Short Rates, Zero Rates, and Option Pricing with Stochastic Rates

# Valuing derivatives under stochastic interest rates


# Valuing derivatives under stochastic interest rates












I would like to price a European option with maturity equals to 5 years. To do this, I'm using the Black-Scholes model with stochastic interest rates.

Suppose I choose the CIR model for the risk-free rate. My question is: should I model the entire term structure of interest rates, or I can just model the 5-year rate?

As a side question, which one would be considered a good proxy for the 5-year risk-free rate in the US?

## Answer by Gordon (score 3, accepted)

https://quant.stackexchange.com/a/26360

A few points can be noted.

- The CIR model is usually for a short, or instantaneous, spot rate $r_t$, which is the forward rate over an infinitesimal interval. That is, \begin{align*} r_t = \lim_{\Delta \rightarrow 0}\frac{1}{\Delta}\left(\frac{1}{P(t, t+\Delta)}-1 \right), \end{align*} where $P(t, u)$ is the price at time $t$ of a zero-coupon bond with maturity $u$ and unit face value.

- The $T$-year rate is usually the zero rate $R_T$, defined by \begin{align*} P(0, T) = e^{-R_T T},\tag{1} \end{align*} which is not the short rate.

- Hull-White model may be better as the initial term structure of zero rates, or correspondingly, bond prices, can be matched.

- For a vanilla European option with a payoff of the form \begin{align*} \max(S_T-K, \, 0), \end{align*} the value is given by the Black's formula \begin{align*} P(0, T)\big[F_TN(d_1) -KN(d_2) \big].\tag{2} \end{align*} Here, $F_T=S_0/P(0, T)$ is the forward price, $d_1 = \frac{\ln F_T/K + \frac{1}{2}\sigma^2 T}{\sigma \sqrt{T}}$, and $d_2 = d_1 - \sigma \sqrt{T}$. Note that, in Formula $(2)$, the volatility $\sigma$ is Black's implied volatility, which can usually be obtained from the market quote. In this case, the stochastic interest rate model is not really needed. That is, only the $T$-year zero rate $R_T$ is needed to compute the bond price $P(0, T)$ by Formula $(1)$. Here, in your case, the 5-year zero rate is needed. However, we note that the Black's implied volatility is different from the Black-Scholes' implied volatility, if stochastic interest rate is assumed. See this question for a detailed exposition.