Partial hedging of American options in discrete time and complete markets: convex duality and optimal Markov policies

We consider the super-hedging price of an American option in a discrete-time market in which stocks are available for dynamic trading and European options are available for static trading. We show that the super-hedging price [Formula: see text] is given by the supremum over the prices of the American option under randomized models. That is, [Formula: see text], where [Formula: see text] and the martingale measure [Formula: see text] are chosen such that [Formula: see text] and [Formula: see text] prices the European options correctly, and [Formula: see text] is the price of the American option under the model [Formula: see text]. Our result generalizes the example given in Hobson & Neuberger (2016) that the highest model-based price can be considered as a randomization over models.

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Asymptotic synthesis of contingent claims with controlled risk in a sequence of discrete‐time markets

Theoretical Economics ◽

10.3982/te4034 ◽

2021 ◽

Vol 16 (1) ◽

pp. 25-47

Author(s):

David M. Kreps ◽

Walter Schachermayer

Keyword(s):

Financial Markets ◽

Discrete Time ◽

Continuous Time ◽

Contingent Claims ◽

Contingent Claim ◽

Complete Markets ◽

Continuous Time Model ◽

Time Model ◽

Black Scholes ◽

Discrete Time Models

We examine the connection between discrete‐time models of financial markets and the celebrated Black–Scholes–Merton (BSM) continuous‐time model in which “markets are complete.” Suppose that (a) the probability law of a sequence of discrete‐time models converges to the law of the BSM model and (b) the largest possible one‐period step in the discrete‐time models converges to zero. We prove that, under these assumptions, every bounded and continuous contingent claim can be asymptotically synthesized, controlling for the risks taken in a manner that implies, for instance, that an expected‐utility‐maximizing consumer can asymptotically obtain as much utility in the (possibly incomplete) discrete‐time economies as she can at the continuous‐time limit. Hence, in economically significant ways, many discrete‐time models with frequent trading resemble the complete‐markets model of BSM.

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Pricing and hedging American options in complete markets

Translations of Mathematical Monographs - Financial Markets ◽

10.1090/mmono/184/05 ◽

1999 ◽

pp. 47-53

Keyword(s):

American Options ◽

Complete Markets

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Optimal Hedging of American Options in Discrete Time

SSRN Electronic Journal ◽

10.2139/ssrn.1729421 ◽

2010 ◽

Cited By ~ 2

Author(s):

Bruno Remillard ◽

Hugues Langlois ◽

Alexandre Hocquard ◽

Nicolas A. Papageorgiou

Keyword(s):

Discrete Time ◽

American Options ◽

Optimal Hedging

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A note on American options with varying exercise price

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000007566 ◽

1995 ◽

Vol 37 (1) ◽

pp. 45-57

Author(s):

J. N. Dewynne ◽

P. Wilmott

Keyword(s):

Time Series ◽

Random Walk ◽

Discrete Time ◽

Continuous Time ◽

A Priori ◽

American Options ◽

Share Price ◽

Exercise Price ◽

Black Scholes

AbstractWe examine the valuation of American options in a discrete time setting where the exercise price is known a priori but varies with time. (This is in contrast with the classical Black-Scholes [2] analysis, which lies in a continuous time framework and with constant exercise price.) In particular we consider a time series of exercise prices which are themselves a realisation of the share price random walk — that of the previous year, say.

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