according to Brigo & Mercurio (2006):
But how is the Zero bond Put of the CIR model? I couldn't find any information about that.
Thanks in advance.
Regards Chris
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Sign up to join this communityYou can simply apply formula (3.4) in Brigo and Mercurio's book (page 56). There is a simple put-call parity for the prices of European-style options written on zero-coupon bonds, i.e. \begin{align*} \mathbf{ZBP}(t,T,S,X) = \mathbf{ZBC}(t,T,S,X) -P(t,S)+XP(t,T). \end{align*} The formula is kind of identical to the standard equity put call parity where you discount the strike $X$ with the discount factor $P(t,T)$ and where $P(t,S)$ is the price of the underlying asset (a bond itself). It also follows directly from the Law of One Price (or no-arbitrage). Based on this observation, there are many parities relating interest rate derivatives (e.g. caps and floors as well as payer swaptions and receiver swaptions).
Please also note that the zero-coupon bond formula follows the 'Black-Scholes-style' \begin{align*} \mathbf{ZBC}(t,T,S,X) = P(t,S)\Pi_1 - XP(t,T) \Pi_2, \end{align*} where $\Pi_1$ and $\Pi_2$ are two probabilities (here related to the $\chi^2$ distribution of the short rate in the CIR model). Thus, the price of a zero-coupon bond put option equals \begin{align*} \mathbf{ZBP}(t,T,S,X) = XP(t,T) (1-\Pi_2) - P(t,S)(1-\Pi_1). \end{align*} The probabilities $\Pi_1$ and $\Pi_2$ can be seen as exercise probabilities using a change of numeraire (i.e. using the bonds maturing at time $T$ and $S$ as numeraires).