three dimensional Bessel process, has been provided. Our main tool is based on a general htransform technique (due to Delbaen and Schachermayer) to generate positive strict local martingales. This gives the basis for a statistical test to verify a suspected bubble is indeed one (or not).
The Economic Plausibility of Strict Local Martingales in Financial Modelling
, 2010
"... The context for this article is a continuous financial market consisting of a riskfree savings account and a single nondividendpaying risky security. We present two concrete models for this market, in which strict local martingales play decisive roles. The first admits an equivalent riskneutral ..."
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Cited by 7 (0 self)
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The context for this article is a continuous financial market consisting of a riskfree savings account and a single nondividendpaying risky security. We present two concrete models for this market, in which strict local martingales play decisive roles. The first admits an equivalent riskneutral probability measure under which the discounted price of the risky security is a strict local martingale, while the second model does not even admit an equivalent riskneutral probability measure, since the putative density process for such a measure is itself a strict local martingale. We highlight a number of apparent anomalies associated with both models that may offend the sensibilities of the classicallyeducated reader. However, we also demonstrate that these issues are easily resolved if one thinks economically about the models in the right way. In particular, we argue that there is nothing inherently objectionable about either model.