The BDT Model: Calibration

The Black-Derman-Toy Model

• This model is extensively used by practitioners.

• The BDT short rate process is the lognormal binomial interest rate process described on pp. 1002ff.^b

• The volatility structure^c is given by the market.

• From it, the short rate volatilities (thus v_i) are determined together with the baseline rates r_i.

aBlack, Derman, & Toy (BDT) (1990), but essentially finished in 1986 according to Mehrling (2005).

bRepeated on next page.

cRecall Eq. (136) on p. 1053.

r₄

 r₃

 

r₂ r₄v₄

  

r₁ r₃v₃

  

r₂v₂ r₄v₄²

 

r₃v₃²



r₄v₄³

The Black-Derman-Toy Model (concluded)

• Our earlier binomial interest rate tree, in contrast, assumes v_i are given a priori.

• Lognormal models preclude negative short rates.

The BDT Model: Volatility Structure

• The volatility structure defines the yield volatilities of zero-coupon bonds of various maturities.

• Let the yield volatility of the i-period zero-coupon bond be denoted by κ_i.

• P_u is the price of the i-period zero-coupon bond one period from now if the short rate makes an up move.

• P_d is the price of the i-period zero-coupon bond one period from now if the short rate makes a down move.

The BDT Model: Volatility Structure (concluded)

• Corresponding to these two prices are the following yields to maturity,

y_u =^Δ P_u^−1/(i−1) − 1, y_d =^Δ P_d^−1/(i−1) − 1.

• The yield volatility is defined as^a κ_i =^Δ ln(y_u/y_d)

2 .

aRyecall Eq. (136) on p. 1053.

The BDT Model: Calibration

• The inputs to the BDT model are riskless zero-coupon bond yields and their volatilities.

• For economy of expression, all numbers are period based.

• Suppose inductively that we have calculated (r₁, v₁), (r₂, v₂), . . . , (r_i−1, v_i−1).

– They define the binomial tree up to time i − 2 (thus period i − 1).^a

– Thus the spot rates up to time i − 1 have been matched.

aRecall that (r_i−1, v_i−1) defines i−1 short rates at time i−2, which are applicable to period i − 1.

The BDT Model: Calibration (continued)

• We now proceed to calculate r_i and v_i to extend the tree to cover period i.

• Assume the price of the i-period zero can move to P_u or P_d one period from now.

• Let y denote the current i-period spot rate, which is known.

• In a risk-neutral economy, P_u + P_d

2(1 + r_i) = 1

(1 + y)ⁱ. (155)

• Obviously, P_u and P_d are functions of the unknown r_i and v_i.

The BDT Model: Calibration (continued)

• Viewed from now, the future (i − 1)-period spot rate at time 1 is uncertain.

• Recall that y_u and y_d represent the spot rates at the up node and the down node, respectively.^a

• With κ²_i denoting their variance, we have

κ_i = 1 2 ln



P_u^−1/(i−1) − 1 P_d^−1/(i−1) − 1



. (156)

aRecall p. 1162.

The BDT Model: Calibration (continued)

• Solving Eqs. (155)–(156) for r_i and v_i with backward induction takes O(i²) time.

– That leads to a cubic-time algorithm.

• We next employ forward induction to derive a quadratic-time calibration algorithm.^a

• Forward induction figures out, by moving forward in time, how much $1 at a node contributes to the price.^b

• This number is called the state price and is the price of the claim that pays $1 at that node and zero elsewhere.

aW. J. Chen (R84526007) & Lyuu (1997); Lyuu (1999).

bReview p. 1030(a).

The BDT Model: Calibration (continued)

• Let the unknown baseline rate for period i be r_i = r.

• Let the unknown multiplicative ratio be v_i = v.

• Let the state prices at time i − 1 be P₁, P₂, . . . , P_i.

• They correspond to rates

r, rv, . . . , rvⁱ⁻¹ for period i, respectively.

• One dollar at time i has a present value of f (r, v) =^Δ P₁

1 + r + P₂

1 + rv + P₃

1 + rv² + · · · + P_i

1 + rvⁱ⁻¹.

The BDT Model: Calibration (continued)

• By Eq. (156) on p. 1165, the yield volatility is

g(r, v) ^Δ= 1 2 ln

⎛

⎜⎝

 _P

u,1

1+rv + ^Pu,2

1+rv² + · · · + _1+rv^Pu,i−1_i−1_−1/(i−1)

_P − 1

d,1

1+r + ^Pd,2

1+rv + · · · + _1+rv^Pd,i−1_i−2_−1/(i−1)

− 1

⎞

⎟⎠ .

• Above, P_u,1, P_u,2, . . . denote the state prices at time i − 1 of the subtree rooted at the up node.^a

• And P_d,1, P_d,2, . . . denote the state prices at time i − 1 of the subtree rooted at the down node.^b

aLike r₂v₂ on p. 1159.

bLike r₂ on p. 1159.

The BDT Model: Calibration (concluded)

• Note that every node maintains three state prices:

P_i, P_u,i, P_d,i.

• Now solve

f (r, v) = 1

(1 + y)ⁱ, g(r, v) = κ_i,

for r = r_i and v = v_i.

• This O(n²)-time algorithm appears on p. 382 of the textbook.

Calibrating the BDT Model with the Differential Tree (in seconds)

Number Running Number Running Number Running

of years time of years time of years time

3000 398.880 39000 8562.640 75000 26182.080 6000 1697.680 42000 9579.780 78000 28138.140 9000 2539.040 45000 10785.850 81000 30230.260 12000 2803.890 48000 11905.290 84000 32317.050 15000 3149.330 51000 13199.470 87000 34487.320 18000 3549.100 54000 14411.790 90000 36795.430 21000 3990.050 57000 15932.370 120000 63767.690 24000 4470.320 60000 17360.670 150000 98339.710 27000 5211.830 63000 19037.910 180000 140484.180 30000 5944.330 66000 20751.100 210000 190557.420 33000 6639.480 69000 22435.050 240000 249138.210 36000 7611.630 72000 24292.740 270000 313480.390

75MHz Sun SPARCstation 20, one period per year.

aLyuu (1999).

The BDT Model: Continuous-Time Limit

• The continuous-time limit of the BDT model is^a d ln r =



θ(t) + σ^(t)

σ(t) ln r



dt + σ(t) dW.

• The short rate volatility σ(t) should be a declining

function of time for the model to display mean reversion.

– That makes σ^(t) < 0.

• In particular, constant σ(t) will not attain mean reversion.

aJamshidian (1991).

The Black-Karasinski Model

• The BK model stipulates that the short rate follows d ln r = κ(t)(θ(t) − ln r) dt + σ(t) dW.

• This explicitly mean-reverting model depends on time through κ(· ), θ( · ), and σ( · ).

• The BK model hence has one more degree of freedom than the BDT model.

• The speed of mean reversion κ(t) and the short rate volatility σ(t) are independent.

aBlack & Karasinski (1991).

The Black-Karasinski Model: Discrete Time

• The discrete-time version of the BK model has the same representation as the BDT model.

• To maintain a combining binomial tree, however, requires some manipulations.

• The next plot illustrates the ideas in which t₂ =^Δ t₁ + Δt₁,

t₃ =^Δ t₂ + Δt₂.

 ln rd(t2)

 

ln r(t1) ln rdu(t3) = ln rud(t3)

 

ln ru(t2)



The Black-Karasinski Model: Discrete Time (continued)

• Note that

ln r_d(t₂) = ln r(t₁) + κ(t₁)(θ(t₁) − ln r(t1)) Δt₁ − σ(t1)

Δt₁ , ln r_u(t₂) = ln r(t₁) + κ(t₁)(θ(t₁) − ln r(t1)) Δt₁ + σ(t₁)

Δt₁ .

• To make sure an up move followed by a down move coincides with a down move followed by an up move,

ln rd(t2) + κ(t2)(θ(t2) − ln rd(t2)) Δt2 + σ(t2)√

Δt2 ,

= ln ru(t2) + κ(t2)(θ(t2) − ln ru(t2)) Δt2 − σ(t2)√

Δt2 .

The Black-Karasinski Model: Discrete Time (continued)

• They imply

κ(t₂) = 1 − (σ(t2)/σ(t₁))

Δt₂/Δt₁

Δt₂ .

(157)

• So from Δt₁, we can calculate the Δt₂ that satisfies the combining condition and then iterate.

– t₀ → Δt₁ → t₁ → Δt₂ → t₂ → Δt₃ → · · · → T (roughly).^a

aAs κ(t), θ(t), σ(t) are independent of r, the Δt_i will not depend on r either.

The Black-Karasinski Model: Discrete Time (concluded)

• Unequal durations Δt_i are often necessary to ensure a combining tree.^a

aAmin (1991); C. I. Chen (R98922127) (2011); Lok (D99922028) &

Lyuu (2016, 2017).

Problems with Lognormal Models in General

• Lognormal models such as BDT and BK share the problem that E^π[ M (t) ] = ∞ for any finite t if they model the continuously compounded rate.^a

• So periodically compounded rates should be modeled.^b

• Another issue is computational.

• Lognormal models usually do not admit of analytical solutions to even basic fixed-income securities.

• As a result, to price short-dated derivatives on long-term bonds, the tree has to be built over the life of the

underlying asset instead of the life of the derivative.

aHogan & Weintraub (1993).

bSandmann & Sondermann (1993).

Problems with Lognormal Models in General (concluded)

• This problem can be somewhat mitigated by adopting variable-duration time steps.^a

– Use a fine time step up to the maturity of the short-dated derivative.

– Use a coarse time step beyond the maturity.

• A down side of this procedure is that it has to be tailor-made for each derivative.

• Finally, empirically, interest rates do not follow the lognormal distribution.

aHull & White (1993).

The Extended Vasicek Model

• Hull and White proposed models that extend the Vasicek model and the CIR model.

• They are called the extended Vasicek model and the extended CIR model.

• The extended Vasicek model adds time dependence to the original Vasicek model,

dr = (θ(t) − a(t) r) dt + σ(t) dW.

• Like the Ho-Lee model, this is a normal model.

• The inclusion of θ(t) allows for an exact fit to the current spot rate curve.

aHull & White (1990).

The Extended Vasicek Model (concluded)

• Function σ(t) defines the short rate volatility, and a(t) determines the shape of the volatility structure.

• Many European-style securities can be evaluated analytically.

• Efficient numerical procedures can be developed for American-style securities.

The Hull-White Model

• The Hull-White model is the following special case,

dr = (θ(t) − ar) dt + σ dW. (158)

• When the current term structure is matched,^a θ(t) = ∂f (0, t)

∂t + af (0, t) + σ² 2a

1 − e^−2at . – Recall that f(0, t) defines the forward rate curve.

aHull & White (1993).

The Extended CIR Model

• In the extended CIR model the short rate follows dr = (θ(t) − a(t) r) dt + σ(t)√

r dW.

• The functions θ(t), a(t), and σ(t) are implied from market observables.

• With constant parameters, there exist analytical solutions to a small set of interest rate-sensitive securities.

The Hull-White Model: Calibration

• We describe a trinomial forward induction scheme to calibrate the Hull-White model given a and σ.

• As with the Ho-Lee model, the set of achievable short rates is evenly spaced.

• Let r0 be the annualized, continuously compounded short rate at time zero.

• Every short rate on the tree takes on a value r₀ + jΔr

for some integer j.

aHull & White (1993).

The Hull-White Model: Calibration (continued)

• Time increments on the tree are also equally spaced at Δt apart.

• Hence nodes are located at times iΔt for i = 0, 1, 2, . . . .

• We shall refer to the node on the tree with t_i =^Δ iΔt,

r_j =^Δ r₀ + jΔr, as the (i, j) node.

• The short rate at node (i, j), which equals r_j, is effective for the time period [ t_i, t_i+1).

The Hull-White Model: Calibration (continued)

• Use

μ_i,j = θ(t^Δ _i) − ar_j (159) to denote the drift rate^a of the short rate as seen from node (i, j).

• The three distinct possibilities for node (i, j) with three branches incident from it are displayed on p. 1187.

• The middle branch may be an increase of Δr, no change, or a decrease of Δr.

aOr, the annualized expected change.

The Hull-White Model: Calibration (continued)

(i, j)

(i + 1, j + 2)

*(i + 1, j + 1)

-_{(i + 1, j)}

(i, j)

*(i + 1, j + 1)

-_{(i + 1, j)} j(i + 1, j − 1)

(i, j) -_{(i + 1, j)} j(i + 1, j − 1)

R(i + 1, j − 2)

The Hull-White Model: Calibration (continued)

• The upper and the lower branches bracket the middle branch.

• Define

p1(i, j) Δ

= the probability of following the upper branch from node (i, j), p2(i, j) Δ

= the probability of following the middle branch from node (i, j), p3(i, j) Δ

= the probability of following the lower branch from node (i, j).

• The root of the tree is set to the current short rate r₀.

• Inductively, the drift μ_i,j at node (i, j) is a function of (the still unknown) θ(t_i).

– It describes the expected change from node (i, j).

The Hull-White Model: Calibration (continued)

• Once θ(t_i) is available, μ_i,j can be derived via Eq. (159) on p. 1186.

• This in turn determines the branching scheme at every node (i, j) for each j, as we will see shortly.

• The value of θ(ti) must thus be made consistent with the spot rate r(0, t_i+2).^a

aNot r(0, t_i+1)!

The Hull-White Model: Calibration (continued)

• The branches emanating from node (i, j) with their probabilities^a must be chosen to be consistent with μ_i,j and σ.

• This is done by selecting the middle node to be as

closest to the current short rate r_j plus the drift μ_i,jΔt.^b

aThat is, p₁(i, j), p₂(i, j), and p₃(i, j).

bA precursor of Lyuu and C. Wu’s (R90723065) (2003, 2005) mean- tracking idea, which is the precursor of the binomial-trinomial tree of Dai (B82506025, R86526008, D8852600) & Lyuu (2006, 2008, 2010).

The Hull-White Model: Calibration (continued)

• Let k be the number among { j − 1, j, j + 1 } that

makes the short rate reached by the middle branch, r_k, closest to

r_j + μ_i,jΔt.

– But note that μ_i,j is still not computed yet.

• Then the three nodes following node (i, j) are nodes (i + 1, k + 1), (i + 1, k), (i + 1, k − 1).

• See p. 1192 for a possible geometry.

• The resulting tree combines.

* - j

(0, 0)

* - j

(1, 1)

* - j

(1, 0) 

*

(1, −1) -

* - j

* - j

* - j

* - j

- j R

* - j

* - j

* - j

* - j



* --



Δt

6

?^Δr

The Hull-White Model: Calibration (continued)

• The probabilities for moving along these branches are functions of μ_i,j, σ, j, and k:

p₁(i, j) = σ²Δt + η²

2(Δr)² + η

2Δr, (160)

p₂(i, j) = 1 − σ²Δt + η²

(Δr)² , (160^) p₃(i, j) = σ²Δt + η²

2(Δr)² − η

2Δr, (160^) where

η = μ^Δ _i,jΔt + (j − k) Δr.

The Hull-White Model: Calibration (continued)

• As trinomial tree algorithms are but explicit methods in disguise,^a certain relations must hold for Δr and Δt to guarantee stability.

• It can be shown that their values must satisfy σ√

3Δt

2 ≤ Δr ≤ 2σ√ Δt

for the probabilities to lie between zero and one.

– For example, Δr can be set to σ√

3Δt .^b

• Now it only remains to determine θ(t_i).

aRecall p. 826.

bHull & White (1988).

The Hull-White Model: Calibration (continued)

• At this point at time t_i,

r(0, t₁), r(0, t₂), . . . , r(0, t_i+1) have already been matched.

• Let Q(i, j) be the state price at node (i, j).

• By construction, the state prices Q(i, j) for all j are known by now.

• We begin with state price Q(0, 0) = 1.

The Hull-White Model: Calibration (continued)

• Let ˆr(i) refer to the short rate value at time t_i.

• The value at time zero of a zero-coupon bond maturing at time t_i+2 is then

e^{−r(0,ti+2)(i+2) Δt}

=

j

Q(i, j) e^−rjΔt E^π

e−ˆr(i+1) Δt ˆr(i) = rj

.(161)

• The right-hand side represents the value of $1 at time t_i+2 as seen at node (i, j) at time^a t_i before being discounted by Q(i, j).

aThus ˆr(i + 1) is stochastic.

The Hull-White Model: Calibration (continued)

• The expectation in Eq. (161) can be approximated by^a E^π

e−ˆr(i+1) Δt  ˆr(i) = rj

≈ e^−rjΔt



1 − μ_i,j(Δt)² + σ²(Δt)³ 2



. (162) – This solves the chicken-egg problem!

• Substitute Eq. (162) into Eq. (161) and replace μ_i,j with θ(t_i) − ar_j to obtain

θ(ti) ≈

j Q(i, j) e−2rjΔt 1 +arj(Δt)2 + σ2(Δt)3/2

− e−r(0,ti+2)(i+2) Δt (Δt)2

j Q(i, j) e−2rjΔt .

aSee Exercise 26.4.2 of the textbook.

The Hull-White Model: Calibration (continued)

• For the Hull-White model, the expectation in Eq. (162) is actually known analytically by Eq. (30) on p. 179:

E^π

e−ˆr(i+1) Δt ˆr(i) = rj

= e^{−rjΔt+(−θ(ti)+arj+σ2Δt/2)(Δt)2}.

• Therefore, alternatively,

θ(t_i) = r(0, t_i+2)(i + 2)

Δt +σ²Δt

2 +ln

j Q(i, j) e^{−2rjΔt+arj(Δt)2}

(Δt)² .

• With θ(t_i) in hand, we can compute μ_i,j.^a

aSee Eq. (159) on p. 1186.

The Hull-White Model: Calibration (concluded)

• With μ_i,j available, we compute the probabilities.^a

• Finally the state prices at time t_i+1:

Q(i + 1, j)

= 

(i, j^∗) is connected to (i + 1, j) with probability p_j∗

pj^∗e^−rj∗ΔtQ(i, j^∗).

• There are at most 5 choices for j^∗ (why?).

• The total running time is O(n²).

• The space requirement is O(n) (why?).

aSee Eqs. (160) on p. 1193.

Comments on the Hull-White Model

• One can try different values of a and σ for each option.

• Or have an a value common to all options but use a different σ value for each option.

• Either approach can match all the option prices exactly.

• But suppose the demand is for a single set of parameters that replicate all option prices.

• Then the Hull-White model can be calibrated to all the observed option prices by choosing a and σ that

minimize the mean-squared pricing error.^a

aHull & White (1995).

The Hull-White Model: Calibration with Irregular Trinomial Trees

• The previous calibration algorithm is quite general.

• For example, it can be modified to apply to cases where the diffusion term has the form σr^b.

• But it has at least two shortcomings.

• First, the resulting trinomial tree is irregular (p. 1192).

– So it is harder to program (for nonprogrammers).

• The second shortcoming is a consequence of the tree’s irregular shape.

The Hull-White Model: Calibration with Irregular Trinomial Trees (concluded)

• Recall that the algorithm figured out θ(t_i) that matches the spot rate r(0, t_i+2) in order to determine the

branching schemes for the nodes at time t_i.

• But without those branches, the tree was not specified, and backward induction on the tree was not possible.

• To avoid this chicken-egg dilemma, the algorithm turned to the continuous-time model to evaluate Eq. (161) on p. 1196 that helps derive θ(t_i).

• The resulting θ(t_i) hence might not yield a tree that matches the spot rates exactly.

The Hull-White Model: Calibration with Regular Trinomial Trees

• The next, simpler algorithm exploits the fact that the Hull-White model has a constant diffusion term σ.

• The resulting trinomial tree will be regular.

• All the θ(t_i) terms can be chosen by backward induction to match the spot rates exactly.

• The tree is constructed in two phases.

aHull & White (1994).

The Hull-White Model: Calibration with Regular Trinomial Trees (continued)

• In the first phase, a tree is built for the θ(t) = 0 case, which is an Ornstein-Uhlenbeck process:

dr = −ar dt + σ dW, r(0) = 0.

– The tree is dagger-shaped (see p. 1205).

– The number of nodes above the r₀-line is j_max, and that below the line is j_min.

– They will be picked so that the probabilities (160) on p. 1193 are positive for all nodes.

* - j

(0, 0) r0

* - j

(1, 1)

* - j

(1, 0)

* -

(1, −1) j

* - j

* - j

* - j

* - j



* -

- j R

* - j

* - j

* - j

* - j



* -

- j R

* - j

* - j

* - j

* - j



* -

- j R

* - j

* - j

* - j

* - j



* - -



Δt

6?^Δr

The short rate at node (0, 0) equals r₀ = 0; here j_max = 3 and j_min = 2.

The Hull-White Model: Calibration with Regular Trinomial Trees (concluded)

• The tree’s branches and probabilities are now in place.

• Phase two fits the term structure.

– Backward induction is applied to calculate the β_i to add to the short rates on the tree at time t_i so that the spot rate r(0, t_i+1) is matched.^a

aContrast this with the previous algorithm, where it was r(0, t_i+2) that was matched!

The Hull-White Model: Calibration

• Set Δr = σ√

3Δt and assume that a > 0.

• Node (i, j) is a top node if j = j_max and a bottom node if j = −j_min.

• Because the root of the tree has a short rate of r₀ = 0, phase one adopts r_j = jΔr.

• Hence the probabilities in Eqs. (160) on p. 1193 use η =^Δ −ajΔrΔt + (j − k) Δr.

• Recall that k tracks the middle branch.

The Hull-White Model: Calibration (continued)

• The probabilities become

p1(i, j)

= 1

6

+ a2j2(Δt)2 − 2ajΔt(j − k) + (j − k)2 − ajΔt + (j − k)

2 , (163)

p2(i, j)

= 2

3 −

a2 j2(Δt)2 − 2ajΔt(j − k) + (j − k)2 

, (164)

p3(i, j)

= 1 6

+ a2j2(Δt)2 − 2ajΔt(j − k) + (j − k)2 + ajΔt − (j − k)

2 . (165)

• p₁: up move; p₂: flat move; p₃: down move.

The Hull-White Model: Calibration (continued)

• The dagger shape dictates this:

– Let k = j − 1 if node (i, j) is a top node.

– Let k = j + 1 if node (i, j) is a bottom node.

– Let k = j for the rest of the nodes.

• Note that the probabilities are identical for nodes (i, j) with the same j.

• Furthermore, p₁(i, j) = p₃(i,−j).

The Hull-White Model: Calibration (continued)

• The inequalities

3 − √ 6

3 < jaΔt <

2

3 (166)

ensure that all the branching probabilities are positive in the upper half of the tree, that is, j > 0 (verify this).

• Similarly, the inequalities

−

2

3 < jaΔt < −3 − √ 6 3

ensure that the probabilities are positive in the lower half of the tree, that is, j < 0.

The Hull-White Model: Calibration (continued)

• To further make the tree symmetric across the r₀-line, we let j_min = j_max.

• As

3 − √ 6

3 ≈ 0.184, a good choice is

j_max = 0.184/(aΔt).

The Hull-White Model: Calibration (continued)

• Phase two computes the β_is to fit the spot rates.

• We begin with state price Q(0, 0) = 1.

• Inductively, suppose that spot rates

r(0, t₁), r(0, t₂), . . . , r(0, t_i) have already been matched.

• By construction, the state prices Q(i, j) for all j are known by now.

The Hull-White Model: Calibration (continued)

• The value of a zero-coupon bond maturing at time t_i+1 equals

e^{−r(0,ti+1)(i+1) Δt} =

j

Q(i, j) e^{−(βi+rj)Δt} by risk-neutral valuation.

• Hence

β_i = r(0, t_i+1)(i + 1) Δt + ln

j Q(i, j) e^−rjΔt

Δt .

(167)

The Hull-White Model: Calibration (concluded)

• The short rate at node (i, j) now equals β_i + r_j.

• The state prices at time t_i+1,

Q(i + 1, j), − min(i + 1, j_max) ≤ j ≤ min(i + 1, j_max), can now be calculated as before.^a

• The total running time is O(nj_max).

• The space requirement is O(n).

aRecall p. 1199.

A Numerical Example

• Assume a = 0.1, σ = 0.01, and Δt = 1 (year).

• Immediately, Δr = 0.0173205 and j_max = 2.

• The plot on p. 1216 illustrates the 3-period trinomial tree after phase one.

• For example, the branching probabilities for node E are calculated by Eqs. (163)–(165) on p. 1208 with j = 2 and k = 1.

* - j

A

* - j

B *

- j

C *

- j

D

- j R

E *

- j

F *

- j

G *

- j

H 

* -

I

Node A, C, G B, F E D, H I

r (%) 0.00000 1.73205 3.46410 −1.73205 −3.46410 p1 0.16667 0.12167 0.88667 0.22167 0.08667 p2 0.66667 0.65667 0.02667 0.65667 0.02667 p3 0.16667 0.22167 0.08667 0.12167 0.88667

A Numerical Example (continued)

• Suppose that phase two is to fit the spot rate curve 0.08 − 0.05 × e^−0.18×t.

• The annualized continuously compounded spot rates are r(0, 1) = 3.82365%, r(0, 2) = 4.51162%, r(0, 3) = 5.08626%.

• Start with state price Q(0, 0) = 1 at node A.

A Numerical Example (continued)

• Now, by Eq. (167) on p. 1213,

β₀ = r(0, 1) + ln Q(0, 0) e^−r0 = r(0, 1) = 3.82365%.

• Hence the short rate at node A equals β₀ + r₀ = 3.82365%.

• The state prices at year one are calculated as Q(1, 1) = p₁(0, 0) e^−(β0+r0) = 0.160414, Q(1, 0) = p₂(0, 0) e^−(β0+r0) = 0.641657, Q(1,−1) = p₃(0, 0) e^−(β0+r0) = 0.160414.

A Numerical Example (continued)

• The 2-year rate spot rate r(0, 2) is matched by picking

β₁ = r(0, 2)×2+ln

Q(1, 1) e^−Δr + Q(1, 0) + Q(1,−1) e^Δr 

= 5.20459%.

• Hence the short rates at nodes B, C, and D equal β₁ + r_j,

where j = 1, 0,−1, respectively.

• They are found to be 6.93664%, 5.20459%, and 3.47254%.

A Numerical Example (continued)

• The state prices at year two are calculated as

Q(2, 2) = p₁(1, 1) e^−(β1+r1)Q(1, 1) = 0.018209,

Q(2, 1) = p₂(1, 1) e^−(β1+r1)Q(1, 1) + p₁(1, 0) e^−(β1+r0)Q(1, 0)

= 0.199799,

Q(2, 0) = p₃(1, 1) e^−(β1+r1)Q(1, 1) + p₂(1, 0) e^−(β1+r0)Q(1, 0) +p₁(1,−1) e^{−(β1+r−1)}Q(1, −1) = 0.473597,

Q(2, −1) = p3(1, 0) e^−(β1+r0)Q(1, 0) + p₂(1,−1) e^{−(β1+r−1)}Q(1, −1)

= 0.203263,

Q(2, −2) = p3(1,−1) e^{−(β1+r−1)}Q(1, −1) = 0.018851.

A Numerical Example (concluded)

• The 3-year rate spot rate r(0, 3) is matched by picking

β2 = r(0, 3) × 3 + ln

Q(2, 2) e^−2×Δr + Q(2, 1) e^−Δr + Q(2, 0) +Q(2, −1) e^Δr + Q(2, −2) e^2×Δr 

= 6.25359%.

• Hence the short rates at nodes E, F, G, H, and I equal β₂ + r_j, where j = 2, 1, 0, −1, −2, respectively.

• They are found to be 9.71769%, 7.98564%, 6.25359%, 4.52154%, and 2.78949%.

• The figure on p. 1222 plots β_i for i = 0, 1, . . . , 29.

      <HDU +L/







 -L+/

The (Whole) Yield Curve Approach

• We have seen several Markovian short rate models.

• The Markovian approach is computationally efficient.

• But it is difficult to model the behavior of yields and bond prices of different maturities.

• The alternative yield curve approach regards the whole term structure as the state of a process and directly specifies how it evolves.

The Heath-Jarrow-Morton (HJM) Model

• This influential model is a forward rate model.

• The HJM model specifies the initial forward rate curve and the forward rate volatility structure.

– The volatility structure describes the volatility of each forward rate for a given maturity date.

• Like the Black-Scholes option pricing model, neither risk preference assumptions nor the drifts of forward rates are needed.

aHeath, Jarrow, & Morton (1992).

The HJM Model (continued)

• Within a finite-time horizon [ 0, U ], we take as given the time-zero forward rate curve f (0, T ) for T ∈ [ 0, U ].

• Since this curve is used as the boundary value at t = 0, perfect fit to the observed term structure is automatic.

• The forward rates are driven by k stochastic factors.

The HJM Model (continued)

• Specifically the forward rate movements are governed by the stochastic process,

df (t, T ) = μ(t, T ) dt +

k i=1

σ_i(t, T ) dW_i,

(168) where μ and σ_i may depend on the past history of the independent Wiener processes W₁, W₂, . . . , W_k.

• One-factor models seem to perform better than multifactor models empirically, at least for pricing short-dated options.^a

aAmin & Morton (1994).

The HJM Model (continued)

• But two-factor models perform better in hedging caps and floors.^a

• Kamakura (2019) has a 10-factor^b (14-factor^c) HJM model for the U.S. Treasuries (German bonds,

respectively).

• A unique equivalent martingale measure π can be established under which the prices of interest rate

derivatives do not depend on the market prices of risk.

aGupta & Subrahmanyam (2001, 2005).

bSee http://www.kamakuraco.com/KamakuraReleasesNewStochasticVolatilityModel

cSee http://www.kamakuraco.com/KamakuraReleases14FactorHeathJarrowandMorto.

The HJM Model (continued)

Theorem 22 (1) For all 0 < t ≤ T , μ(t, T ) =

k i=1

σ_i(t, T )

 _T

t

σ_i(t, u) du (169) holds under π almost surely. (2) The bond price dynamics under π is given by

dP (t, T )

P (t, T ) = r(t) dt −

k i=1

σ_p,i(t, T ) dW_i, (170)

where σ_p,i(t, T ) ≡ _T

t σ_i(t, u) du.

The HJM Model (concluded)

• Hence choosing the volatility functions σ_i(t, T ) of the forward rate dynamics under π uniquely determines the drift parameters under π and the prices of all claims.

The Use of the HJM Model

• Take the one-factor model,

df (t, T ) = μ(t, T ) dt + σ(t, T ) dW_t.

• To use the HJM model, we first pick σ(t, T ).

• This is the modeling part.

• The drift parameters are then determined by Eq. (169) on p. 1228.

• Now fetch today’s forward rate curve { f(0, T ), T ≥ 0 } and integrate it to obtain the forward rates,

f (t, T ) = f (0, T ) +

 _t

0

μ(s, T ) ds +

 _t

0

σ(s, T ) dW_s.

The Use of the HJM Model (concluded)

• Compute the future bond prices by P (t, T ) = e⁻

_T

t f (t,s) ds

if necessary.

• European-style derivatives can be priced by simulating many paths and taking average.

Short Rate under the HJM Model

• From Eq. (26.19) of the textbook, the short rate follows the following SDE,

dr(t) = ∂f (0, t)

∂t dt +

  _t

0



σ_p(s, t) ∂σ(s, t)

∂t + σ(s, t)²

 ds

 dt +

 _t

0

∂σ(s, t)

∂t dW_s



dt + σ(t, t) dW_t. (171)

• Since the second and the third terms on the right-hand side depend on the history of σ_p and/or dW , they can make r non-Markovian.

Short Rate under the HJM Model (concluded)

• If σ_p(t, T ) = σ(T − t) for a constant σ, the short rate process r becomes Markovian.

• Then Eq. (171) on p. 1232 is reduced to dr =

∂f (0, t)

∂t + σ²t



dt + σ dW.

• This is the continuous-time Ho-Lee model (154) on p.

1154.^a

• See Carverhill (1994) and Jeffrey (1995) for conditions for the short rate to be Markovian.

aSee p. 392 of the textbook.

The Alternative HJM Model

• Alternatively, we can start with the bond process under π:

dP (t, T )

P (t, T ) = r(t) dt +

k i=1

σ_p,i(t, T ) dW_i. (172)

• Then^a

df (t, T ) =

k i=1

σ_p,i(t, T ) ∂σ_p,i(t, T )

∂T dt

−

k i=1

∂σ_p,i(t, T )

∂T dW_i.

aCarverhill (1995); Musiela & Rutkowski (1997); Hull (1999).

Gaussian HJM Models

• A nonstochastic volatility depends on only t and T .

• When the forward rate volatilities σ_i(t, T ) are nonstochastic, we have a Gaussian HJM model.

• For Gaussian HJM models, the bond price volatilities σ_p,i(t, T ) must also be nonstochastic.

• The forward rates have a normal distribution, whereas the bond prices have a lognormal distribution.

aMusiela & Rutkowski (1997).

Gaussian HJM Models (concluded)

• σ(t, T ) = σ: The Ho-Lee model (154) on p. 1154 obtains.

• σ(t, T ) = σe^{−a(T −t)}: The Hull-White model (158) on p.

1182 obtains.

• σ(t, T ) = σ₀ + σ₁(T − t): The linear absolute model.^a

• σ(t, T ) = σ [ γ(T − t) + 1 ] e−(λ/2)(T −t): The Mercurio-Moraleda (2000) model.

aGupta & Subrahmanyam (2001, 2005).

Local-Volatility HJM Models

• If the forward rate volatilities σ_i(t, T, f (t, T )) depend on t, T , and f (t, T ) only, we have a local-volatility HJM model.

• The same term may also apply to HJM models whose bond price volatilities σ_p,i(t, T, P (t, T )) depend on t, T , and P (t, T ) only.

aBrigo & Mercurio (2006).

Local-Volatility HJM Models (continued)

• The (nearly) proportional volatility model:^a

σ(t, T, f (t, T )) = σ₀ min(κ, f (t, T )), σ₀, κ > 0.

• The proportional volatility model:^b

σ(t, T, f (t, T )) = σ₀f (t, T ). (173)

• The linear proportional model:^c

σ(t, T, f (t, T )) = [ σ₀ + σ₁(T − t) ] f(t, T ).

aHeath, Jarrow, & Morton (1992); Jarrow (1996). The large positive constant κ prevents explosion in finite time.

bGupta & Subrahmanyam (2001, 2005).

cGupta & Subrahmanyam (2001, 2005).

Local-Volatility HJM Models (continued)

• Exponentially dampened volatility proportional to the short rate:^a

σ(t, T ) = σf (t, t) e^{−a(T −t)}.

• The Ritchken-Sankarasubramanian (1995) model:^b σ(t, T ) = σ(t, t) e⁻

_T

t κ(x) dx. – For example,^c

σ(t, t) = σr(t)^γ.

aGrant & Vora (1999).

bThe short rate volatility σ(t, t) may depend on the short rate r(t).

cRitchken & Sankarasubramanian (1995); Li, Ritchken, & Sankara- subramanian (1995).

Local-Volatility HJM Models (concluded)

• A model attributed to Ian Cooper (1993):^a σ_p(t, T, P (t, T )) = ψ(t) ln P (t, T ) in Eq. (172) on p. 1234:

aRebonato (1996). It is equivalent to the proportional volatility model (173) when ψ(t) is a constant.

Trees for HJM Models

• Obtain today’s forward rate curve:

f (0, 0), f (0, Δt), f (0, 2Δt), f (0, 3Δt), . . . , f (0, T ).

• For binomial trees, generate the two forward rate curves at time Δt:

f_u(Δt, Δt), f_u(Δt, 2Δt), f_u(Δt, 3Δt), . . . , f_u(Δt, T ), f_d(Δt, Δt), f_d(Δt, 2Δt), f_d(Δt, 3Δt), . . . , f_d(Δt, T ).

by Eq. (168) on p. 1226 with μ(t, T ) from Eq. (169) on p. 1228.

Trees for HJM Models (continued)

• Iterate until the maturity t ≤ T of the derivative.

• A straightforward implementation of the HJM model results in noncombining trees.

– For a binomial tree with n time steps, O(2ⁿ) nodes for one-factor HJM models; O(3ⁿ) or O(4ⁿ) for

two-factor models.^a

aClewlow & Strickland (1998); Hull (1999); Nawalkha, Beliaeva, &

Soto (2007).

H(0) B(0.1) B(0,2) B(0,3)

H(1) B(1,2) B(1,3)

H(1) B(1,2) B(1,3)

H(2) B(2,3)

H(2) B(2,3)

H(2) B(2,3)

H(2) B(2,3)

H(3) H(3) H(3) H(3) H(3) H(3) H(3) H(3)