Hull-White Model

The Hull-White model (1990), an interest rate model, is a yield-based no-arbitrage model. This is extension of the Vasicek model. The model allows closed-form solutions for European options on zero-coupon bonds.

TheoV

Call

Put

Where

L – bond principal (i.e. face value),

τ – bond time maturity,

,

,

P(T) - the price at time zero of a zero-coupon bond that pays $1 at time T,

a – the speed of the mean reversion.

Unlike Vasicek model, P_T and P_τ are input parameters.

Delta

Call

Put

Gamma

Gamma is identical for put and call options.

Vega

Because

Theta

Call

Put

where

Rho

Since, the price at time zero of a zero-coupon bond that pays $1 at time t is

then

Call

Put

Implied volatility

The system finds implied volatility numerically.