1. Definitions and notation for the i

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1. Definitions and notation for the i
Stochastic Calculus, Week 10
Definitions and Notation
Term-Structure Models & Interest Rate Derivatives
• Zero-coupon bond, or discount bond: asset which pays,
Topics:
with certainty, 1 at time T . Price at time t is P (t, T ).
1. Definitions and notation for the interest rate market
With fixed rate r: P (t, T ) = exp(−r[T − t]).
Pull-to-par: limt→T P (t, T ) = P (T, T ) = 1.
• Yield-to-maturity: average continuously compounded
2. Term-structure models
return on zero-coupon bond, over remaining lifetime:
3. Interest rate derivatives
log P (T, T ) − log P (t, T )
T −t
log P (t, T )
= −
.
T −t
R(t, T ) =
With fixed rate, R(t, T ) = r.
• Yield curve, term structure: plot of R(t, T ) against (T −
t). With fixed rate, yield curve is flat line.
• Instantaneous rate, short rate:
∂ log P (t, T ) rt = lim R(t, T ) = −
.
T ↓t
∂T
T =t
1
• Forward contract: agreement at time t < T , to buy an S-
• The definitions of R(t, T ) and f (t, T ) in terms of P (t, t)
bond at time T < S at price K. Replicated by portfolio
may be inverted to obtain
of +1 S-bond and −K T -bonds. Value at time t should
P (t, T ) = exp (−(T − t)R(t, T ))
Z T
= exp −
f (t, s)ds .
be zero, so K = P (t, S)/P (t, T ). Return (yield) over
period [T, S] is therefore
t
log P (S, S) − log K
log P (t, S) − log P (t, T )
=−
.
S−T
S−T
In general, P (t, T ) can not be obtained from rt alone!
Note that this return is known at time t! The limit, as S
approaches T , of this return is the instantaneous forward
rate,
log P (t, S) − log P (t, T )
S↓T
S−T
∂ log P (t, T )
= −
.
∂T
f (t, T ) = lim −
Note that f (t, t) = rt . Also,
f (t, T ) = R(t, T ) + (T − t)
∂R(t, T )
.
∂T
The term structure may also be described by f (t, T ) as a
function of (T − t).
2
Term structure models
and
∂µP (t, T )
− σ(t, T )Σ(t, T ),
∂T
Z T
1
µP (t, T ) = rt −
α(t, s)ds + Σ(t, T )2 .
2
t
α(t, T ) = −
We only consider single-factor models (one Brownian motion). Ingredients:
• The market price of risk does not depend on T :
Z t
W̃t = Wt +
γ s ds,
• A cash bond Bt paying the short rate (rt ): dBt = rt Bt dt,
R
t
so Bt = exp 0 rs ds .
0
• An Itô process for the zero-coupon bonds:
with
dP (t, T ) = P (t, T ) [µP (t, T )dt + Σ(t, T )dWt ]
h
i
= P (t, T ) rt dt + Σ(t, T )dW̃t ,
µ (t, T ) − rt
1
1
γt = P
= Σ(t, T )−
Σ(t, T )
2
Σ(t, T )
the term structure of volatilities σ(t, T ).
motion. Under Q, P (t, T )/Bt is a martingale.
• The SDE for P (t, T ) implies, via Itô’s lemma, SDE’s for
R(t, T ), f (t, T ) and rt . Heath-Jarrow-Morton start with
Itô processes for f (t, T ):
df (t, T ) = α(t, T )dt + σ(t, T )dWt
= −σ(t, T )Σ(t, T )dt + σ(t, T )dW̃t ,
∂Σ(t, T )
,
σ(t, T ) = −
∂T
Z
Σ(t, T ) = −
t
t
T
α(t, s)ds.
• Note that under risk-neutrality, everything depends on
with Wt a P-Brownian motion and W̃t a Q-Brownian
where
Z
T
σ(t, s)ds,
3
Short-rate models
We distinguish:
Usually the single factor is associated with the short rate
• Endogenous term structure models: here the current
rt . Short-rate models are usually formulated in “risk-neutral
term structure follows from the SDE for rt , which may
form”, i.e., as a SDE
be different from the actual term structure at time t.
drt = ρ(rt , t)dt + ν(rt , t)dW̃t ,
Well-known examples are the Vasicek model:
where W̃t is a Q-Brownian motion. (Baxter and Rennie de-
drt = (θ − αrt )dt + σdW̃t ,
note this by Wt in Section 5.4).
Let Bt = exp
R
and the Cox-Ingersoll-Ross model
t
0 rs ds be the cash bond price. Under the
√
drt = (θ − αrt )dt + σ rt dW̃t .
risk-neutral measure, P (t, T )/Bt is a martingale, so that
P (t, T ) = Bt EQ BT−1 P (T, T ) Ft
= EQ Bt BT−1 Ft
Z T
= EQ exp −
rs ds Ft .
These are also called equilibrium models.
• Exogenous term structure models: here the drift ρ(rt , t)
is adjusted such that the current term structure is fitted
exactly. The simplest example is the Ho-Lee model:
t
drt = θt dt + σdW̃t ,
Therefore, a single-factor (risk-neutral) model for rt implies
where θt is such that f (0, T ) = r0 − 12 σ 2 T 2 +
P (t, T ), and hence the term structure, as a function of rt .
matches the initial term structure.
These are also called no-arbitrage models.
4
RT
0
θs ds
Interest rate derivatives
Forward contracts
The forward price of a zero-coupon T2 -bond, to be delivered
We assume again the general single-factor model
at T1 , is simply F (t, T1 , T2 ) = P (t, T2 )/P (t, T1 ).
dP (t, T ) = rt P (t, T )dt + Σ(t, T )P (t, T )dW̃t ,
Coupon-bearing bonds
where W̃t is a Q-Brownian motion. We consider:
Suppose that a bond has a principal of 1 dollar, and pays
coupons at times Ti = iδ, i = 1, . . . , n, at rate k; at time Tn
• Forward contracts
the principal amount is repaid. This means that the payoff
• Coupon-bearing bonds
is kδ at time Ti , i < n, and 1 + kδ at time Tn . Since the noarbitrage value at time t, of a certain payment of x at time
• Floating rate bonds
Ti is
• Bond options
EQ
• Swaps
Z
exp −
Ti
t
rs ds x Ft = P (t, Ti )x,
the bond value at time t between Tj and Tj+1 becomes
• Swaptions
Vt =
• Caps, floors, collars
n
X
kδP (t, Ti ) + P (t, Tn ).
i=j+1
• Stock options with stochastic interest rates
Since we expect that V0 = 1 (the par value), we find that the
coupon rate should be
1 − P (t, Tn )
k = Pn
.
i=1 δP (t, Ti )
Some of these turn out to have a price that depend only on
P (t, T ), independent of model assumptions (i.e., independent of Σ(t, T )).
5
Floating-rate bonds
Bond options
Suppose that the coupon rate it not fixed, but equal to the
A European call option on a zero-coupon T -bond, struck at
LIBOR rate at the previous payment time Ti−1 . The δ-period
k with exercise time S, is worth
Ct = Bt EQ BS−1 [P (S, T ) − k]+ Ft .
LIBOR rate L(t) is defined by P (t, t + δ)(1 + δL(t)) = 1,
1
1
L(t) =
−1 .
δ P (t, t + δ)
Hence the payoff at time Ti is 1/P (Ti−1 , Ti ) − 1 for i < n,
or
When the short rate rt is a Gaussian process, then both Bt
and P (t, T ) will have a log-normal distribution. This can be
used to obtain a Black-Scholes-type formula
and the payoff at time Tn is 1/P (Tn−1 , Tn ).
Ct = P (t, S) {F (t, S, T )Φ(d1 ) − kΦ(d2 )} ,
Then the no-arbitrage value V0 of this bond should be 1. The
where F (t, S, T ) is the forward price, and
replicating strategy is:
d1,2 =
• At time 0, buy 1/P (0, T1 ) zero coupon bonds maturing
at T1 (costing 1);
log(F (t, S, T )/k) 1 √
√
± σ̄ S − t,
2
σ̄ S − t
where σ̄ 2 is the conditional variance (conditional on Ft ) of
√
log(P (S, T )/P (t, T ))/ S − t, i.e.,
Z S
1
σ̄ 2 =
Σ(u, T )2 du.
S−t t
• At time T1 = δ, sell the T1 -bonds (yielding 1/P (0, T1 )),
and buy 1/P (T1 , T2 ) of T2 -bonds (costing 1). Hence the
payoff is 1/P (0, T1 ) − 1;
For example, in the Ho-Lee model, σ̄ = σ(T − S).
• At time Ti , sell the current position of 1/P (Ti−1 , Ti ) of
Ti -bonds, and buy 1/P (Ti , Ti+1 ) of Ti+1 -bonds.
Options on coupon-bearing bonds will only have a closed-
This gives exactly the right payoff, and costs 1.
form price in single-factor models where rt is a diffusion
process. See Baxter & Rennie, p.170, or Hull, pp.568-570.
6
Swaps
Caps, floors, collars
A fixed-versus variable interest rate swap (on a principal of
For a cap, the payoff at Ti is δ[L(Ti−1 ) − k]+ . The payoff
1) has payoff, at times Ti , equal to δ[L(Ti−1 ) − k], where
of one caplet can be replicated as follows: Buy (1 + kδ) put
k is the fixed swap rate. This means that the swap can be
options on a Ti -bond, struck at 1/(1+kδ) with exercise time
replicated by a portfolio of a short position in a fixed coupon
Ti−1 .
bond, and a long position of a floating coupon bond. Since
At time Ti−1 , this yields (1 + kδ)[1/(1 + kδ) −
the swap should have intial value of zero, we have
1−
n
X
P (Ti−1 , Ti )]+ = [1 − (1 + kδ)P (Ti−1 , Ti )]+ . Then put this
kδP (t, Ti ) − P (t, Tn ) = 0,
money in the bank, to yield the LIBOR rate L(Ti−1 ), so that
i=1
the payoff at time Ti becomes [1 + δL(Ti−1 ) − (1 + kδ)]+ =
or
δ[L(Ti−1 )−k]+ . Thus the value of a cap can be derived from
1 − P (t, Tn )
k = Pn
,
δP
(t,
T
)
i
i=1
which is also the coupon rate that sets the bond value equal
the value of the put option.
to par.
Similarly, the payoff of a floor is δ[k − L(Ti−1 )]+ at time
Ti , the value of which can be derived from a call option. A
Swaptions
swap is essentially the sum of a cap and a floor, at the same
A swaption is an option to enter a swap. Its value is the same
rate k; this results in a put-call parity. A collar is also a
as the value of a call option on a fixed-coupon bearing bond,
combination of a cap and a floor, but at different rates (cap
struck at 1.
rate bigger than floor rate).
7
Stock options with stochastic interest rates
Suppose we wish to price a European call option on a stock
with price St , and that the interest rate is stochastic. In particular, let
dSt = rt St dt + σSt dW̃1t
dP (t, T ) = rt P (t, T )dt + Σ(t, T )P (t, T )dW̃2t ,
and furthermore define the cash bond as usual.
Here
(W̃1t , W̃2t ) is a bivariate Q-Brownian motion, with correlation ρ. If rt is a Gaussian process, such that Bt and P (t, T )
are log-normal, then the price of a European call option becomes
Ct = P (t, T ) {F (t, T )Φ(d1 ) − kΦ(d2 )} ,
where F (t, T ) = P (t, T )−1 St is the forward stock price,
and
log(F (t, T )/k) 1 √
√
± σ̂ T − t,
2
σ̂ T − t
d1,2 =
where
1
σ̂ =
T −t
2
Z
t
T
2
σ + Σ(s, T )2 − 2ρσΣ(s, T ) ds.
This result was derived by Merton (1973).
8

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