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In this work, we expand the idea of Samuelson[3] and Shepp[2,5,6] for stockoptimization using the Bachelier model [4] as our models for the stock price atthe money (X[stock price]= K[strike price]) for the American call and putoptions [1]. At the money (X= K) for American options, the expected payoff ofboth the call and put options is zero. Shepp investigated several stochasticoptimization problems using martingale and stopping time theories [2,5,6]. Oneof the problems he investigated was how to optimize the stock price using boththe Black-Scholes (multiplicative) and the Bachelier (additive) models [7,6]for the American option above the strike price K (exercise price) to a stoppingpoint. In order to explore the non-relativistic quantum effect on the expectedpayoff for both the call and put options at the money, we assumed the stockprice to undergo a stochastic process governed by the Bachelier (additive)model [4]. Further, using Ito calculus and martingale theory, we obtained adifferential equation for the expected payoff for both the call and put optionsin terms of delta and gamma. We also obtained the solution to thenon-relativistic Schroedinger equation as the expected payoff for both the calland put options. Then, we expressed the stochastic process that is the expectedpayoff for both the call and put options at the money in terms of the solutionto the Schroedinger equation. We concluded the stochastic process that is theexpected payoff at the money for both options to be an oscillatory functionwith quantized interest rates.

Quantized Interest Rate at the Money for American Options