May 15, 2023
The Ho-Lee model assumes the stochastic short rate process is f_t = r + \sigma B_t where r and \sigma are constant and B_t is standard Brownian motion.
The stochastic discount is D_t = \exp(-\int_0^t f_s\,ds) = \exp(-\int_0^t r + \sigma B_s\,ds) = \exp(-rt - \sigma\int_0^t B_s\,ds).
The discount to time t is D(t) = E[D_t]. The forward curve f(t) is defined by D(t) = \exp(-\int_0^t f(s)\,ds).
Exercise. Show f(t) = -(d/dt)\log D(t).
Hint: Using Ito’s formula, d(t B_t) = B_t\,dt + t dB_t since dt\,dB = 0.
Exercise. If dX_t/X_t = \Lambda(t)\,dB_t and X_0 = 1 then X_t = \exp(-\frac{1}{2}\int_0^t \Lambda(s)^2\,ds + \int_0^t \Lambda(s)\,dB_s).
Hint: d\log X_t = (1/X_t)\,dX_t + \frac{1}{2}(-1/X_t^2)\,dX_t^2 = \Lambda(t)\,dB_t - \frac{1}{2}\Lambda(t)^2\,dt using Ito’s calculus.
The discount at time t of a zero coupon bond maturing at u is D_t(u) = E_t[D_u/D_t]. The stochastic forward curve f_t(u) is defined by D_t(u) = \exp(-\int_t^u f_t(s)\,ds).
Exercise. Show \int_0^t B_s\,ds = -\int_0^t s dB_s + t B_t.
Hint: d(t B_t) = t dB_t + B_t\,dt + dt\,dB_t and dt\,dB_t = 0.
This shows D_t = \exp(-rt + \sigma \int_0^t s\,dB_s - \sigma tB_t).
Exercise. If dX_t/X_t = \Lambda(t)\,dB_t and X_0 = 1 then X_t = \exp(-\frac{1}{2}\int_0^t \Lambda(s)^2\,ds + \int_0^t \Lambda(s)\,dB_s).
Hint: d\log X_t = (1/X_t)\,dX_t + \frac{1}{2}(-1/X_t^2)\,dX_t^2 = \Lambda(t)\,dB_t - \frac{1}{2}\Lambda(t)^2\,dt by Ito’s formula.
Exercise. Show E_t[\exp(\int_t^u \Lambda(s)\,dB_s)] = \exp(\frac{1}{2}\int_t^u \Lambda(s)^2\,ds).
Hint: X_t is a martingale. Note the right-hand side is not random.
X_t = E_t[\exp(\int_0^u \Lambda(s)\,dB_s - \frac{1}{2}\Lambda(s)^2\,ds)]
1= E_t[\exp(\int_t^u \Lambda(s)\,dB_s - \frac{1}{2}\Lambda(s)^2\,ds)]
The price at time t of a zero coupon bond maturing at time u, D_t(u), satisfies D_t(u)D_t = E_t[D_u] so
\begin{aligned} D_t(u) &= E_t[D_u/D_t] \\ &= E_t[\exp(-\int_t^u r + \sigma B_s\,ds)] \\ &= E_t[\exp(-r(t - u) - \int_t^u \sigma B_s\,ds)] \\ &= E_t[\exp(-r(u - t) + \int_t^u \sigma s\,dB_s - \sigma (u B_u - t B_t))] \\ &= E_t[\exp(-r(u - t) + \int_t^u \sigma s\,dB_s - \sigma (u B_u - u B_t + u B_t - t B_t))] \\ &= E_t[\exp(-r(u - t) + \int_t^u \sigma s\,dB_s - \int_t^u \sigma u\,dB_s + \sigma(u - t) B_t)] \\ &= E_t[\exp(-r(u - t) - \int_t^u \sigma (u - s)\,dB_s + \sigma(u - t) B_t)] \\ &= E_t[\exp(-r(u - t) - \frac{1}{2}\int_t^u \sigma^2 (u - s)^2\,ds + \sigma(u - t) B_t)] \\ \end{aligned}
This show f_t(u) = r(u - t) + \frac{1}{2}\int_t^u \sigma^2 (u - s)^2\,ds - \sigma(u - t) B_t).
Exercise. Show \int_t^u (s - u)^2\,ds = (u - t)^3/3.In the Ho-Lee model the dynamics of zero coupon bond prices are D_t(u) = \exp(-r(u - t) + \frac{1}{6}\sigma^2(u - t)^3 - \sigma(u - t) B_t). In particular, the discount to time t is D(t) = D_0(t) = \exp(-rt + \sigma^2 t^3/6).
Exercise. Show the forward curve is f(t) = r - \sigma^2 t^2/2.Hint: D(t) = \exp(-\int_0^t f(s)\,ds).
Define the stochastic forward curve at time t, f_t(u), by D_t(u) = \exp(-\int_t^u f_t(s)\,ds). Note f_t(t) = f_t is the stochastic short rate. For each t there is a futures contract expiring at t on f_t. The futures quote at s is \phi_s(t) = E_s[f_t] since futures quotes are a martingale.
Exercise. Show f_t(u) = r - \sigma^2 (u - t)^2/2 + \sigma B_t.
Exercise. Show the stochastic forward curve is f_t(u) = r - \sigma^2 (u - t)^2/2 - \sigma B_t.
Note f_t(t) = r + \sigma B_t = f_t.
Exercise. Show E[f_t] - f(t) = \sigma^2t^2/2.The difference between the futures quote and forward rate is called convexity.
Exercise. Derive the formula for D_t(u) when r = r(t) is a function of time.
Exercise. Find the formula for D_t(u) when r = r(t) is a function of time.We can also allow \sigma = \sigma(t) to be a function of time. Let f_t = r(t) + \sigma(s) B_t. Since d(\Sigma(t)B_t) = \Sigma'(t)B_t\,dt + \Sigma(t)\,dB_t and taking \sigma(s) = \Sigma'(s) we have \begin{aligned} E_t[D_u/D_t] &= E_t[\exp(-\int_t^u r(s) + \sigma(s) B_s\,ds)] \\ &= E_t[\exp(-\int_t^u r(s)\,ds + d(\Sigma(s)B_s) - \Sigma(s)\,dB_s)] \\ &= E_t[\exp(-\int_t^u r(s)\,ds + \Sigma(u)B_u - \Sigma(t)B_t - \int_u^t \Sigma(s)\,dB_s)] \\ &= E_t[\exp(-\int_t^u r(s)\,ds + (\Sigma(u)B_u - \Sigma(u)B_t + \Sigma(u)B_t - \Sigma(t)B_t) - \int_u^t \Sigma(s)\,dB_s)] \\ &= E_t[\exp(-\int_t^u r(s)\,ds + \Sigma(u)\int_t^u dB_s + (\Sigma(u) - \Sigma(t))B_t - \int_u^t \Sigma(s)\,dB_s)] \\ &= E_t[\exp(-\int_t^u r(s)\,ds + \int_t^u (\Sigma(u) - \Sigma(s))\,dB_s + (\Sigma(u) - \Sigma(t))B_t )] \\ &= \exp(-\int_t^u r(s)\,ds + \frac{1}{2}\int_t^u (\Sigma(u) - \Sigma(s))^2\,ds + (\Sigma(u) - \Sigma(t))B_t ) \\ \end{aligned}
Since \int_t^u f_t(s)\,ds = \int_t^u r(s)\,ds + \frac{1}{2}\int_t^u (\Sigma(u) - \Sigma(s))^2\,ds + (\Sigma(u) - \Sigma(t))B_t we have f_t(u) = r(u) + \sigma(u) \int_t^u (\Sigma(u) - \Sigma(s))\,ds + \sigma(u) B_t using (d/dx) \int_a^x g(x,s)\,ds = g(x,x) + \int_a^x (\partial/\partial x)g(x,s)\,ds.
Exercise. If \sigma(t) = \sigma is constant then f_t(u) = r(u) - \sigma^2 (u - t)^2/2 + \sigma B_t.
A forward contract is specified by an interval [t,u], a forward rate f, and a day count basis \delta. It has cash flows -1 at t and 1 + f\delta(t,u) at u where the day count fraction \delta(t,u) is approximately equal to the time in years from t to u.
Exercise. The price of the forward contract is zero at time s \le t if and only if f = (D_s(t)/D_s(u) - 1)/\delta(t,u)
Hint: 0 = E_s[-D_t + (1 + f\delta)D_u].
We call F_s^\delta(t,u) = (D_s(t)/D_s(u) - 1)/\delta(t,u) the par forward at time s over [t,u] for day count basis \delta.
Exercise Show E_s[F_t(t,u))\delta(t,u)D_u]] = E_s[D_t - D_u].
A forward contract paying in arrears is also specified by an interval [t,u], a forward rate f, and a day count basis \delta. It has a single cash flow (f - F_t^\delta(t,u))\delta(t,u) at u. Note F_t(t,u) is the forward rate at t over the interval [t, u]. The effective date of the contract is t and the termination date is u.
Exercise. Show E_s[-D_t + (1 + f\delta)D_u] = E_s[(f - F_t(t,u))\delta(t,u)D_u].
Hint: Use the previous exercise.
Both contracts have the same risk-neutral value, but they have very different risk profiles.
A forward contract involves the exchange of a notional amount at the beginning and end of the contract. We have been using unit notional, but real-world contracts specify a notional N with cash flows -N at t and N(1 + f\delta(t,u)) at u. If one counterparty defaults during the interval [t,u] then the other counterparty will not get paid what they expect. If the absolute value of N is large both counterparties have to pay attention to this contingency. Collateral accounts are used to mitigate this risk. These are similar to margin accounts used by exchanges.
Forward contracts paying in arrears are less risky. The cash flow N(f - F_t(t,u))\delta(t,u) at u involves the difference of similar amounts.
A caplet with strike k is a call option on a forward rate. It has cash flow \max\{F_t^\delta(t,u) - k, 0\} at time u. A floolet is a put option on a forward rate. It has cash flow \max\{k - F_t^\delta(t,u), 0\} at time u.
Exercise. Find a closed form solution for the value of a caplet and floorlet in the Ho-Lee model.
Hint: F - k = fe^{sZ - s^2/2} - h for some constants f, s, h where Z is standard normal. The value of a caplet involves the Black put formula and the value of a floorlet involes the Black call formula.
An objection to the Ho-Lee model is that it allows negative interest rates. This is unusual, but not a violation of arbitrage and has occurred in the real world.
The parameterization f_t = r(1 + \sigma B_t) is closer to a lognormal model since 1 + x\approx e^x for small x. When using this replace \sigma by r\sigma in the equations above.
A multi-factor model can be specified using multi-dimensional independent Brownian motions and vector-valued volatility.
Exercise. Show \operatorname{Cov}(\sigma(t)\cdot B_t, \sigma(u)\cdot B_u) = \sigma(t)\cdot\sigma(u)\min\{t,u\}.
Hint: \operatorname{Cov}(B_t, B_u) = \min\{t,u\}I where I is the identity matrix.
A common choice for the 2-dimensional case is \sigma(t) = \sigma(\cos\alpha t, \sin\alpha t) for some constants \sigma and \alpha.
Exercise. Show \sigma(t)\cdot\sigma(u) = \sigma^2\cos(\alpha(t - u)).