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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?

Egodym
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1 Answers1

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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.
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Gordon
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  • Thank you, but my question was more a follow-up of one you answered a year ago: http://quant.stackexchange.com/questions/18289/black-scholes-under-stochastic-interest-rates . Under the framework described, can I model the 5-year rate using CIR just by considering the 5-year gov yield time series or should I model the entire yield curve? – Egodym May 31 '16 at 20:12
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    @Egodym: You do not need to model the 5-year rate. What you need is the 5-year zero rate, which you can obtain from a given yield curve. The previous question is based on expectation with risk-neutral measure. However, with the $T$-forward measure, the option is given by (2) above. – Gordon May 31 '16 at 20:47
  • Hence, I use the 5-year zero given by the yield curve I can observe in the market. However, this would mean that interest rates are going to be constant for the whole life of the option, which is not the case for options with longer maturities (cfr. http://www.actuaires.org/AFIR/colloquia/Toronto/Lin_Tan.pdf ). Thus, the need to model the short rate. What would you do in this case? – Egodym May 31 '16 at 20:57
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    @Egodym: The 5-year zero rate is a constant. Note that the zero rates form the initial term structure, which does not mean that the short interest rate $r_t$ is constant. – Gordon May 31 '16 at 21:02
  • Yes, short rate is random, but by doing so we use a constant zero rate to price the option. My objective was to account for the dynamic of interest rates in the option price, since the maturity is much longer than that of usually traded options. – Egodym May 31 '16 at 21:12
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    @Egodym What Gordon is telling you, and he is perfectly right, is that when you price a European equity option, only the zero rate at the maturity date of the option matters in the end (or equivalently the price of a zero coupon bonds $P(0,T)$), even when short interest rates are stochastic. In other words, the short rate stochasticity does not transpire in the price of European contingent claims. This stochasticity can however be crucial when pricing path-dependent derivatives. – Quantuple May 31 '16 at 21:47
  • Thanks Quantuple. Two follow-up questions: 1) if that's the case, why the paper I metioned above develops a model for interest rates too? 2) Could you provide me with an example of when stochasticity is crucial? – Egodym May 31 '16 at 21:49
  • Thanks @Quantuple. for a vanilla European option where the maturity and payment dates are the same, the stochastic nature of the underlying short interest rate does not play any role for valuation, while only the $T$-year zero rate matters. For path-dependent derivatives, such as Asian options, the valuation whether the underlying interest rate is deterministic or stochastic will then matter. – Gordon May 31 '16 at 22:30
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    Asian option are a good example. Bermudan options as well. Any path-dependent option. – Quantuple May 31 '16 at 22:32
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    In addition, if the payment is much later than the option maturity (e.g., paid at $T+ \Delta$), the stochastic nature will be matter for the option value. – Gordon Jun 01 '16 at 01:25