We address the problem of optimally exercising American options based on the assumption that the underlying stock's price follows a Brownian bridge whose final value coincides with the strike price. In order to do so, we solve the discounted optimal stopping problem endowed with the gain function G(x) = (S - x)^+ and a Brownian bridge whose final value equals S. These settings came up as a first approach of optimally exercising an option within the so-called " stock pinning " scenario. The optimal stopping boundary for this problem isproved to be the unique solution , up to certain regularity conditions, of an integral equation, which is then numerically solved by an algorithm hereby exposed. We face the case where the volatility is unspecified by providing an estimated optimal stopping boundary that, alongside with pointwise confidence intervals, provide alternative stopping rules . Finally, we demonstrate the usefulness of our method within the stock pinning scenario through a comparison with the optimal exercise time based on a geometric Brownian motion . We base our comparison on the contingent claims and the 5-minutes intraday stock price data of Apple and IBM for the period 2011-2018. Supplementary materials with the main proofs and auxiliary lemmas are available online .
Mathematically, the execution of an American-style financial derivative is commonly reduced to solving an optimal stopping problem. Breaking the general assumption that the knowledge of the holder is restricted to the price history of the underlying asset, we allow for the disclosure of future information about the terminal price of the asset by modeling it as a Brownian bridge . This model may be used under special market conditions, in particular we focus on what in the literature is known as the " pinning effect " , that is, when the price of the asset approaches the strike price of a highly-traded option close to its expiration date. Our main mathematical contribution is in characterizing the solution to the optimal stopping problem when the gain function includes the discount factor. We show how to numerically compute the solution and we analyze the effect of the volatility estimation on the strategy by computing the confidence curves around the optimal stopping boundary . Finally, we compare our method with the optimal exercise time based on a geometric Brownian motion by using real data exhibiting pinning .
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We address the problem of optimally exercising American options based on the
Mathematically, the execution of an American-style financial derivative is commonly reduced to solving an optimal stopping problem. Breaking the general
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underlying stock's price follows a Brownian bridge whose final value coincides with the strike price. In order to do so, we solve the discounted optimal stopping problem endowed with the gain function G(x) = (S - x)^+ and
knowledge of the holder is restricted to the price history of the underlying asset, we allow for the disclosure of future information about the terminal price of the asset by modeling it as
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whose final value equals S. These settings came up as a first approach of optimally exercising an option within the so-called
. This model may be used under special market conditions, in particular we focus on what in the literature is known as the
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stock pinning
pinning effect
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scenario. The optimal stopping boundary for this problem isproved to be the unique solution , up to certain regularity conditions, of an integral equation, which is then numerically solved by an algorithm hereby exposed. We face the case where the volatility is unspecified by providing an estimated
, that is, when the price of the asset approaches the strike price of a highly-traded option close to its expiration date. Our main mathematical contribution is in characterizing the solution to the optimal stopping problem when the gain function includes the discount factor. We show how to numerically compute the solution and we analyze the effect of the volatility estimation on the strategy by computing the confidence curves around the
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that, alongside with pointwise confidence intervals, provide alternative stopping rules
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demonstrate the usefulness of our method within the stock pinning scenario through a comparison
compare our method
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. We base our comparison on the contingent claims and the 5-minutes intraday stock price data of Apple and IBM for the period 2011-2018. Supplementary materials with the main proofs and auxiliary lemmas are available online
by using real data exhibiting pinning