Risk-Neutral Pricing •

Risk-Neutral Pricing

• Assume the local expectations theory.

• The expected rate of return of any riskless bond over a single period equals the prevailing one-period spot rate.

– For all t + 1 < T ,

E_t[ P (t + 1, T ) ]

P (t, T ) = 1 + r(t). (101) – Relation (101) in fact follows from the risk-neutral

valuation principle.^a

aTheorem 16 on p. 457.

Risk-Neutral Pricing (continued)

• The local expectations theory is thus a consequence of the existence of a risk-neutral probability π.

• Rewrite Eq. (101) as

E_t^π[ P (t + 1, T ) ]

1 + r(t) = P (t, T ).

– It says the current market discount function equals the expected market discount function one period from now discounted by the short rate.

Risk-Neutral Pricing (continued)

• Apply the above equality iteratively to obtain

P (t, T )

= E_t^π

[ P (t + 1, T ) 1 + r(t)

]

= E_t^π

[ E_t+1^π [ P (t + 2, T ) ] (1 + r(t))(1 + r(t + 1))

]

= · · ·

= E_t^π

[ 1

(1 + r(t))(1 + r(t + 1))· · · (1 + r(T − 1)) ]

. (102)

Risk-Neutral Pricing (concluded)

• Equation (101) on p. 889 can also be expressed as E_t[ P (t + 1, T ) ] = F (t, t + 1, T ).

– Verify that with, e.g., Eq. (96) on p. 884.

• Hence the forward price for the next period is an unbiased estimator of the expected bond price.

Continuous-Time Risk-Neutral Pricing

• In continuous time, the local expectations theory implies P (t, T ) = E_t

[ e⁻

∫_T

t r(s) ds ]

, t < T. (103)

• Note that e^{∫tT r(s) ds} is the bank account process, which denotes the rolled-over money market account.

Interest Rate Swaps

• Consider an interest rate swap made at time t with payments to be exchanged at times t₁, t₂, . . . , t_n.

• The fixed rate is c per annum.

• The floating-rate payments are based on the future annual rates f₀, f₁, . . . , f_n−1 at times t₀, t₁, . . . , t_n−1.

• For simplicity, assume tⁱ⁺¹ − tⁱ is a fixed constant ∆t for all i, and the notional principal is one dollar.

• If t < t⁰, we have a forward interest rate swap.

• The ordinary swap corresponds to t = t .

Interest Rate Swaps (continued)

• The amount to be paid out at time tⁱ⁺¹ is (f_i − c) ∆t for the floating-rate payer.

• Simple rates are adopted here.

• Hence fⁱ satisfies

P (t_i, t_i+1) = 1

1 + f_i∆t.

Interest Rate Swaps (continued)

• The value of the swap at time t is thus

∑n i=1

E_t^π [

e⁻

∫_ti

t r(s) ds

(f_i−1 − c) ∆t]

=

∑n i=1

E_t^π [

e⁻

∫_ti

t r(s) ds

( 1

P (t_i−1, t_i) − (1 + c∆t) )]

=

∑n i=1

[ P (t, t_i−1) − (1 + c∆t) × P (t, tⁱ) ]

= P (t, t₀) − P (t, tn) − c∆t

∑n i=1

P (t, t_i).

Interest Rate Swaps (concluded)

• So a swap can be replicated as a portfolio of bonds.

• In fact, it can be priced by simple present value calculations.

Swap Rate

• The swap rate, which gives the swap zero value, equals S_n(t) ≡ P (t, t₀) − P (t, tⁿ)

∑n

i=1 P (t, t_i) ∆t . (104)

• The swap rate is the fixed rate that equates the present values of the fixed payments and the floating payments.

• For an ordinary swap, P (t, t⁰) = 1.

The Term Structure Equation

• Let us start with the zero-coupon bonds and the money market account.

• Let the zero-coupon bond price P (r, t, T ) follow dP

P = µ_p dt + σ_p dW.

• At time t, short one unit of a bond maturing at time s¹ and buy α units of a bond maturing at time s₂.

The Term Structure Equation (continued)

• The net wealth change follows

−dP (r, t, s1) + α dP (r, t, s₂)

= (−P (r, t, s1) µ_p(r, t, s₁) + αP (r, t, s₂) µ_p(r, t, s₂)) dt + (−P (r, t, s1) σ_p(r, t, s₁) + αP (r, t, s₂) σ_p(r, t, s₂)) dW.

• Pick

α ≡ P (r, t, s₁) σ_p(r, t, s₁) P (r, t, s₂) σ_p(r, t, s₂).

The Term Structure Equation (continued)

• Then the net wealth has no volatility and must earn the riskless return:

−P (r, t, s¹) µ_p(r, t, s₁) + αP (r, t, s₂) µ_p(r, t, s₂)

−P (r, t, s¹) + αP (r, t, s₂) = r.

• Simplify the above to obtain

σ_p(r, t, s₁) µ_p(r, t, s₂) − σ^p(r, t, s₂) µ_p(r, t, s₁)

σ_p(r, t, s₁) − σ^p(r, t, s₂) = r.

• This becomes

µ_p(r, t, s₂) − r

σ_p(r, t, s₂) = µ_p(r, t, s₁) − r σ_p(r, t, s₁)

The Term Structure Equation (continued)

• Since the above equality holds for any s¹ and s₂, µ_p(r, t, s) − r

σ_p(r, t, s) ≡ λ(r, t) (105) for some λ independent of the bond maturity s.

• As µ^p = r + λσ_p, all assets are expected to appreciate at a rate equal to the sum of the short rate and a constant times the asset’s volatility.

• The term λ(r, t) is called the market price of risk.

• The market price of risk must be the same for all bonds

The Term Structure Equation (continued)

• Assume a Markovian short rate model, dr = µ(r, t) dt + σ(r, t) dW.

• Then the bond price process is also Markovian.

• By Eq. (14.15) on p. 202 in the text,

µ_p = (

−∂P

∂T + µ(r, t) ∂P

∂r + σ(r, t)² 2

∂²P

∂r² )

/P,

(106)

σ_p = (

σ(r, t) ∂P

∂r )

/P, (106^′)

subject to P (· , T, T ) = 1.

The Term Structure Equation (concluded)

• Substitute µ^p and σ_p into Eq. (105) on p. 902 to obtain

− ∂P

∂T + [ µ(r, t)− λ(r, t) σ(r, t) ] ∂P

∂r + 1

2 σ(r, t)² ∂²P

∂r² = rP.

(107)

• This is called the term structure equation.

• Once P is available, the spot rate curve emerges via r(t, T ) = −ln P (t, T )

T − t .

• Equation (107) applies to all interest rate derivatives, the difference being the terminal and the boundary

The Binomial Model

• The analytical framework can be nicely illustrated with the binomial model.

• Suppose the bond price P can move with probability q to P u and probability 1 − q to P d, where u > d:

P

* P d 1 − q

q j Pu

The Binomial Model (continued)

• Over the period, the bond’s expected rate of return is bµ ≡ qP u + (1 − q) P d

P − 1 = qu + (1 − q) d − 1.

(108)

• The variance of that return rate is

bσ² ≡ q(1 − q)(u − d)². (109)

The Binomial Model (continued)

• In particular, the bond whose maturity is one period

away will move from a price of 1/(1 + r) to its par value

$1.

• This is the money market account modeled by the short rate r.

• The market price of risk is defined as λ ≡ (bµ − r)/bσ.

• As in the continuous-time case, it can be shown that λ is independent of the maturity of the bond (see text).

The Binomial Model (concluded)

• Now change the probability from q to p ≡ q − λ√

q(1 − q) = (1 + r) − d

u − d , (110) which is independent of bond maturity and q.

– Recall the BOPM.

• The bond’s expected rate of return becomes pP u + (1 − p) P d

P − 1 = pu + (1 − p) d − 1 = r.

• The local expectations theory hence holds under the

Numerical Examples

• Assume this spot rate curve:

Year 1 2

Spot rate 4% 5%

• Assume the one-year rate (short rate) can move up to 8% or down to 2% after a year:

4%

* 8%

j 2%

Numerical Examples (continued)

• No real-world probabilities are specified.

• The prices of one- and two-year zero-coupon bonds are, respectively,

100/1.04 = 96.154, 100/(1.05)² = 90.703.

• They follow the binomial processes on p. 911.

Numerical Examples (continued)

90.703

* 92.593 (= 100/1.08)

j 98.039 (= 100/1.02) 96.154

* 100 j 100

The price process of the two-year zero-coupon bond is on the left; that of the one-year zero-coupon bond is on the right.

Numerical Examples (continued)

• The pricing of derivatives can be simplified by assuming investors are risk-neutral.

• Suppose all securities have the same expected one-period rate of return, the riskless rate.

• Then

(1 − p) × 92.593

90.703 + p × 98.039

90.703 − 1 = 4%,

where p denotes the risk-neutral probability of a down move in rates.

Numerical Examples (concluded)

• Solving the equation leads to p = 0.319.

• Interest rate contingent claims can be priced under this probability.

Numerical Examples: Fixed-Income Options

• A one-year European call on the two-year zero with a

$95 strike price has the payoffs, C

* 0.000 j 3.039

• To solve for the option value C, we replicate the call by a portfolio of x one-year and y two-year zeros.

Numerical Examples: Fixed-Income Options (continued)

• This leads to the simultaneous equations, x × 100 + y × 92.593 = 0.000, x × 100 + y × 98.039 = 3.039.

• They give x = −0.5167 and y = 0.5580.

• Consequently,

C = x × 96.154 + y × 90.703 ≈ 0.93 to prevent arbitrage.

Numerical Examples: Fixed-Income Options (continued)

• This price is derived without assuming any version of an expectations theory.

• Instead, the arbitrage-free price is derived by replication.

• The price of an interest rate contingent claim does not depend directly on the real-world probabilities.

• The dependence holds only indirectly via the current bond prices.

Numerical Examples: Fixed-Income Options (concluded)

• An equivalent method is to utilize risk-neutral pricing.

• The above call option is worth

C = (1 − p) × 0 + p × 3.039

1.04 ≈ 0.93,

the same as before.

• This is not surprising, as arbitrage freedom and the existence of a risk-neutral economy are equivalent.

Numerical Examples: Futures and Forward Prices

• A one-year futures contract on the one-year rate has a payoff of 100 − r, where r is the one-year rate at

maturity:

F

* 92 (= 100 − 8) j 98 (= 100 − 2)

• As the futures price F is the expected future payoff (see text or p. 458), F = (1 − p) × 92 + p × 98 = 93.914.

• The forward price for a one-year forward contract on a one-year zero-coupon bond is 90.703/96.154 = 94.331%.

• The forward price exceeds the futures price.^a

Numerical Examples: Mortgage-Backed Securities

• Consider a 5%-coupon, two-year mortgage-backed

security without amortization, prepayments, and default risk.

• Its cash flow and price process are illustrated on p. 920.

• Its fair price is

M = (1 − p) × 102.222 + p × 107.941

1.04 = 100.045.

• Identical results could have been obtained via arbitrage considerations.

105

↗ 5

↗ ↘ 102.222 (= 5 + (105/1.08))

105 ↗

0 M

105 ↘

↘ ↗ 107.941 (= 5 + (105/1.02))

5

↘

105

The left diagram depicts the cash flow; the right diagram

Numerical Examples: MBSs (continued)

• Suppose that the security can be prepaid at par.

• It will be prepaid only when its price is higher than par.

• Prepayment will hence occur only in the “down” state when the security is worth 102.941 (excluding coupon).

• The price therefore follows the process, M

* 102.222

j 105

• The security is worth

M = (1 − p) × 102.222 + p × 105

1.04 = 99.142.

Numerical Examples: MBSs (continued)

• The cash flow of the principal-only (PO) strip comes from the mortgage’s principal cash flow.

• The cash flow of the interest-only (IO) strip comes from the interest cash flow (p. 923(a)).

• Their prices hence follow the processes on p. 923(b).

• The fair prices are

PO = (1 − p) × 92.593 + p × 100

1.04 = 91.304,

IO = (1 − p) × 9.630 + p × 5

1.04 = 7.839.

PO: 100 IO: 5

↗ ↗

0 5

↗ ↘ ↗ ↘

100 5

0 0

0 0

↘ ↗ ↘ ↗

100 5

↘ ↘

0 0

(a)

92.593 9.630

↗ ↗

po io

↘ ↘

100 5

(b)

Numerical Examples: MBSs (continued)

• Suppose the mortgage is split into half floater and half inverse floater.

• Let the floater (FLT) receive the one-year rate.

• Then the inverse floater (INV) must have a coupon rate of

(10% − one-year rate) to make the overall coupon rate 5%.

• Their cash flows as percentages of par and values are shown on p. 925.

FLT: 108 INV: 102

↗ ↗

4 6

↗ ↘ ↗ ↘

108 102

0 0

0 0

↘ ↗ ↘ ↗

104 106

↘ ↘

0 0

(a)

104 100.444

↗ ↗

flt inv

↘ ↘

104 106

(b)

Numerical Examples: MBSs (concluded)

• On p. 925, the floater’s price in the up node, 104, is derived from 4 + (108/1.08).

• The inverse floater’s price 100.444 is derived from 6 + (102/1.08).

• The current prices are

FLT = 1

2 × 104

1.04 = 50,

INV = 1

2 × (1 − p) × 100.444 + p × 106

1.04 = 49.142.

Equilibrium Term Structure Models

8. What’s your problem? Any moron can understand bond pricing models.

— Top Ten Lies Finance Professors Tell Their Students

Introduction

• This chapter surveys equilibrium models.

• Since the spot rates satisfy

r(t, T ) = −ln P (t, T ) T − t ,

the discount function P (t, T ) suffices to establish the spot rate curve.

• All models to follow are short rate models.

• Unless stated otherwise, the processes are risk-neutral.

The Vasicek Model

• The short rate follows

dr = β(µ − r) dt + σ dW.

• The short rate is pulled to the long-term mean level µ at rate β.

• Superimposed on this “pull” is a normally distributed stochastic term σ dW .

• Since the process is an Ornstein-Uhlenbeck process, E[ r(T )| r(t) = r ] = µ + (r − µ) e^{−β(T −t)} from Eq. (55) on p. 517.

The Vasicek Model (continued)

• The price of a zero-coupon bond paying one dollar at maturity can be shown to be

P (t, T ) = A(t, T ) e−B(t,T ) r(t), (111) where

A(t, T ) =













exp [

(B(t,T )−T +t)(β2µ−σ2/2)

β2 − σ2 B(t,T )2

4β

]

if β ̸= 0,

exp [

σ2 (T−t)3 6

]

if β = 0.

and

B(t, T ) =





1−e^{−β(T −t)}

β if β ̸= 0, T − t if β = 0.

The Vasicek Model (concluded)

• If β = 0, then P goes to infinity as T → ∞.

• Sensibly, P goes to zero as T → ∞ if β ̸= 0.

• Even if β ̸= 0, P may exceed one for a finite T .

• The spot rate volatility structure is the curve (∂r(t, T )/∂r) σ = σB(t, T )/(T − t).

• When β > 0, the curve tends to decline with maturity.

• The speed of mean reversion, β, controls the shape of the curve.

• Indeed, higher β leads to greater attenuation of

2 4 6 8 10 Term 0.05

0.1 0.15 0.2

Yield

humped

inverted

normal

The Vasicek Model: Options on Zeros

• Consider a European call with strike price X expiring at time T on a zero-coupon bond with par value $1 and maturing at time s > T .

• Its price is given by

P (t, s) N (x) − XP (t, T ) N(x − σ^v).

aJamshidian (1989).

The Vasicek Model: Options on Zeros (concluded)

• Above

x ≡ 1

σ_v ln

( P (t, s) P (t, T ) X

)

+ σ_v 2 , σ_v ≡ v(t, T ) B(T, s),

v(t, T )² ≡





σ²[^{1−e−2β(T −t)}]

2β , if β ̸= 0 σ²(T − t), if β = 0

.

• By the put-call parity, the price of a European put is XP (t, T ) N (−x + σ^v) − P (t, s) N(−x).

Binomial Vasicek

• Consider a binomial model for the short rate in the time interval [ 0, T ] divided into n identical pieces.

• Let ∆t ≡ T/n and

p(r) ≡ 1

2 + β(µ − r)√

∆t

2σ .

• The following binomial model converges to the Vasicek model,^a

r(k + 1) = r(k) + σ√

∆t ξ(k), 0 ≤ k < n.

a

Binomial Vasicek (continued)

• Above, ξ(k) = ±1 with

Prob[ ξ(k) = 1 ] =









p(r(k)) if 0 ≤ p(r(k)) ≤ 1 0 if p(r(k)) < 0

1 if 1 < p(r(k))

.

• Observe that the probability of an up move, p, is a decreasing function of the interest rate r.

• This is consistent with mean reversion.

Binomial Vasicek (concluded)

• The rate is the same whether it is the result of an up move followed by a down move or a down move followed by an up move.

• The binomial tree combines.

• The key feature of the model that makes it happen is its constant volatility, σ.

• For a general process Y with nonconstant volatility, the resulting binomial tree may not combine, as we will see next.

The Cox-Ingersoll-Ross Model

• It is the following square-root short rate model:

dr = β(µ − r) dt + σ√

r dW. (112)

• The diffusion differs from the Vasicek model by a multiplicative factor √

r .

• The parameter β determines the speed of adjustment.

• The short rate can reach zero only if 2βµ < σ².

• See text for the bond pricing formula.

aCox, Ingersoll, and Ross (1985).

Binomial CIR

• We want to approximate the short rate process in the time interval [ 0, T ].

• Divide it into n periods of duration ∆t ≡ T/n.

• Assume µ, β ≥ 0.

• A direct discretization of the process is problematic because the resulting binomial tree will not combine.

Binomial CIR (continued)

• Instead, consider the transformed process x(r) ≡ 2√

r/σ.

• It follows

dx = m(x) dt + dW, where

m(x) ≡ 2βµ/(σ²x) − (βx/2) − 1/(2x).

• Since this new process has a constant volatility, its associated binomial tree combines.

Binomial CIR (continued)

• Construct the combining tree for r as follows.

• First, construct a tree for x.

• Then transform each node of the tree into one for r via the inverse transformation r = f (x) ≡ x²σ²/4 (p. 943).

x + 2√

∆t f (x + 2√

∆t)

↗ ↗

x +√

∆t f (x +√

∆t)

↗ ↘ ↗ ↘

x x f (x) f (x)

↘ ↗ ↘ ↗

x−√

∆t f (x−√

∆t)

↘ ↘

x− 2√

∆t f (x− 2√

∆t)

Binomial CIR (concluded)

• The probability of an up move at each node r is p(r) ≡ β(µ − r) ∆t + r − r⁻

r⁺ − r⁻ . (113)

– r⁺ ≡ f(x + √

∆t) denotes the result of an up move from r.

– r⁻ ≡ f(x − √

∆t) the result of a down move.

• Finally, set the probability p(r) to one as r goes to zero to make the probability stay between zero and one.

Numerical Examples

• Consider the process,

0.2 (0.04 − r) dt + 0.1√

r dW,

for the time interval [ 0, 1 ] given the initial rate r(0) = 0.04.

• We shall use ∆t = 0.2 (year) for the binomial approximation.

• See p. 946(a) for the resulting binomial short rate tree with the up-move probabilities in parentheses.

0.04 (0.472049150276)

0 . 0 5 9 8 8 8 5 4 3 8 2 (0.44081188025)

0.03155572809 (0.489789553691)

0.02411145618 (0.50975924867)

0.0713328157297 (0.426604457655)

0 . 0 8 3 7 7 7 0 8 7 6 4

0.01222291236 0.01766718427

(0.533083330907) 0.04

(0.472049150276) 0.0494442719102

(0.455865503068)

0.0494442719102 (0.455865503068)

0.03155572809 (0.489789553691)

0 . 0 5 9 8 8 8 5 4 3 8 2

0.04

0.02411145618

(a)

0.992031914837 0.984128889634 0 . 9 7 6 2 9 3 2 4 4 4 0 8 0.968526861261 0.960831229521

0.992031914837 0.984128889634 0 . 9 7 6 2 9 3 2 4 4 4 0 8

0.992031914837 0 . 9 9 0 1 5 9 8 7 9 5 6 5

0.980492588317 0.970995502019 0.961665706744

0 . 9 9 3 7 0 8 7 2 7 8 3 1 0.987391576942 0.981054487259 0 . 9 7 4 7 0 2 9 0 7 7 8 6

0 . 9 8 8 0 9 3 7 3 8 4 4 7 0 . 9 7 6 4 8 6 8 9 6 4 8 5 0.965170249273

0 . 9 9 0 1 5 9 8 7 9 5 6 5 0.980492588317

0.995189317343 0.990276851751

0 . 9 9 3 7 0 8 7 2 7 8 3 1 0.987391576942 0 . 9 8 5 8 3 4 7 2 2 0 3 0.972116454453

0 . 9 8 3 3 8 4 1 7 3 7 5 6

0 . 9 8 8 0 9 3 7 3 8 4 4 7

0.995189317343

Numerical Examples (continued)

• Consider the node which is the result of an up move from the root.

• Since the root has x = 2√

r(0)/σ = 4, this particular node’s x value equals 4 + √

∆t = 4.4472135955.

• Use the inverse transformation to obtain the short rate x² × (0.1)²/4 ≈ 0.0494442719102.

Numerical Examples (concluded)

• Once the short rates are in place, computing the probabilities is easy.

• Note that the up-move probability decreases as interest rates increase and decreases as interest rates decline.

• This phenomenon agrees with mean reversion.

• Convergence is quite good (see text).

A General Method for Constructing Binomial Models

• We are given a continuous-time process, dy = α(y, t) dt + σ(y, t) dW.

• Make sure the binomial model’s drift and diffusion

converge to the above process by setting the probability of an up move to

α(y, t) ∆t + y − y^d y_u − y^d .

• Here y^u ≡ y + σ(y, t)√

∆t and y_d ≡ y − σ(y, t)√

∆t represent the two rates that follow the current rate y.

• The displacements are identical, at σ(y, t)√

∆t .

A General Method (continued)

• But the binomial tree may not combine as σ(y, t)√

∆t − σ(y^u, t + ∆t)√

∆t

̸= −σ(y, t)√

∆t + σ(y_d, t + ∆t)√

∆t in general.

• When σ(y, t) is a constant independent of y, equality holds and the tree combines.

A General Method (continued)

• To achieve this, define the transformation x(y, t) ≡

∫ y

σ(z, t)⁻¹ dz.

• Then x follows dx = m(y, t) dt + dW for some m(y, t) (see text).

• The key is that the diffusion term is now a constant, and the binomial tree for x combines.

• The transformation that turns a 1-dim stochastic process into one with a constant diffusion term is unique.^a

aChiu (R98723059) (2012).

A General Method (concluded)

• The probability of an up move remains

α(y(x, t), t) ∆t + y(x, t) − y^d(x, t) y_u(x, t) − y^d(x, t) ,

where y(x, t) is the inverse transformation of x(y, t) from x back to y.

• Note that yu(x, t) ≡ y(x + √

∆t, t + ∆t) and y_d(x, t) ≡ y(x − √

∆t, t + ∆t) .

Examples

• The transformation is

∫ r

(σ√

z)⁻¹ dz = 2√ r/σ for the CIR model.

• The transformation is

∫ S

(σz)⁻¹ dz = (1/σ) ln S for the Black-Scholes model.

• The familiar binomial option pricing model in fact discretizes ln S not S.

On One-Factor Short Rate Models

• By using only the short rate, they ignore other rates on the yield curve.

• Such models also restrict the volatility to be a function of interest rate levels only.

• The prices of all bonds move in the same direction at the same time (their magnitudes may differ).

• The returns on all bonds thus become highly correlated.

• In reality, there seems to be a certain amount of independence between short- and long-term rates.

On One-Factor Short Rate Models (continued)

• One-factor models therefore cannot accommodate

nondegenerate correlation structures across maturities.

• Derivatives whose values depend on the correlation structure will be mispriced.

• The calibrated models may not generate term structures as concave as the data suggest.

• The term structure empirically changes in slope and curvature as well as makes parallel moves.

• This is inconsistent with the restriction that all

segments of the term structure be perfectly correlated.

On One-Factor Short Rate Models (concluded)

• Multi-factor models lead to families of yield curves that can take a greater variety of shapes and can better

represent reality.

• But they are much harder to think about and work with.

• They also take much more computer time—the curse of dimensionality.

• These practical concerns limit the use of multifactor models to two-factor ones.

Options on Coupon Bonds

• Assume a one-factor short rate model.

• The price of a European option on a coupon bond can be calculated from those on zero-coupon bonds.

• Consider a European call expiring at time T on a bond with par value $1.

• Let X denote the strike price.

• The bond has cash flows c¹, c₂, . . . , c_n at times t₁, t₂, . . . , t_n, where t_i > T for all i.

aJamshidian (1989).

Options on Coupon Bonds (continued)

• The payoff for the option is max

( _n

∑

i=1

c_iP (r(T ), T, t_i) − X, 0 )

.

• At time T , there is a unique value r^∗ for r(T ) that renders the coupon bond’s price equal the strike price X.

• This r^∗ can be obtained by solving X =

∑n i=1

c_iP (r, T, t_i)

Options on Coupon Bonds (continued)

• The solution is unique for one-factor models whose bond price is a monotonically decreasing function of r.

• Let

X_i ≡ P (r^∗, T, t_i),

the value at time T of a zero-coupon bond with par value $1 and maturing at time t_i if r(T ) = r^∗.

• Note that P (r(T ), T, tⁱ) ≥ Xⁱ if and only if r(T ) ≤ r^∗.

Options on Coupon Bonds (concluded)

• As X = ∑

i c_iX_i, the option’s payoff equals max

( _n

∑

i=1

c_iP (r(T ), T, t_i) − ∑

i

c_iX_i, 0 )

=

∑n i=1

c_i × max(P (r(T ), T, tⁱ) − Xⁱ, 0).

• Thus the call is a package of n options on the underlying zero-coupon bond.

• Why can’t we do the same thing for Asian options?^a

No-Arbitrage Term Structure Models

How much of the structure of our theories really tells us about things in nature, and how much do we contribute ourselves?

— Arthur Eddington (1882–1944)

Motivations

• Recall the difficulties facing equilibrium models mentioned earlier.

– They usually require the estimation of the market price of risk.

– They cannot fit the market term structure.

– But consistency with the market is often mandatory in practice.

No-Arbitrage Models

• No-arbitrage models utilize the full information of the term structure.

• They accept the observed term structure as consistent with an unobserved and unspecified equilibrium.

• From there, arbitrage-free movements of interest rates or bond prices over time are modeled.

• By definition, the market price of risk must be reflected in the current term structure; hence the resulting

interest rate process is risk-neutral.

aHo and Lee (1986). Thomas Lee is a “billionaire founder” of Thomas

No-Arbitrage Models (concluded)

• No-arbitrage models can specify the dynamics of

zero-coupon bond prices, forward rates, or the short rate.

• Bond price and forward rate models are usually non-Markovian (path dependent).

• In contrast, short rate models are generally constructed to be explicitly Markovian (path independent).

• Markovian models are easier to handle computationally.

The Ho-Lee Model

• The short rates at any given time are evenly spaced.

• Let p denote the risk-neutral probability that the short rate makes an up move.

• We shall adopt continuous compounding.

aHo and Lee (1986).

↗ r₃

↗ ↘

r2

↗ ↘ ↗

r₁ r₃ + v₃

↘ ↗ ↘

r2 + v2

↘ ↗

r₃ + 2v₃

↘

The Ho-Lee Model (continued)

• The Ho-Lee model starts with zero-coupon bond prices P (t, t + 1), P (t, t + 2), . . . at time t identified with the root of the tree.

• Let the discount factors in the next period be

P_d(t + 1, t + 2), P_d(t + 1, t + 3), . . . if short rate moves down P_u(t + 1, t + 2), P_u(t + 1, t + 3), . . . if short rate moves up

• By backward induction, it is not hard to see that for n ≥ 2,

P_u(t + 1, t + n) = P_d(t + 1, t + n) e^{−(v2+···+vn)}

(114)

The Ho-Lee Model (continued)

• It is also not hard to check that the n-period zero-coupon bond has yields

y_d(n) ≡ −ln P_d(t + 1, t + n) n − 1

y_u(n) ≡ −ln P_u(t + 1, t + n)

n − 1 = y_d(n) + v₂ + · · · + vⁿ n − 1

• The volatility of the yield to maturity for this bond is therefore

κn ≡ √

pyu(n)² + (1 − p) y^d(n)² − [ py^u(n) + (1 − p) y^d(n) ]²

= √

p(1 − p) (y^u(n) − y^d(n))

= √

p(1 − p) v2 + · · · + vⁿ

− 1 .

The Ho-Lee Model (concluded)

• In particular, the short rate volatility is determined by taking n = 2:

σ = √

p(1 − p) v². (115)

• The variance of the short rate therefore equals p(1 − p)(ru − rd)², where r_u and r_d are the two successor rates.^a

aContrast this with the lognormal model.

The Ho-Lee Model: Volatility Term Structure

• The volatility term structure is composed of κ², κ₃, . . . . – It is independent of the r_i.

• It is easy to compute the vⁱs from the volatility structure, and vice versa.

• The rⁱs can be computed by forward induction.

• The volatility structure is supplied by the market.

The Ho-Lee Model: Bond Price Process

• In a risk-neutral economy, the initial discount factors satisfy

P (t, t+n) = (pP_u(t+1, t+n)+(1−p) P^d(t+1, t+n)) P (t, t+1).

• Combine the above with Eq. (114) on p. 968 and assume p = 1/2 to obtain^a

P_d(t + 1, t + n) = P (t, t + n) P (t, t + 1)

2 × exp[ v2 + · · · + vⁿ ] 1 + exp[ v₂ + · · · + vⁿ ],

(116)

P_u(t + 1, t + n) = P (t, t + n) P (t, t + 1)

2

1 + exp[ v₂ + · · · + vⁿ ].

(116^′)

The Ho-Lee Model: Bond Price Process (concluded)

• The bond price tree combines.

• Suppose all vⁱ equal some constant v and δ ≡ e^v > 0.

• Then

Pd(t + 1, t + n) = P (t, t + n) P (t, t + 1)

2δⁿ⁻¹ 1 + δⁿ⁻¹ , P_u(t + 1, t + n) = P (t, t + n)

P (t, t + 1)

2

1 + δⁿ⁻¹ .

• Short rate volatility σ equals v/2 by Eq. (115) on p. 970.

• Price derivatives by taking expectations under the

The Ho-Lee Model: Yields and Their Covariances

• The one-period rate of return of an n-period zero-coupon bond is

r(t, t + n) ≡ ln

(P (t + 1, t + n) P (t, t + n)

) .

• Its value is either ln ^Pd_{P (t,t+n)}^(t+1,t+n) or ln ^Pu_{P (t,t+n)}^(t+1,t+n).

• Thus the variance of return is

Var[ r(t, t + n) ] = p(1 − p)((n − 1) v)² = (n − 1)²σ².

The Ho-Lee Model: Yields and Their Covariances (concluded)

• The covariance between r(t, t + n) and r(t, t + m) is (n − 1)(m − 1) σ² (see text).

• As a result, the correlation between any two one-period rates of return is unity.

• Strong correlation between rates is inherent in all one-factor Markovian models.

The Ho-Lee Model: Short Rate Process

• The continuous-time limit of the Ho-Lee model is dr = θ(t) dt + σ dW.

• This is Vasicek’s model with the mean-reverting drift replaced by a deterministic, time-dependent drift.

• A nonflat term structure of volatilities can be achieved if the short rate volatility is also made time varying, i.e., dr = θ(t) dt + σ(t) dW .

• This corresponds to the discrete-time model in which vⁱ are not all identical.

The Ho-Lee Model: Some Problems

• Future (nominal) interest rates may be negative.

• The short rate volatility is independent of the rate level.

Problems with No-Arbitrage Models in General

• Interest rate movements should reflect shifts in the model’s state variables (factors) not its parameters.

• Model parameters, such as the drift θ(t) in the

continuous-time Ho-Lee model, should be stable over time.

• But in practice, no-arbitrage models capture yield curve shifts through the recalibration of parameters.

– A new model is thus born everyday.

Problems with No-Arbitrage Models in General (concluded)

• This in effect says the model estimated at some time does not describe the term structure of interest rates and their volatilities at other times.

• Consequently, a model’s intertemporal behavior is

suspect, and using it for hedging and risk management may be unreliable.