# Jamshidian's trick for Swaptions

Following Brigo$^1$ p.77, we can decompose the price of a swaption as a sum of Zero-Coupon bond options (Jamshidian's Trick).

To do so, the authors suggest to find $r^*$ the value of the spot rate at $t$ for which $\sum_{i=1}^n c_i P(t,T_i, r*)= 1$

As an example they show this for the Vasicek Model, where $A()$ and $B()$ give the analytical ZCB Prices.

$\sum_{i=1}^n c_i A(t,T_i) e^{-B(t,T_i)r*} = 1$

Assume we have a short-rate model for which we do not have an analytical expression for the ZCB prices (e.g. Exponential Vasicek), how can we still find the value of $r^*$?

See another $^2$ step-by-step explanation on the problem

• If you do not have an analytical bond price formula, you won't be able to find $r^*$ specifically, though you may be able to show the existence if the bond price $P(t, T)$ is a monotonic function of $r_t$ for any $T\ge t$. – Gordon Jan 27 '17 at 15:56
• @Gordon Does that help if I do simulations to find out the optimal value? – SmallChess Jan 28 '17 at 4:15
• are there any numerical implications/advantages for this trick? is it performant to price/risk a series of options using analytic formulae rather than doing a bump and reval on the portfolio? are there any studies on this aspect? thank you – satishv Jul 1 '18 at 13:12