On the Option Pricing Formula Based on the Bachelier Model

Abstract

Under the recent negative interest rate situation, the Bachelier model has been attracting attention and adopted for evaluating the price of interest rate options. In this paper, we will derive an option pricing formula based on the Bachelier model and compare it with the prior researches. We will derive it by eight methods and clarify the property of the Bachelier model. Then we will confirm the validity of the Normal model that is actually used in the valuation of interest rate options under negative interest rate, while comparing it with the Bachelier model for stocks. We start from the natural setting of modeling the undiscounted stock price by the Ornstein=Uhlenbeck process, and derive the Bachelier formula in consideration of discount. On the other hand, since the major prior researches start from modeling the discounted stock price by the Brownian motion, their models of the undiscounted stock price has an unnatural setting that the price of the numeraire asset is included. Furthermore, It has been confirmed that their formulas are not consistent among them. During the derivation process, we have obtained various results concerning the Bachelier model. In particular, in the case of the Bachelier model, it has been confirmed that the utility function of a representative agent is the CARA utility function unlike the Black-Scholes model. The assumption of the exponential type utility function is quite natural setting. In addition, we have derived other expressions of the Bachelier's formula (the formula decomposed into the intrinsic value and the time value and the formula using a characteristic function). As for the Normal model used for pricing interest rate options, we have derived an original pricing formula (Modified Normal model) in which the unnatural points of the Normal model of the forward LIBOR and forward swap rate have been partially corrected.