Reading the CIR Bond Call Formula’s Noncentral Chi-Square Terms

Summary

The document presents the closed-form price of a call option on a zero-coupon bond under the Cox-Ingersoll-Ross interest-rate model. The formula expresses the price as a difference of bond-price terms weighted by noncentral chi-square distribution functions. It identifies the distribution’s degrees of freedom and noncentrality parameter, then asks how to interpret the remaining leading argument when simulating the distribution.

The question is specifically whether that argument should be applied as a multiplier or divisor after generating a random variable. No response or derivation is supplied, so the document does not resolve the interpretation or explain the formula’s distribution-function convention. It provides a pricing context and parameterization issue, but cannot serve alone as an implementation recipe. Users would need to consult the model derivation and verify whether the chi-square notation denotes a cumulative probability or another quantity.

Key ideas

• The CIR model supplies a closed-form expression for pricing a call option on a bond.
• The formula uses noncentral chi-square terms with degrees of freedom and noncentrality parameters.
• The question concerns how to interpret the leading argument in the distribution notation when simulating.
• No answer clarifies whether that argument is a multiplier or divisor, or specifies a simulation procedure.

Tags

# Pricing call option on bond under CIR model by simulating noncentral chi square distribution


# Pricing call option on bond under CIR model by simulating noncentral chi square distribution












In the original paper of CIR model, there is a pricing formula about call option on bond $$ \begin{array}{l}{C(r, t, T ; s, K)} \\ {=P(r, t, s) \chi^{2}\left(2 r^{*}[\phi+\psi+B(T, s)] ; \frac{4 \kappa \theta}{\sigma^{2}}, \frac{2 \phi^{2} r e^{\gamma(T-t)}}{\phi+\psi+B(T, s)}\right)} \\ {-K P(r, t, T) \chi^{2}\left(2 r^{*}[\phi+\psi] ; \frac{4 \kappa \theta}{\sigma^{2}}, \frac{2 \phi^{2} r e^{\gamma(T-t)}}{\phi+\psi}\right)}\end{array} $$ However, I can't understand the first parameter of the noncentral chi square. The second and the third parameter is the degree of freedom and the noncentral parameter. After simulating the random variable with these two parameters, what should I to get the value of $$ \chi^{2}\left(2 r^{*}[\phi+\psi+B(T, s)] ; \frac{4 \kappa \theta}{\sigma^{2}}, \frac{2 \phi^{2} r e^{\gamma(T-t)}}{\phi+\psi+B(T, s)}\right) $$ Should I divide or multiply it?

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