Abstract
The main result of this paper is a probabilistic proof of the penalty method for approximating the price of an American put in the Black-Scholes market. The method gives a parametrized family of partial differential equations, and by varying the parameter the corresponding solutions converge to the price of an American put. For each PDE the parameter may be interpreted as a rationality parameter of the holder of the option. The method may be extended to other valuation situations given as an optimal stopping problem with no explicit solution. The method may also be used for valuations where actors do not behave completely rationally but instead have randomness affecting their choices. The rationality parameter is a measure for this randomness.
Highlights
The buyer of an American put can exercise at any time of his choice within the time of the contract.The arbitrage-free value of the American put is formulated as an optimal stopping problem, where the optimal stopping time is an optimal exercise rule for the buyer of the American put
The irrationality may be due to that the buyer does not have the correct input for the model, he does not monitor his position sufficiently, or he holds the American put as part of a hedge where it might not be
The main contribution of the present paper is a probabilistic proof of the following convergence result: Under mild restrictions the value of the American put in the intensity-based model converges to the arbitrage-free value when the rationality parameter converges to infinity
Summary
The buyer of an American put can exercise at any time of his choice within the time of the contract. The main contribution of the present paper is a probabilistic proof of the following convergence result: Under mild restrictions the value of the American put in the intensity-based model converges to the arbitrage-free value when the rationality parameter converges to infinity. The proof decomposes the value of the American put in the intensity-based model into the arbitrage-free price and losses coming from respectively exercising when it is not optimal and not exercising when it is optimal.
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