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Pricing CIR Zero-Coupon Bond Puts with Put-Call Parity

Article Quant Q&A · Author: Christian M
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Summary

The document explains how to price a European put option on a zero-coupon bond under the Cox-Ingersoll-Ross (CIR) interest-rate model. It derives the put price from the corresponding bond call price using put-call parity: the put equals the call less the current bond value, plus the discounted strike payment. This relation follows from no-arbitrage and is analogous to parity for equity options.

It then applies the CIR bond-call formula, which is expressed using two probabilities tied to the noncentral chi-squared distribution of the short rate. Substitution gives the put price in terms of the complements of those probabilities. The probabilities can be interpreted as exercise-related quantities under changes of numeraire using bonds maturing at the option and underlying bond dates. The document presents a formula-based explanation, but provides no numerical example or discussion of calibration and model risk.

Key ideas

  • European options on zero-coupon bonds satisfy put-call parity through no-arbitrage.
  • The bond put price can be obtained from the call price, the underlying bond price, and the discounted strike.
  • The CIR bond-call formula uses probabilities associated with the short-rate distribution.
  • Changing numeraire gives an exercise-related interpretation of the probabilities in the pricing formula.

Tags

Fixed incomeOptionsDerivatives pricing
Full text
# Cox-Ingersoll-Ross Zero Bond Put Option


# Cox-Ingersoll-Ross Zero Bond Put Option












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

## Answer by Kevin (score 1, accepted)

https://quant.stackexchange.com/a/49679

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.