We compare the option pricing formulas of Louis Bachelier and Black-Merton-Scholes and observe – theoretically and by typical data – that the prices coincide very well. We illustrate Louis Bachelier’s efforts to obtain applicable formulas for option pricing in pre-computer time. Furthermore we explain – by simple methods from chaos expansion – why Bachelier’s model yields good short-time approx...

This paper investigates the situation in Merton (1969) model that volatility is a constant rather than stochastic process, then points out this misspecification since it doesn't match real market. Next, HJB equation with derived through control, thereby calibrate misspecification.

A perturbation involving a small parameter of the Black-Scholes-Merton model with time dependent volatility is considered. A pricing formula is derived as an asymptotic series in powers of the small parameter. The summation of this series is performed, using methods of the theory of Borel summability in a suitable direction.

The valuation of counterparty risk for single name credit derivatives requires the computation of joint distributions of default times of two default-prone entities. For a Merton-type model, we derive some formulas for these joint distributions. As an application, closed formulas for coun-terparty risk on a CDS or for a first-to-default swap on two underlyings are obtained.