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