FI-3 Lecture 3

Term Structure Models

Multiple Interest Rate Factors

Key Rates & Perturbations

We assume that

P=f(y_1,y_2,y_3,y_4)

where

y_1,y_2,y_3,y_4

are the 2-year, 5-year, 10-year and 30-year par coupon rates.

For some security with deterministic cash flows, the first-order approximation can be expressed as

\Delta P=\frac{\partial f}{\partial y_1}\Delta y_1 +\frac{\partial f}{\partial y_2}\Delta y_2 + \frac{\partial f}{\partial y_3}\Delta y_3 + \frac{\partial f}{\partial y_4}\Delta y_4

The quantities

-\frac{\frac{\partial f}{\partial y_i}}{10,000}

can be thought of as the DV01s corresponding to the i-th key rate.

The quantities

-\frac{\frac{\partial f}{\partial y_i}}{P}

can be thought of as the durations corresponding to the i-th key rate.

The second-order approximation takes the form

\Delta P=\sum_{i=1}^4 \frac{\partial f}{\partial y_i}\Delta y_i+\frac{1}{2}\sum_{i,j=1}^4\frac{\partial^2 f}{\partial y_i\partial y_j}\Delta y_i\Delta y_j

with multiple factors, duration is described by a vector and convexity is described by a matrix.

Key Rate Perturbations

Let’s assume that a change of one basis point in each individual key rate

y_i

leads to a change in the par-coupon yield curve of

Y_i(t)

basis points where the function

Y_i

are as shown in the graphs below (Note that all the impacts in the graphs are our assumptions.)

Par Coupon yields and Discount Factors

Note that we can derive the discount factors from the par-coupon yields.

Recall that the par coupon yield

y(\frac{n+1}{2})

corresponding to maturity

\frac{n+1}{2}

is given by

y\Big(\frac{n+1}{2}\Big)=\frac{2[1-d(\frac{n+1}{2})]}{\sum_{i=1}^{n+1}d(\frac{i}{2})}

starting with a par-coupon yield curve

y(t)

, the discount factors can be computed recursively by

\begin{aligned} d\Big(\frac{1}{2}\Big) &=\frac{1}{1+\frac{y(\frac{1}{2})}{2}}\\ d\Big(\frac{n+1}{2}\Big) &=\frac{2-y(\frac{n+1}{2})\sum_{i=1}^nd(\frac{i}{2})}{2+y(\frac{n+1}{2})} \end{aligned}

Construct a Hedge using Key Rates

Generalization of the Method

Suppose we have sold a security

S

whose price is given by

P^{(S)}=f(x_1,x_2,...,x_N)

and we wish to hedge the sale of this security using a portfolio of basic securities. (Typically we will want

N

securities in the hedging portfolio). If the price of the hedging portfolio is given by

P^{(H)}=g(x_1,x_2,...,x_N)

then we choose the amounts of each of the securities in the hedging portfolio so that

\frac{\partial }{\partial x_i}f(0,0,...,0)=\frac{\partial}{\partial x_i}g(0,0,...,0),i=1,2,...,N

This is analogous to matching all of the key rate DV01s. (note that

x_i=0

for

i=1,2,...,N

corresponding to the original yield curve, i.e., the perturbation is zero).

Risk-Neutral Measure for Binomial Framework

Suppose that there are two trading times

t=0,1

. Assume that there is a zero-coupon bond that will pay $1 at time 1 (i.e. the bond has face value

F=1

and maturity

T=1

.) Denote the price of this bond at time 0 by

B_{0,1}

. Assume that there is another basic (traded) security that has initial price

S_0 > 0

and price

S_1>0

at time 1 taking one of two possible different values, each with prob > 0.

\Omega=\{H,T\}

Besides, assume

S(T)<S(H)

Consider a strategy described by

(X_0,\Delta_0)

, which means that the total initial capital is

X_0

and that the portfolio will hold

\Delta_0

shares of the risky asset.

Denote the capital of the strategy at time 1 by

X_1

, which is a random variable. The initial capital invested in the risky asset is

\Delta_0 S_0

and the initial capital invested in the bond is

X_0-\Delta_0S_0

. The terminal capital is given by

\begin{aligned} X_1(w) &=(X_0-\Delta_0 S_0)\frac{1}{B_{0,1}}+\Delta_0 S_1(w)\\ &=(X_0-\Delta_0S_0)(1+R_0)+\Delta_0 S_1(w) \end{aligned}

for all

w\in\{H, T\}

An arbitrage strategy means

$X_0=0$

$P(X_1\ge0)=1$

$P(X_1>0)>0$

That is, a strategy is an arbitrage iff

$X_0=0$

$X_1(T)\ge0,X_1(H)\ge0$

$X_1(T)+X_1(H)>0$

Absence of Arbitrage

\frac{S_1(T)}{S_0}<\frac{1}{B_{0,1}}<\frac{S_1(H)}{S_0}

If we put

u=\frac{S_1(H)}{S_0},\ \ d=\frac{S_1(T)}{S_0},\ \ R_0=\frac{1}{B_{0,1}}-1

then

d<1+R_0<u

Replication and Pricing

Consider a derivative security that pays the amount

V_1(w)

at time 1. Use it the replicate the payoff of the security by a portfolio with initial capital

X_0

and

\Delta_0

shares of the risky asset. Thus

X_0,\Delta_0

satisfies

\begin{aligned} (X_0-\Delta_0S_0)(1+R_0)+\Delta_0S_1(H)&=V_1(H),\\ (X_0-\Delta_0S_0)(1+R_0)+\Delta_0S_1(T)&=V_1(T) \end{aligned}

we can solve for

\Delta_0

\Delta_0=\frac{V_1(H)-V_1(T)}{S_1(H)-S_1(T)}

thus

\begin{aligned} X_0(1+R_0) &=V_1(H)+\Delta_0[S_0(1+R_0)-S_1(H)]\\ &=\tilde{p}V_1(H)+\tilde{q}V_1(T) \end{aligned}

where

\begin{aligned} \tilde{p}&=\frac{S_0(1+R_0)-S_1(T)}{S_1(H)-S_1(T)}\\ \tilde{q}&=\frac{S_1(H)-S_0(1+R_0)}{S_1(H)-S_1(T)} \end{aligned}

and

\tilde{p}+\tilde{q}=1

We can define another probability measure

\tilde{P}

\Omega

\tilde{P}(H)=\tilde{p},\ \ \tilde{P}(T)=\tilde{q}

which is called the risk-neutral measure or pricing measure.

Since

X_0(1+R_0)=\tilde{E}(V_1)

thus the arbitrage-free time-0 price of

V

is given by

V_0=\tilde{E}\Big[\frac{V_1}{1+R_0}\Big]

Remark:

It is customary to define

u=\frac{S_1(H)}{S_0},\ \ d=\frac{S_1(T)}{S_0}

thus the no aribitrage condition is

d<1+R_0<u

and the risk-neutral probabilities are given by

\tilde{p}=\frac{1+R_0-d}{u-d},\ \ \tilde{q}=\frac{u-(1+R_0)}{u-d}

For simplicity, questions in this course all assume that

\tilde{p}=\tilde{q}=0.5

. But in more complex situations, we may calculate different

\tilde{p},\tilde{q}

N-Period Binomial Models

In the N-period binomial model, there will be N+1 trading dates 0, 1, 2, …, N. Denote

B_{n,m}

for the price at time n of a ZCB that pays $1 at time

m\ge n

. Denote

R_n

for the spot rate that will prevail at time

n

for loans to be settled at time

n+1

(one-period compounding).

B_{n,n+1}=\frac{1}{1+R_n}

Note that the interest rates

R_n

are known at time

n

, but not earlier.

Short Rate Models v.s. Whole-Yield Models

Short rate models, also called equilibrium models, have no arbitrage (although they may give rise to negative interest rates). The spot rate curve is an output of the mode, rather than an input. Typically there are parameters in the model that are chosen to try to match the spot-rate curve as closely as possible.

Whole-Yield models, also called arbitrage-free models, take the current spot-rate as an input, and model the random evolution of the spot-rate curve. A key idea is to use forward rates. Prices are computed using a risk-neutrail measure.

Binomial Short Rate Models

Notations:

Dates: $n=0,1,...,N$

Sample Sapce: $\Omega=\{H,T\}^N$

Interest Rate Process: $(R_n)_{0\le n\le N-1}$

adapted process (values of the process at time n depend only on the information available at time n).

Pricing Measure: $\tilde{P}$ (risk-neutral measure)

Discount Process: $(D_n)_{0\le n\le N}$

predictable (values of the process at time n are determined by the information available at time n-1)

Risk-Neutral Pricing Formula

For a security that makes a single payment of amount

V_m(w_1,...,w_m)

at a given date

m

, the prices

V_n

of the security at the dates

n=0,1,...,m

are given by

V_n=\frac{1}{D_n}\tilde{E}_n[D_mV_m]=\tilde{E}_n[\frac{D_m}{D_n}V_m]

where

\begin{aligned} D_m&=\frac{1}{(1+R_0)(1+R_1)...(1+R_{m-1})}\\ \frac{D_m}{D_n} &=\frac{1}{(1+R_n)(1+R_{n+1})...(1+R_{m-1})} \end{aligned}

In particular, the time-0 price is given by

V_0=\tilde{E}[D_mV_m]

Remark:

Note $D_n \ne d_n$ , because $d_n$ is known at time 0, but as for $D_n$ , $R_1,...,R_{n-1}$ are not known at time 0.

the discount factor for time $n$ is given by

d(n)=\tilde{E}[D_n]

Given an interest rate process and a pricing measure, we can compute the zero-coupon yield curve (spot-rate curve).

Four Binomial Models

In all 4 of these models, we take the risk-neutral measure to be a binomial product measure (be H or T with prob of p & q), with probability of heads equal to 0.5. We put

\Delta R_n=R_{n+1}-R_n

Ho-Lee Model

\begin{aligned} \Delta R_n &=\lambda_{n+1}+\sigma X_{n+1}\\ R_n &=R_0+\sum_{i=1}^n\lambda_i + \sigma M_n \end{aligned}

Remark:

$\lambda_1,\lambda_2,...$ are called drift parameters and $\sigma$ is called the volatility parameter.

The Ho-Lee Model can be written in the form

R_n(w_1,...,w_n)=a_n+b·\#H(w_1,...,w_n)

where

a_n=R_0-\sigma n +\sum_{i=1}^n \lambda_i,\ \ b=2\sigma

Ho-Lee with Constant Drift:

\begin{aligned} \Delta R_n &=\lambda+\sigma X_{n+1}\\ R_n &=R_0+\lambda n + \sigma M_n\\ &=R_0+\lambda n + \sigma [(n-\#H_n)*(-1)+1*\# H_n]\\ &=R_0+(\lambda-\sigma)n + 2\sigma\# H_n \end{aligned}

Vasicek Model

Binomial Verision of Vasicek Model:

0<k<1

\Delta R_n=k(\theta-R_n)+\sigma X_{n+1}

The solution of this difference equation is given by

R_n=(1-k)^n(R_0-\theta)+\theta+\sigma\sum_{j=1}^n(1-k)^{n-j}X_j

where

X_j

is a random walk.

Remarks:

This model exhibits mean reversion

$\theta$ is called the long-term mean, and $k$ is called the speed of mean reversion

Hull-White Model

Binomial Version of Hull-White Model:

0<k<1

\Delta R_n=k(\theta_{n+1}-R_n)+\sigma_{n+1}X_{n+1}

Black-Derman-Toy Model

\Delta (\ln R_n)=-\frac{\Delta \sigma_n}{\sigma_n}(\ln \theta_{n+1}-\ln R_n)+\sigma_{n+1}X_{n+1}

which can be rewritten as

R_n(w_1,...,w_n)=a_nb_n^{\#H(w_1,...,w_n)}

Pricing

Zero Coupon Bonds

For each

n,m\in \{0,1,...,N\}

with

0\le n\le m

, let

B_{n,m}

denote the price at time

n

of a zero coupon bond with maturity

m

and face value $1. Then

B_{n,m}=\frac{1}{D_n}\tilde{E}_n[D_m]

where

(D_n)_{0\le n\le N}

is the discount process. In particular,

B_{n,n}=1

Coupon Bonds

Given

q>-1

and

0\le n\le m\le N

, let

C_{n,m}^q

denote the price at time

n

of a coupon bond with maturity

m

, face value $1 and one-period coupon rate

q

. If

n>0

, we make the convention that

C_{n,m}^q

is the price after the coupon payment is received at time

n

. Then

C_{n,m}^q=B_{n,m}+q\sum_{i=n+1}^m B_{n,i}

provided that

m\ge n+1

General Security with Adapted Cash Flows

Let

(A_n)_{1\le n\le m}

be an adapted process. The price at time 0 of a security that pays the amount

A_n

at each of the times

n=1,2,...,N

is given by

\tilde{E}\Big[\sum_{n=1}^mD_nA_n\Big]

One-Period Forward Rates

For

0\le n\le m\le N-1

, denote

F_{n,m}

as the interest rate agreed upon at time

n

for a loan initiated at time

m

and settled at time

m+1

. If $1 is the amount borrowed at time

m

then the amount to be repaid at time

m+1

1+F_{n,m}

. Then

B_{n,m}=(1+F_{n,m})B_{n,m+1}

note

F_{m,m}=R_m

Backward Induction

Sinlge Payment

Consider a security that makes a single payment of amount

V_m

at time

m

. Start at time

m

and work backward by using

V_n=\frac{1}{1+R_n}\tilde{E}_n[V_{n+1}]

for

n=m-1,m-2,...,1,0

Multiple Payments

Consider a security that makes payments

A_n(w1,...,w_n)

at each of the times

n=1,2,...,m

. We can determine the price at time 0 by viewing this as a string of

m

securties each making one payment and pasting together the results.

Let

V_n

be the price of the security at time

n

after the payment

A_n

has been made. The backward induction is

\begin{cases} V_n=\frac{1}{1+R_n}\tilde{E}_n[A_{n+1}+V_{n+1}]=\frac{A_{n+1}}{1+R_n}+\frac{1}{1+R_n}\tilde{E}[V_{n+1}]\\ V_m=0 \end{cases}

for

n=0,1,...,m-1

Notice that with multiple payments, the discounted price process will not be a martingale under

\tilde{P}

Number of Heads as a State Variable

Simplification is possible when the short spot rate at time

n

depends only on

w

only through the number of heads up to time

n

, i.e.

R_n(w_1,...,w_n)=r_n(\#H(w_1,...,w_n))

and the pricing measure is a binomial product measure (

\tilde{P}

For this scenario, for a security that makes a single payment

V_m

at time

m

and the payment can be expressed as function of

\# H(w_1,...,w_m)

V_n(w_1,...,w_n)=v_n(\# H(w_1,...,w_n)),\ \ \forall n=0,1,...,m-1

The functions

v_n

can be computed by backward induction:

v_n(k)=\frac{1}{1+r_n(k)}[\tilde{p}v_{n+1}(k+1)+\tilde{q}v_{n+1}(k)]

where

\tilde{p},\tilde{q}

are the risk-neutral probabilities of heads and tails,

k

is the number of heads up to time

n

For a security that pays

A_n

at each of the times

n=1,2,3,...,m

where each

A_n

can be expressed as a function of either

\#H(w_1,...,w_n)

\# H(w_1,...,w_{n-1})

, i.e.

A_n=a_n(\# H(w_1,...,w_{n-1})),\ \ n=1,2,....,m

Then the function

v_n

can be computed by backward induction:

v_n(k)=\frac{a_{n+1}(k)}{1+r_n(k)}+\frac{1}{1+r_n(k)}[\tilde{p}v_{n+1}(k+1)+\tilde{q}v_{n+1}(k)]