Bond Pricing in Continuous Time

Stochastic Foundations for Finance

Lorenzo Naranjo

Fall 2024

The Vasicek Model

• The model proposed by Vasicek (1977) assumes that the short-term interest rate follows an Ornstein–Uhlenbeck process, dr = \kappa (\theta - r) dt + \sigma dz_{r}.
  • The parameter \kappa determines the speed of mean-reversion towards the long-run mean \theta.
• When r is above \theta, \operatorname{E}(dr) < 0 so that the next rate is expected to be lower than the current one.
• Similarly, when r is below \theta, \operatorname{E}(dr) > 0 and the next rate is expected to be higher.

The figure plots a simulated path for the short-term interest rate r_{t_{n}} = r_{t_{n-1}} + \kappa (\theta - r_{t_{n-1}}) \Delta t + \sigma \sqrt{\Delta t} \varepsilon_{i} where 1 \leq n \leq 320000, t_{n} = n \Delta t, r_{0} = 0.08, \kappa = 0.15, \theta = 0.05, \sigma = 0.01, \{\varepsilon_{i}\} is a white noise process with unit variance and \Delta t = 40 / 320000 = 0.000125 years. The dashed line denotes \theta = 0.05.

Solving the Model

• Write u = r e^{\kappa t}, so that du = e^{\kappa t} dr + \kappa r e^{\kappa t} dt = \kappa \theta e^{\kappa t} dt + \sigma e^{\kappa t} dz_{r}.
• Integrating from 0 to T u(T) - u(0) = \theta (e^{\kappa T} - 1) + \sigma \int_{0}^{T} e^{\kappa t} dz_{r}(t). so that r(T) = e^{-\kappa T} r(0) + \theta (1 - e^{- \kappa T}) + \sigma e^{- \kappa T} \int_{0}^{T} e^{\kappa t} dz_{r}(t). \tag{1}

The Distribution of the Short-Rate

• Equation (1) implies that \operatorname{E}(r(T)) = e^{-\kappa T} r(0) + \theta (1 - e^{- \kappa T}). \tag{2}
  • We have that \lim_{T \rightarrow \infty} \operatorname{E}(r(T)) = \theta, so \theta is effectively the long term value of the short-term rate.
  • How fast this happens depends on large \kappa is.
• We also can compute \operatorname{Var}(r(T)) = \sigma^{2} e^{-2 \kappa T} \int_{0}^{T} e^{2 \kappa t} dt = \frac{\sigma^{2}}{2 \kappa} (1 - e^{-2 \kappa T}). \tag{3}
• Thus, r(T) is normally distributed with the mean and variance computed above.

The Stochastic Discount Factor

• The dynamics of the SDF are given by \frac{d\Lambda}{\Lambda} = - r dt - \lambda dz_{m}, where (dz_{m})(dz_{r}) = \rho_{rm} dt,
  • As usual, dz_{m} is generated from the Brownian motions driving the risk-factors in the economy.
• The interest-rate risk is priced as long as \rho_{rm} \neq 0.

Pricing of Bonds

• Consider a zero-coupon bond P(r, T) with face value 1 and expiring in T years.
• The fundamental pricing equation is \operatorname{E}\left(\frac{dP}{P}\right) - r dt = - \frac{d\Lambda}{\Lambda} \frac{dP}{P}, or \operatorname{E}(dP) - r P dt = - \frac{d\Lambda}{\Lambda} dP.

The Bond Price Evolution

• Since the zero-coupon bond price is P(r, T), Ito’s lemma implies that \begin{aligned} dP & = P_{r} dr + \frac{1}{2} P_{rr} (dr)^{2} - P_{T} dt \\ & = \left( \kappa (\theta - r) P_{r} + \frac{1}{2} \sigma^{2} P_{rr} - P_{T} \right) dt + P_{r} \sigma dz. \end{aligned}
• Thus, \frac{d\Lambda}{\Lambda} dP = - \sigma \rho \lambda P_{r} dt.

The Fundamental Pricing Equation

• The covariance between the SDF returns and the changes in the bond price implies that \kappa (\theta^{*} - r) P_{r} + \frac{1}{2} \sigma^{2} P_{rr} - P_{T} - r P = 0. \tag{4} where \theta^{*} = \theta - \frac{\sigma \rho \lambda}{\kappa}.
  • The equation requires that P(r, 0) = 1.
• To solve it, we can guess that P(r, T) = \exp(A(T) + r B(T)).
• We find that P_{r} = B P, P_{rr} = B^{2} P, and P_{T} = (A' + r B') P, so that \kappa (\theta^{*} - r) B + \frac{1}{2} \sigma^{2} B^{2} - (A' + r B') - r = 0. \tag{5}

Solving for the Bond Price

• Since (5) must be valid for any r, it must be the case that \begin{aligned} A' & = \kappa \theta^{*} B + \frac{1}{2} \sigma^{2} B^{2}, \\ B' & = - 1 - \kappa B, \end{aligned} subject to A(0) = 0 and B(0) = 0, implying that \begin{aligned} B(T) & = - \xi(T) T, \\ A(T) & = - \theta^{*} T (1 - \xi(T)) + \frac{1}{2} \frac{\sigma^{2}}{\kappa^{2}} T (1 - 2 \xi(T) + \xi(2 T)), \end{aligned} where \xi(T) = \frac{1 - e^{-\kappa T}}{\kappa T}.

The Zero-Coupon Rate

• The price of the zero-coupon bond can be written as P(r, T) = \exp(- y(T) T), where y(T) = \theta^{*} (1 - \xi(T)) - \frac{1}{2} \frac{\sigma^{2}}{\kappa^{2}} (1 - 2 \xi(T) + \xi(2 T)) + \xi(T) r, is the zero-coupon rate for maturity T.
• We note that \lim_{T \rightarrow 0} \xi(T) = 1, and \lim_{T \rightarrow \infty} \xi(T) = 0, implying \lim_{T \rightarrow 0} y(T) = r, \quad \lim_{T \rightarrow \infty} y(T) = \theta^{*} - \frac{1}{2} \frac{\sigma^{2}}{\kappa^{2}}.

The Instantaneous Forward Rate

• We can recover the zero-rate from instantaneous forward rates as y(T) = \frac{1}{T} \int_{0}^{T} f(s) ds.
• Thus, (y(T) T)' = y(T) + y'(T) T = f(T).
• We can use this expression to compute the instantaneous forward rate in the Vasicek model f(T) = \theta^{*} (1 - e^{-\kappa T}) - \frac{1}{2} \frac{\sigma^{2}}{\kappa^{2}} (1 - e^{-\kappa T})^{2} + e^{-\kappa T} r.

Interest Rate Futures

• The futures rate \varphi(T) is equal to the expected risk-adjusted short-rate \varphi(T) = \theta^{*} (1 - e^{- \kappa T}) + e^{-\kappa T} r, which implies that forward rates are downward biased estimates of futures rates since f(T) = \varphi(T) - \frac{1}{2} \frac{\sigma^{2}}{\kappa^{2}} (1 - e^{-\kappa T})^{2} < \varphi(T).
• The term \frac{1}{2} \frac{\sigma^{2}}{\kappa^{2}} (1 - e^{-\kappa T})^{2} > 0 is a convexity adjustment.
  • The convexity adjustment increases with the maturity of the forward rate (T \uparrow), the volatility of the short rate (\sigma \uparrow) and the persistence of the short-rate (K \downarrow).

The Convexity Adjustment

• In general, we can define the forward rate as the rate f(T) that solves \operatorname{E}^{*}\left([r(T) - f(T)] e^{-\int_{0}^{T} r(s) ds}\right) = 0, implying that f(T) = \varphi(T) + \frac{\operatorname{Cov}^{*}\left(r(T), e^{-\int_{0}^{T} r(s) ds}\right)}{B(T)}.
• Since \operatorname{Cov}^{*}\left(r(T), e^{-\int_{0}^{T} r(s) ds}\right) < 0, we must have that f(T) < \varphi(T).
  • The forward rate is a downward-biased estimate of the futures rate.

The CIR Model

• The model proposed by Cox, Ingersoll, and Ross (1985), usually denoted as the CIR model, assumes that the short-term interest rate follows a square-root process, dr = \kappa (\theta - r) dt + \sigma \sqrt{r} dz_{r}.
  • As with the Vasicek model, the parameter \kappa determines the speed of mean-reversion towards the long-run mean \theta.
• The main difference with the Vasicek model is that the variance of the short rate depends on its level.
  • As r approaches zero the variance goes down to zero as well.
• Thus, in the CIR model the short rate cannot be negative.

The Stochastic Discount Factor

• For the CIR model, the dynamics of the SDF can be specified as \frac{d\Lambda}{\Lambda} = - r dt - \left(\frac{\lambda_{1}}{\sqrt{r}} + \lambda_{2} \sqrt{r} \right) dz_{m}, where (dz_{m})(dz_{r}) = \rho_{rm} dt.
• The discount factor implies the following risk-neutral dynamics for the short rate, dr = \kappa^{*} (\theta^{*} - r) dt + \sigma \sqrt{r} dz_{r}^{*}.

Pricing a Zero-Coupon Bond

• The price P(r, T) of a zero-coupon bond expiring at T satisfies the following PDE \kappa^{*} (\theta^{*} - r) P_{r} + \frac{1}{2} \sigma^{2} r P_{rr} - P_{T} - r P = 0, with boundary condition P(r, 0) = 1.

References

Cox, John C., Jonathan E. Ingersoll, and Stephen A. Ross. 1985. “A Theory of the Term Structure of Interest Rates.” Econometrica, 385–407.

Vasicek, Oldrich. 1977. “An Equilibrium Characterization of the Term Structure.” Journal of Financial Economics 5 (2): 177–88.