The problem of valuation for contingent claims that can be exercised at any time before or at maturity, such as American options, is discussed in the manner of Bensoussan [1]. We offer an approach which both simplifies and extends the results of existing theory on this topic.

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On the pricing of American options

https://www.math.columbia.edu/~ik/DayKar.pdf

On the optimal stopping problem for one-dimensional diffusions

This paper studies contingent claim valuation of risky assets in a stochastic interest rate economy. The model employed generalizes the approach utilized by Heath, Jarrow, and Morton (1992) by imbedding their stochastic interest rate economy into one containing an arbitrary number of additional risky assets. We derive closed form formulae for certain types of European options in this context, notably call and put options on risky assets, forward contracts, and futures contracts. We also value American contingent claims whose payoffs are permitted to be general functions of both the term structure and asset prices generalizing Bensoussan (1984) and Karatzas (1988) in this regard. Here, we provide an example where an American call’s value is well defined, yet there does not exist an optimal trading strategy which attains this value. Furthermore, this example is not pathological as it is a generalization of Roll’s (1977) formula for a call option on a stock that pays discrete dividends.

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Pricing options on risky assets in a stochastic interest rate economy

The valuation theory for American Contingent Claims, due to Bensous- san (1984) and Karatzas (1988), is extended to deal with constraints on portfolio choice, including incomplete markets and borrowing/short-selling constraints, or with different interest rates for borrowing and lending. In the unconstrained case, the classical theory provides a single arbitrage-free price u0; this is expressed as the supremum, over all stopping times, of the claim's expected discounted value under the equivalent martingale measure. In the presence of constraints, fu0g is replaced by an entire interval (hlow; hup) of arbitrage-free prices, with endpoints characterized as hlow = inf2D u; hup = sup 2D u . Here u is the analogue of u0, the arbitrage-free price with unconstrained portfolios, in an auxiliary market model M ; and the family fMg2D is suitably chosen, to contain the original model and to reflect the constraints on portfolios. For several such constraints, explicit computations of the endpoints are carried out in the case of the American call-option. The analysis involves novel results in martingale theory (including simultaneous Doob-Meyer decompositions), optimal stopping and stochastic con- trol problems, stochastic games, and uses tools from convex analysis.

/pdf/hedging-american-contingent-claims-with-constrained-3yuhotvnl3.pdf

Hedging American contingent claims with constrained portfolios

This idea is employed to show that direct integration of the optimal risk in a stopping problem for Brownian motion, yields the value function of the so-called monotone follower stochastic control problem and provides an explicit construction of its optimal process. Ideas from the theory of balayage for continuous semimartingales are employed, in order to find novel and useful representations for the value functions of these problems.

A. G. Fakeev

Papers

Optimal Stopping Rules for Stochastic Processes with Continuous Parameter