A T -maturity zero coupon bond (ZCB) pays 1 at time T (and only - - PowerPoint PPT Presentation

A T-maturity zero coupon bond (ZCB) pays 1 at time T (and only that); its time-t price denoted P(t; T). As a fct of T: Smooth. As a fct of t: Erratic (Ito-process). Continuously compounded ZC yield y(t, τ) is defined by P(t; t + τ) = exp(−τy(t; τ)) ⇔ y(t; τ) = −ln P(t; t + τ) τ . Note the shift from time of maturity to time to maturity. Instantaneous forward rates (mathematically convenient) f(t, T) = −∂ ln P(t; T) ∂T .

• Interpretation. Why does this make sense? BLACKBOARD

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

1

The term structure of interest rates at date t is the mapping τ → y(t; τ)

• r some translation of it (eg. into ZCB prices or forward rates). In theory this

curve is observable in practice. In practice, well . . . Short rate: r(t) = f(t; t) Bank account: β(t) = exp t r(s)ds

• ,

so dβ(t) = r(t)β(t)dt, and we say/note that this is a locally risk-free asset.

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

2

Dynamics equations (Bj¨

• rk equations 22.1-3)

Short rate (⊤ means transposition) dr(t) = a(t)dt + b⊤(t)dW(t) (1) ZCB prices (one eqn’ for each T; note shift to prop. coefficients) dP(t; T) = P(t; T)m(t; T)dt + P(t; T)v⊤(t; T)dW(t) (2) Forward rates d f(t; T) = α(t; T)dt + σ⊤(t; T)dW(t) (3)

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

3

No financial assumptions yet; W is just BM under some measure. Coefficients are adapted (vector-valued) process, but smooth in T; subscript-T denotes T-differentiation. We have f(t; T) = −∂ ln P(t; T) ∂T ⇔ P(t; T) = exp

• −

T

t

f(t; s)ds

• and

r(t) = f(t, t). So what’s the connection?

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

4

Proposition 22.5 (Quite important result.) 1) If ZCB prices satisfy (2) then forward rates satisfy (3) with α(t; T) = v⊤

T (t; T)v(t; T) − mT(t; T) and σ(t; T) = −vT(t; T).

2) If forward rates satisfy (3) then the short rate satisfies (1) with a(t) = fT(t, t) + α(t, t) and b(t) = σ(t; t). 3) If forward rates satisfy (3) then ZCB prices satisfy (2) with m(t; T) = r(t) + A(t; T) + 1 2S⊤(t; T)S(t; T) and v(t; T) = S(t; T), where A(t; T) = − T

t α(t; s)ds

and S(t; T) = − T

t σ(t; s)ds.

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

5

You’ll forget terms if you aren’t careful, so let’s look at a proof. 1): Take logs, use Ito & differentiate wrt. T The complication with the last two statements is that we have t appearing both under the integral sign and in the limit. Recall the Leibniz rule (where h : R2 → R is a smooth fuction) d dx x h(t, x)dt = h(x, x) + x hx(t, x)dt. 2): r(t) = f(t; t), but by Leibniz, we’re inspired to write dr(t) = dtf(t; T)|T =t + dTf(t; T)

• =fT (t;T )dt

|T =t

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

6

and the result follows. Bj¨

• rk has a proof on integral form — except his “changing the order and

identifying” may leave a little too much to the reader. Details on homepage.

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

7

3): P(t; T) = exp

• −

T

t f(t; s)ds

• , so t enters in two places in an even trickier

way. Bj¨

• rk gives “a heuristic proof”; even if I wanted to, I can’t repeat the arguments

in a way that sounds convincing. So let me sketch a proof. BLACKBOARD and details on homepage.

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

8

An application: The HJM drift condition

Assume that a model of forward rates is given by (3) under some measure P. Suppose further that the model is arbitrage-free. Then there exists an equivalent martingale measure Q ∼ P such that P(t; T) β(t) is a Q-martingale for all T. So dP(t; T) = P(t; T)r(t)dt + P(t; T)S⊤(t; T)dW Q(t).

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

9

Note the subtle application of Girsanov’s theorem: Equivalent changes of measure change drift – not volatility. But from Proposition 20.5 3) we get r(t) = r(t) + AQ(t; T) + 1 2S⊤(t; T)S(t; T) ⇒ −AQ(t; T) = 1 2S⊤(t; T)S(t; T). Differentiate both sides wrt. T and get the Heath-Jarrow-Morton drift condition αQ(t; T) = σ⊤(t; T) T

t

σ(t; s)ds.

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

10

But what about drifts under P? From Girsanov’s theorem we know that there exists a stochastic process λ such that dW Q = dW P − λ(t)dt defines a Q-BM. Important: λ doesn’t depend on T. Not important: Whether I choose to write “+” or “-”. We have dP(t; T) P(t; T) = (r(t) + AP(t; T) + 1 2S⊤(t; T)S(t; T))dt + S⊤(t; T)dW P(t) = r(t)dt + S⊤(t; T)dW Q(t) +

• AP(t; T) + 1

2S⊤(t; T)S(t; T) + S⊤(t; T)λ(t)

• must = 0

dt.

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

11

This means that −AP(t; T) = 1

2S⊤(t; T)S(t; T) + S⊤(t; T)λ(t). Differentiating

• wrt. T gives

αP(t; T) = σ⊤(t; T) T

t

σ(t; s) − λ(t)

• .

In sloppy matrix notation we may write λ(t) = −EP(return on ZCB) − r(t) Vol(ZCB) . If σ (forward rate volatility) is chosen positive then (typically) λ(t) will be positive. Otherwise it’s negative.

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

12

Forward rates are martingales under forward measures The QT-drift of f(t, T) is 0. Expectation hypotheses: No w/ 1-d Brownian motion. Yes, in higher dimensions (find function such that ∇g · g = 0) — but pretty strange models (forward rates move like waves).

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

13

Another application: HJM and the Markov-property The drift condition makes the dynamics of one forward rates dependent on all

• ther forward rates ⇒ Non-markovian,

But sometimes the models are Markovian. And this you get to do in an exercise There’s more literature on this where people do cunning stuff.

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

14

A third application: Musiela formulation/parametrization Change to time to maturity in forward rates: r(t, x) = f(t; t + x). “How rates are quoted” (practice) and we get an object that lives on a rectangular domain (mathematical). By Leibniz dr(t, x) = dtf(t; T)|T =t+x + dTf(t; T)|T =t+x = d f(t, t + x) + ∂ ∂x

• =F

r(t, x), and in fancy notation the drift condition gives dr(t, x) = (Fr(t, x) + D(t, x)) dt + σ0(t, x)

=σ(t,t+x)

dW Q(t)

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

15

The term structure can now be analyzed with tools for infinite dimensional SDEs. Very tricky stuff! Bj¨

• rk et al. study consistency questions.

Asset Pricing 2; Bj¨

• rk Ch. 22, 25

16