Editing Hull–White model (section)

==Derivative pricing==

By selecting as [[numeraire]] the time-''S'' bond (which corresponds to switching to the ''S''-forward measure), we have from the [[fundamental theorem of arbitrage-free pricing]], the value at time ''t'' of a derivative which has payoff at time ''S''.

:<math>V(t) = P(t,S)\mathbb{E}_S[V(S) \mid \mathcal{F}(t)].</math>

Here, <math>\mathbb{E}_S</math> is the expectation taken with respect to the [[forward measure]]. Moreover, standard arbitrage arguments show
that the time ''T'' forward price <math>F_V(t,T)</math> for a payoff at time ''T'' given by ''V(T)'' must satisfy <math>F_V(t,T) = V(t)/P(t,T)</math>, thus
:<math>F_V(t,T) = \mathbb{E}_T[V(T)\mid\mathcal{F}(t)].</math>

Thus it is possible to value many derivatives ''V'' dependent solely on a single bond <math>P(S,T)</math> analytically when working in the Hull–White model. For example, in the case of a [[put option|bond put]]
:<math>V(S) = (K-P(S,T))^+.</math>

Because <math>P(S,T)</math> is lognormally distributed, the general calculation used for the [[Black–Scholes model]] shows that

:<math>{E}_S[(K-P(S,T))^{+}] = KN(-d_2) - F(t,S,T)N(-d_1),</math>

where

:<math>d_1 = \frac{\log(F/K) + \sigma_P^2S/2}{\sigma_P \sqrt{S}}</math>

and

:<math>d_2 = d_1 - \sigma_P \sqrt{S}.</math>

Thus today's value (with the ''P''(0,''S'') multiplied back in and ''t'' set to 0) is:

:<math>P(0,S)KN(-d_2) - P(0,T)N(-d_1).</math>

Here <math>\sigma_P</math> is the standard deviation (relative volatility) of the log-normal distribution for <math>P(S,T)</math>. A fairly substantial amount of algebra shows that it is related to the original parameters via

:<math>\sqrt{S}\sigma_P
=\frac{\sigma}{\alpha}(1-\exp(-\alpha(T-S)))\sqrt{\frac{1-\exp(-2\alpha S)}{2\alpha}}.</math>

Note that this expectation was done in the ''S''-bond measure, whereas we did not specify a measure at all for the original Hull–White process. This does not matter — the volatility is all that matters and is measure-independent.

Because [[interest rate caps/floors]] are equivalent to bond puts and calls respectively, the above analysis shows that caps and floors can be priced analytically in the Hull–White model. [[Jamshidian's trick]] applies to Hull–White (as today's value of a swaption in the Hull–White model is a [[monotonic function]] of today's short rate). Thus knowing how to price caps is also sufficient for pricing swaptions. In the event that the underlying is a compounded backward-looking rate rather than a (forward-looking) LIBOR term rate, Turfus (2020) shows how this formula can be straightforwardly modified to take into account the additional [[convex function|convexity]].

Swaptions can also be priced directly as described in Henrard (2003). Direct implementations are usually more efficient.