Implied remaining variance with application to bachelier model

In this article, we propose a way to model vanilla options implied volatility curve in closed form under the Brownian motion assumption for underlying asset. The work is an extension of the recent article by Carr and Sun [2013]. Using Brownian motion to model the financial assets was first proposed by Bachelier one century ago. Even though in the option pricing world, the usual setting is geometric Brownian motion, which guarantees the positiveness of the underlying price, it is becoming common to use the Brownian motion in some other situations as well, in particular for interest rate derivatives due to the environments of super low or even negative rates in JPY, EUR, and USD currencies. We will use our model to calibrate the implied normal volatilities in the swaption market. Calibration results show that the model proposed in this article works very well. Furthermore we will prove that, under certain conditions, our model could be free of arbitrage. The proofs of the main results are left in the appendix.