according to Brigo & Mercurio (2006):

enter image description here

But how is the Zero bond Put of the CIR model? I couldn't find any information about that.

Thanks in advance.

Regards Chris


1 Answer 1


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).

  • $\begingroup$ 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
  • $\begingroup$ @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. $\endgroup$
    – Kevin
    Nov 13, 2019 at 20:02
  • $\begingroup$ 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
  • $\begingroup$ @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' (: $\endgroup$
    – Kevin
    Nov 13, 2019 at 20:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.