Fixed-Income Options

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Fixed-Income Options

• Consider a two-year 99 European call on the three-year, 5% Treasury.

• Assume the Treasury pays annual interest.

• From p. 739 the three-year Treasury’s price minus the $5 interest could be $102.046, $100.630, or $98.579 two years from now.

• Since these prices do not include the accrued interest, we should compare the strike price against them.

• The call is therefore in the money in the first two scenarios, with values of $3.046 and $1.630, and out of the money in the third scenario.

2006 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 738

A

C B

B

C

D

D D D

105

105 4.00%

101.955 1.458

3.526%

102.716 2.258

2.895%

102.046 3.046

5.289%

99.350 0.774

4.343%

100.630 1.630

6.514%

98.579 0.000

(a)

A

C B

B

C

D

D D D

105

105 4.00%

101.955 0.096

3.526%

102.716 0.000

2.895%

102.046 0.000

5.289%

99.350 0.200

4.343%

100.630 0.000

6.514%

98.579 0.421

(b)

Fixed-Income Options (continued)

• The option value is calculated to be $1.458 on p. 739(a).

• European interest rate puts can be valued similarly.

• Consider a two-year 99 European put on the same security.

• At expiration, the put is in the money only if the Treasury is worth $98.579 without the accrued interest.

• The option value is computed to be $0.096 on p. 739(b).

Fixed-Income Options (concluded)

• The present value of the strike price is PV(X) = 99 × 0.92101 = 91.18.

• The Treasury is worth B = 101.955.

• The present value of the interest payments during the life of the options is

PV(I) = 5 × 0.96154 + 5 × 0.92101 = 9.41275.

• The call and the put are worth C = 1.458 and P = 0.096, respectively.

• Hence the put-call parity is preserved:

C = P + B − PV(I) − PV(X).

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Delta or Hedge Ratio

• How much does the option price change in response to changes in the price of the underlying bond?

• This relation is called delta (or hedge ratio) defined as O_h− Oℓ

P_h− Pℓ

.

• In the above P^h and Pℓ denote the bond prices if the short rate moves up and down, respectively.

• Similarly, Oh and O_ℓ denote the option values if the short rate moves up and down, respectively.

Delta or Hedge Ratio (concluded)

• Since delta measures the sensitivity of the option value to changes in the underlying bond price, it shows how to hedge one with the other.

• Take the call and put on p. 739 as examples.

• Their deltas are

0.774 − 2.258

99.350 − 102.716 = 0.441, 0.200 − 0.000

99.350 − 102.716 = −0.059,

respectively.

Volatility Term Structures

• The binomial interest rate tree can be used to calculate the yield volatility of zero-coupon bonds.

• Consider an n-period zero-coupon bond.

• First find its yield to maturity yh (y_ℓ, respectively) at the end of the initial period if the rate rises (declines, respectively).

• The yield volatility for our model is defined as (1/2) ln(y_h/y_ℓ).

Volatility Term Structures (continued)

• For example, based on the tree on p. 720, the two-year zero’s yield at the end of the first period is 5.289% if the rate rises and 3.526% if the rate declines.

• Its yield volatility is therefore 1

2 ln

0.05289 0.03526

= 20.273%.

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Volatility Term Structures (continued)

• Consider the three-year zero-coupon bond.

• If the rate rises, the price of the zero one year from now will be

1

2× 1

1.05289×

1

1.04343+ 1 1.06514

= 0.90096.

• Thus its yield is q

1

0.90096 − 1 = 0.053531.

• If the rate declines, the price of the zero one year from now will be

1

2× 1

1.03526×

1

1.02895+ 1 1.04343

= 0.93225.

Volatility Term Structures (continued)

• Thus its yield is q

1

0.93225 − 1 = 0.0357.

• The yield volatility is hence 1

2 ln

0.053531 0.0357

= 20.256%, slightly less than the one-year yield volatility.

• This is consistent with the reality that longer-term bonds typically have lower yield volatilities than shorter-term bonds.

• The procedure can be repeated for longer-term zeros to obtain their yield volatilities.

0 100 200 300 400 500

Time period 0.1

0.101 0.102 0.103 0.104

Spot rate volatility

Short rate volatility given flat %10 volatility term structure.

Volatility Term Structures (continued)

• We started with vi and then derived the volatility term structure.

• In practice, the steps are reversed.

• The volatility term structure is supplied by the user along with the term structure.

• The vi—hence the short rate volatilities via Eq. (77) on p. 700—and the r_i are then simultaneously determined.

• The result is the Black-Derman-Toy model.

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Volatility Term Structures (concluded)

• Suppose the user supplies the volatility term structure which results in (v1, v2, v3, . . . ) for the tree.

• The volatility term structure one period from now will be determined by (v₂, v₃, v₄, . . . ) not (v₁, v₂, v₃, . . . ).

• The volatility term structure supplied by the user is hence not maintained through time.

• This issue will be addressed by other types of (complex) models.

Foundations of Term Structure Modeling

[Meriwether] scoring especially high marks in mathematics — an indispensable subject for a bond trader.

— Roger Lowenstein, When Genius Failed

Terminology

• A period denotes a unit of elapsed time.

– Viewed at time t, the next time instant refers to time t + dt in the continuous-time model and time t + 1 in the discrete-time case.

• Bonds will be assumed to have a par value of one unless stated otherwise.

• The time unit for continuous-time models will usually be measured by the year.

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Standard Notations

The following notation will be used throughout.

t: a point in time.

r(t): the one-period riskless rate prevailing at time t for repayment one period later (the instantaneous spot rate, or short rate, at time t).

P (t, T ): the present value at time t of one dollar at time T .

Standard Notations (continued)

r(t, T ): the (T − t)-period interest rate prevailing at time t stated on a per-period basis and compounded once per period—in other words, the (T − t)-period spot rate at time t.

• The long rate is defined as r(t, ∞).

F (t, T, M ): the forward price at time t of a forward contract that delivers at time T a zero-coupon bond maturing at time M ≥ T .

Standard Notations (concluded)

f (t, T, L): the L-period forward rate at time T implied at time t stated on a per-period basis and compounded once per period.

f (t, T ): the one-period or instantaneous forward rate at time T as seen at time t stated on a per period basis and compounded once per period.

• It is f(t, T, 1) in the discrete-time model and f (t, T, dt) in the continuous-time model.

• Note that f(t, t) equals the short rate r(t).

Fundamental Relations

• The price of a zero-coupon bond equals

P (t, T ) =





(1 + r(t, T ))^{−(T −t)} in discrete time, e−r(t,T )(T −t) in continuous time.

• r(t, T ) as a function of T defines the spot rate curve at time t.

• By definition,

f (t, t) =





r(t, t + 1) in discrete time, r(t, t) in continuous time.

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Fundamental Relations (continued)

• Forward prices and zero-coupon bond prices are related:

F (t, T, M ) = P (t, M )

P (t, T ), T ≤ M. (82) – The forward price equals the future value at time T

of the underlying asset (see text for proof).

• Equation (82) holds whether the model is discrete-time or continuous-time, and it implies

F (t, T, M ) = F (t, T, S) F (t, S, M ), T ≤ S ≤ M.

Fundamental Relations (continued)

• Forward rates and forward prices are related definitionally by

f(t, T, L) =

1

F(t, T, T + L)

1/L

− 1 =

P(t, T ) P(t, T + L)

1/L

− 1 (83)

in discrete time.

– f (t, T , L) =_L¹ (_{P (t,T +L)}^{P (t,T )} − 1) is the analog to Eq. (83) under simple compounding.

Fundamental Relations (continued)

• In continuous time,

f (t, T, L) = −ln F (t, T, T + L)

L = ln(P (t, T )/P (t, T + L))

L (84)

by Eq. (82) on p. 758.

• Furthermore,

f (t, T, ∆t) = ln(P (t, T )/P (t, T + ∆t))

∆t → −∂ ln P (t, T )

∂T

= −∂P (t, T )/∂T P (t, T ) .

Fundamental Relations (continued)

• So

f (t, T ) ≡ lim

∆t→0f (t, T, ∆t) = −∂P (t, T )/∂T

P (t, T ) , t ≤ T.

(85)

• Because Eq. (85) is equivalent to

P (t, T ) = e⁻^R^t^T^{f (t,s) ds}, (86) the spot rate curve is

r(t, T ) = 1 T − t

Z T t

f (t, s) ds.

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Fundamental Relations (concluded)

• The discrete analog to Eq. (86) is

P (t, T ) = 1

(1 + r(t))(1 + f (t, t + 1)) · · · (1 + f(t, T − 1)). (87)

• The short rate and the market discount function are related by

r(t) = − ∂P (t, T )

∂T _{T =t}.

– This can be verified with Eq. (85) on p. 761 and the observation that P (t, t) = 1 and r(t) = f (t, t).

Risk-Neutral Pricing

• Under 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 ,

Et[ P (t + 1, T ) ]

P (t, T ) = 1 + r(t). (88) – Relation (88) in fact follows from the risk-neutral

valuation principle, Theorem 14 (p. 419).

Risk-Neutral Pricing (continued)

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

• Rewrite Eq. (88) as

E_t^π[ P (t + 1, T ) ]

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

– It says the current spot rate curve equals the expected spot rate curve 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))

. (89)

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Risk-Neutral Pricing (concluded)

• Equation (88) on p. 763 can also be expressed as E_t[ P (t + 1, T ) ] = F (t, t + 1, T ).

• 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_th

e⁻^R^t^T^{r(s) ds}i

, t < T. (90)

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

• When the local expectations theory holds, riskless arbitrage opportunities are impossible.

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 = t0.

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 + fi∆t.

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Interest Rate Swaps (continued)

• The value of the swap at time t is thus Xn

i=1

E_t^πh

e⁻^R^t^ti^{r(s) ds}(f_i−1− c) ∆ti

= Xn i=1

E_t^π

e⁻^R^t^ti^{r(s) ds}

1

P (t_i−1, t_i)− (1 + c∆t)

= Xn i=1

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

= P (t, t₀) − P (t, tn) − c∆t Xn 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 Sn(t) ≡ P (t, t₀) − P (t, tn)

Pn

i=1P (t, t_i) ∆t . (91)

• 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 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

j P u q

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The Binomial Model (continued)

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

P − 1 = qu + (1 − q) d − 1.

(92)

• The variance of that return rate is

σb² ≡ q(1 − q)(u − d)². (93)

• The bond whose maturity is only 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.

The Binomial Model (continued)

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

• The same arbitrage argument as in the continuous-time case can be employed to show 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 − λp

q(1 − q) = (1 + r) − d

u − d , (94)

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 new probability measure p.

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%

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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. 779.

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 an up move in rates.

Numerical Examples (concluded)

• Solving the equation leads to p = 0.319.

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

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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.

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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, as shown below.

F * 92 (= 100 − 8) j 98 (= 100 − 2)

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

• On the other hand, the forward price for a one-year forward contract on a one-year zero-coupon bond equals 90.703/96.154 = 94.331%.

• The forward price exceeds the futures price.

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. 788.

• 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 ր 5

ր ց 102.222 (= 5 + (105/1.08))

105 ր

0 M

105 ց

ց ր 107.941 (= 5 + (105/1.02))

5 ց

105

The left diagram depicts the cash flow; the right diagram illustrates the price process.

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.

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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. 791(a)).

• Their prices hence follow the processes on p. 791(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

ր ր

0 5

ր ց ր ց

100 5

0 0

ց ր ց ր

100 5

ց ց

0 0

(a)

92.593 9.630

ր ր

po io

ց ց

100 5

(b)

The price 9.630 is derived from 5 + (5/1.08).

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. 793.

FLT: 108 INV: 102

ր ր

4 6

ր ց ր ց

108 102

0 0

ց ր ց ր

104 106

ց ց

0 0

(a)

104 100.444

ր ր

flt inv

ց ց

104 106

(b)

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Numerical Examples: MBSs (concluded)

• On p. 793, 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.

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The Vasicek Model

^a

• 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. (52) on p. 475.

aVasicek (1977).

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), (95) where

A(t, T ) =









 exp

(B(t,T )−T +t)(β2µ−σ2/2)

β2 −σ2 B(t,T )2

4β

if β 6= 0,

exp

σ2(T −t)3 6

if β = 0.

and

B(t, T ) =





1−e^{−β(T −t)}

β if β 6= 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 6= 0.

• Even if β 6= 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 volatility with maturity.

2 4 6 8 10 Term

0.05 0.1 0.15 0.2

Yield

humped

inverted

normal

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The Vasicek Model: Options on Zeros

^a

• 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 − σv).

aJamshidian (1989).

The Vasicek Model: Options on Zeros (concluded)

• Above

x ≡ 1

σ_vln

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 6= 0 σ²(T − t), if β = 0

.

• By the put-call parity, the price of a European put is XP (t, T ) N (−x + σv) − 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.

aNelson and Ramaswamy (1990).

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.

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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.

The Cox-Ingersoll-Ross Model

^a

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

dr = β(µ − r) dt + σ√

r dW. (96)

• 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.

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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. 811).

x + 2√

∆t f (x + 2√

∆t)

ր ր

x +√

∆t f (x +√

∆t)

ր ց ր ց

x x f (x) f (x)

ց ր ց ր

x −√

∆t f (x −√

∆t)

ց ց

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⁻ . (97)

– 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. 814(a) for the resulting binomial short rate tree with the up-move probabilities in parentheses.

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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)

(b) 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 0.985271123591

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 9 6 4 7 2 7 9 8 3 8 8 0.992781347933

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

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

Numerical Examples (continued)

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

• Since the root has x = 2p

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

^a

• 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 − yu

y_u− yd

.

• Here yu≡ 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 .

aNelson and Ramaswamy (1990).

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A General Method (continued)

• But the binomial tree may not combine:

σ(y, t)√

∆t − σ(yu, t)√

∆t 6= −σ(y, t)√

∆t + σ(y_d, t)√

∆t in general.

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

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

Z y

σ(z, t)⁻¹dz.

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

A General Method (continued)

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

• The probability of an up move remains α(y(x, t), t) ∆t + y(x, t) − yd(x, t)

y_u(x, t) − yd(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) .

A General Method (concluded)

• The transformation is Z r

(σ√

z)⁻¹dz = 2√ r/σ for the CIR model.

• The transformation is Z S

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

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

Fixed-Income Options

Fixed-Income Options

Fixed-Income Options (continued)

Fixed-Income Options (concluded)

Delta or Hedge Ratio

Delta or Hedge Ratio (concluded)

Volatility Term Structures

Volatility Term Structures (continued)

Volatility Term Structures (continued)

Volatility Term Structures (continued)

Volatility Term Structures (continued)

Volatility Term Structures (concluded)

Foundations of Term Structure Modeling

Terminology

Standard Notations

Standard Notations (continued)

Standard Notations (concluded)

Fundamental Relations

Fundamental Relations (continued)

Fundamental Relations (continued)

Fundamental Relations (continued)

Fundamental Relations (continued)

Fundamental Relations (concluded)

Risk-Neutral Pricing

Risk-Neutral Pricing (continued)

Risk-Neutral Pricing (continued)

Risk-Neutral Pricing (concluded)

Continuous-Time Risk-Neutral Pricing

Interest Rate Swaps

Interest Rate Swaps (continued)

Interest Rate Swaps (continued)

Interest Rate Swaps (concluded)

Swap Rate

The Binomial Model

The Binomial Model (continued)

The Binomial Model (continued)

The Binomial Model (concluded)

Numerical Examples

Numerical Examples (continued)

Numerical Examples (continued)

Numerical Examples (continued)

Numerical Examples (concluded)

Numerical Examples: Fixed-Income Options

Numerical Examples: Fixed-Income Options (continued)

Numerical Examples: Fixed-Income Options (continued)

Numerical Examples: Fixed-Income Options (concluded)

Numerical Examples: Futures and Forward Prices

Numerical Examples: Mortgage-Backed Securities

Numerical Examples: MBSs (continued)

Numerical Examples: MBSs (continued)

Numerical Examples: MBSs (continued)

Numerical Examples: MBSs (concluded)

Equilibrium Term Structure Models

Introduction

The Vasicek Model

The Vasicek Model (continued)

The Vasicek Model (concluded)

The Vasicek Model: Options on Zeros

The Vasicek Model: Options on Zeros (concluded)

Binomial Vasicek

Binomial Vasicek (continued)

Binomial Vasicek (concluded)

The Cox-Ingersoll-Ross Model

Binomial CIR

Binomial CIR (continued)

Binomial CIR (continued)

Binomial CIR (concluded)

Numerical Examples

Numerical Examples (continued)

Numerical Examples (concluded)

A General Method for Constructing Binomial Models

A General Method (continued)

A General Method (continued)

A General Method (concluded)

Finis