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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You 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).
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$\begingroup$ Thank you very much! If the ZBC is a Floor for example, is Π1 the Chi square in the formula of my post? Can I simply calculate (1-Π1) and (1-Π2) to achieve a Cap valuation? Or is the 1-Π1 and 1-Π2 the term INSIDE the chi square probability distribution. How to the non central parameter and the degree of freedom change? $\endgroup$ Nov 13, 2019 at 14:32
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$\begingroup$ @ChristianM I'm not quite sure I get your question. A floor is a portfolio of European-style zero-coupon bond call options. The bond call options can be prices with the formula cited in your question. The floor price is then a linear combination of these bond option prices. The corresponding cap price can be found by the cap floor parity. The $\Pi_1$ and $\Pi_2$ in my answer correspond to $N(d_1)$ and $N(d_2)$ in the Black-Scholes equity options case or to $\chi^2(a,b,c)$ in the CIR model (see your question). They are just probabilities. $\endgroup$– KevinNov 13, 2019 at 20:02
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$\begingroup$ Alright, thank you! I cannot price over Put-Call parity sincei do not know the swap prices, but I will try to calibrate my caplets via your suggestion of the ZBP with (1-Π1) and (1-Π2). $\endgroup$ Nov 13, 2019 at 20:03
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$\begingroup$ @ChristianM Question fully answered and no open questions left? I hope I didn't confuse you with the comments about $\Pi_1$ and $\Pi_2$. It's just nice to see how these formulae fit into a 'greater picture' (: $\endgroup$– KevinNov 13, 2019 at 20:05