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(1)

Ideas for Binomial Trees To Handle Two Barriers

* j* j* j* j

* j* j* j

-

∆t ∆t-∆t- -

 T

ln(H)

ln S ln(L)

∆t

ln(H/L)

(2)

The Binomial-Trinomial Tree

• Append a trinomial structure to a binomial tree can lead to improved convergence and efficiency.a

• The resulting tree is called the binomial-trinomial tree.b

• Suppose a binomial tree will be built with ∆t as the duration of one period.

• Node X at time t needs to pick three nodes on the binomial tree at time t + ∆t as its successor nodes.

– ∆t ≤ ∆t < 2∆t.

aDai (R86526008, D8852600) and Lyuu (2006, 2008, 2010).

bThe idea first emerged in a hotel in Muroran, Hokkaido, Japan, in May of 2005.

(3)

The Binomial-Trinomial Tree (continued)

pu

pm

pd

-

 ∆t

-

 ∆t -

 ∆t

? 6

∆t

?6| β |

? 6

∆t

? 6

| α |

?

| γ |6

ˆ

µ + 2σ

∆t

µ ˆ µ

0

ˆ

µ − 2σ

∆t X

A

B

C

(4)

The Binomial-Trinomial Tree (continued)

• These three nodes should guarantee:

1. The mean and variance of the stock price are matched.

2. The branching probabilities are between 0 and 1.

• Let S be the stock price at node X.

• Use s(z) to denote the stock price at node z.

(5)

The Binomial-Trinomial Tree (continued)

• Recall that the expected value of the logarithmic return ln(St+∆t/S) at time t + ∆t equalsa

µ (

r − σ2/2)

∆t. (69)

• Its variance equals

Var ≡ σ2∆t. (70)

• Let node B be the node whose logarithmic return ˆ

µ ≡ ln(s(B)/S) is closest to µ among all the nodes on the binomial tree at time t + ∆t.

aSee p. 275.

(6)

The Binomial-Trinomial Tree (continued)

• The middle branch from node X will end at node B.

• The two nodes A and C, which bracket node B, are the destinations of the other two branches from node X.

• Recall that adjacent nodes on the binomial tree are spaced at 2σ√

∆t apart.

• Review the figure on p. 674 for illustration.

(7)

The Binomial-Trinomial Tree (continued)

• The three branching probabilities from node X are obtained through matching the mean and variance of the logarithmic return ln(St+∆t/S).

• Recall that

ˆ

µ ≡ ln (s(B)/S)

is the logarithmic return of the middle node B.

• Let α, β, and γ be the differences between µ and the logarithmic returns

ln(s(Z)/S), Z = A, B, C, in that order.

(8)

The Binomial-Trinomial Tree (continued)

• In other words,

α ≡ ˆµ + 2σ√

∆t − µ = β + 2σ√

∆t , (71)

β ≡ ˆµ − µ, (72)

γ ≡ ˆµ − 2σ√

∆t − µ = β − 2σ√

∆t . (73)

• The three branching probabilities pu, pm, pd then satisfy puα + pmβ + pdγ = 0, (74) puα2 + pmβ2 + pdγ2 = Var, (75) pu + pm + pd = 1. (76)

(9)

The Binomial-Trinomial Tree (concluded)

• Equation (74) matches the mean (69) of the logarithmic return ln(St+∆t/S) on p. 676.

• Equation (75) matches its variance (70) on p. 676.

• The three probabilities can be proved to lie between 0 and 1.

(10)

Pricing Double-Barrier Options

• Consider a double-barrier option with two barriers L and H, where L < S < H.

• We need to make each barrier coincide with a layer of the binomial tree for better convergence.

• This means choosing a ∆t such that κ ln(H/L)

2σ√

∆t is a positive integer.

– The distance between two adjacent nodes such as nodes Y and Z in the figure on p. 682 is 2σ√

∆t .

(11)

Pricing Double-Barrier Options (continued)

* j* j* j* j

* j* j* j

 :

j Y

Z

-

 ∆t ∆t-∆t- -

 T

ln(H/S)

ln(L/S) + 4σ

∆t

ln(L/S) + 2σ

∆t 0

ln(L/S)

∆t

ln(H/L)

A

B

C

(12)

Pricing Double-Barrier Options (continued)

• Suppose that the goal is a tree with ∼ m periods.

• Suppose we pick ∆τ ≡ T/m for the length of each period.

• There is no guarantee that ln(H/L)∆τ is an integer.

• So we pick a ∆t that is close to, but does not exceed,

∆τ and makes ln(H/L)

∆t an integer.

• Specifically, we select

∆t =

(ln(H/L) 2κσ

)2

, where κ =

ln(H/L)

∆τ

⌉ .

(13)

Pricing Double-Barrier Options (continued)

• We now proceed to build the binomial-trinomial tree.

• Start with the binomial part.

• Lay out the nodes from the low barrier L upward and downward.

• Automatically, a layer coincides with the high barrier H.

• It is unlikely that ∆t divides T , however.

• So the position at time 0 and with logarithmic return ln(S/S) = 0 is not occupied by a binomial node to serve as the root node (recall p. 682).

(14)

Pricing Double-Barrier Options (continued)

• The binomial-trinomial structure can address this problem as follows.

• Between time 0 and time T , the binomial tree spans T /∆t periods.

• Keep only the last ⌊T/∆t⌋ − 1 periods and let the first period have a duration equal to

∆t = T

(⌊ T

∆t

− 1 )

∆t.

• Then these ⌊T/∆t⌋ periods span T years.

• It is easy to verify that ∆t ≤ ∆t < 2∆t.

(15)

Pricing Double-Barrier Options (continued)

• Start with the root node at time 0 and at a price with logarithmic return ln(S/S) = 0.

• Find the three nodes on the binomial tree at time ∆t as described earlier.

• Calculate the three branching probabilities to them.

• Grow the binomial tree from these three nodes until time T to obtain a binomial-trinomial tree with

⌊T/∆t⌋ periods.

• See the figure on p. 682 for illustration.

(16)

Pricing Double-Barrier Options (continued)

• Now the binomial-trinomial tree can be used to price double-barrier options by backward induction.

• That takes quadratic time.

• But we know a linear-time algorithm exists for

double-barrier options on the binomial tree (see text).

• Apply that algorithm to price the double-barrier option’s prices at the three nodes at time ∆t.

– That is, nodes A, B, and C on p. 682.

• Then calculate their expected discounted value for the root node.

• The overall running time is only linear!

(17)

Pricing Double-Barrier Options (continued)

• Binomial trees have troubles with pricing barrier options (see p. 375, p. 666, and p. 671).

• Even pit against the much better trinomial tree, the binomial-trinomial tree converges faster and smoother (see p. 689).

• In fact, the binomial-trinomial tree has an error of O(1/n) for single-barrier options.a

• It has an error of O(1/n1−a) for any 0 < a < 1 for double-barrier options.b

aLyuu and Palmer (2010).

bElisa Appolloni, Gaudenziy, and Zanette (2014).

(18)

Pricing Double-Barrier Options (concluded)

10.19 10.195 10.2 10.205 10.21

0.01 0.02 0.03 0.04 0.05 0.06 0.07

Value

Time A

B

The thin line denotes the double-barrier option prices

computed by the trinomial tree against the running time in seconds (such as point A). The thick line denotes those

computed by the binomial-trinomial tree (such as point B).

(19)

Pricing Discrete Barrier Options

• Barrier options whose barrier is monitored only at discrete times are called discrete barrier options.

• They are more common than the continuously monitored versions.

• The main difficulty with pricing discrete barrier options lies in matching the monitored times.

• Here is why.

• Suppose each period has a duration of ∆t and the ℓ > 1 monitored times are

t0 = 0, t1, t2, . . . , t = T.

(20)

Pricing Discrete Barrier Options (continued)

• It is unlikely that all monitored times coincide with the end of a period on the tree, meaning ∆t divides ti for all i.

• The binomial-trinomial tree can handle discrete options with ease, however.

• Simply build a binomial-trinomial tree from time 0 to time t1, followed by one from time t1 to time t2, and so on until time t.

• See p. 692.

(21)

* j* j* j* j

* j* j* j

 : j

R z

*

R z

*

R z

*

R z

*

R z

*

t0

-

∆t1 ∆t-1 ∆t-1 ∆t2-

t1

∆t1

{

∆t2

}

(22)

Pricing Discrete Barrier Options (concluded)

• This procedure works even if each ti is associated with a distinct barrier or if each window [ ti, ti+1) has its own continuously monitored barrier or double barriers.

(23)

Options on a Stock That Pays Known Dividends

• Many ad hoc assumptions have been postulated for option pricing with known dividends.a

1. The one we saw earlier (p. 292) models the stock price minus the present value of the anticipated dividends as following geometric Brownian motion.

2. One can also model the stock price plus the forward values of the dividends as following geometric

Brownian motion.

aFrishling (2002).

(24)

Options on a Stock That Pays Known Dividends (continued)

• Realistic models assume:

– The stock price decreases by the amount of the dividend paid at the ex-dividend date.

– The dividend is part cash and part yield (i.e., α(t)S0 + β(t)St), for practitioners.a

• The stock price follows geometric Brownian motion between adjacent ex-dividend dates.

• But they result in binomial trees that grow exponentially (recall p. 291).

• The binomial-trinomial tree can often avoid this problem.

aHenry-Labord`ere (2009).

(25)

Options on a Stock That Pays Known Dividends (continued)

• Suppose that the known dividend is D dollars and the ex-dividend date is at time t.

• So there are m ≡ t/∆t periods between time 0 and the ex-dividend date.

• To avoid negative stock prices, we need to make sure the lowest stock price at time t is at least D, i.e.,

Se−(t/∆t)σ

∆t ≥ D.

– Equivalently,

∆t

[ ln(S/D)

]2

.

(26)

Options on a Stock That Pays Known Dividends (continued)

• Build a binomial tree from time 0 to time t as before.

• Subtract D from all the stock prices on the tree at time t to represent the price drop on the ex-dividend date.

• Assume the top node’s price equals S.

– As usual, its two successor nodes will have prices Su and Su−1.

• The remaining nodes’ successor nodes will have prices Su−3, Su−5, Su−7, . . . ,

same as the binomial tree.

(27)

Options on a Stock That Pays Known Dividends (concluded)

• For each node at time t below the top node, we build the trinomial connection.

• Note that the binomial-trinomial structure remains valid in the special case when ∆t = ∆t on p. 674.

• Hence the construction can be completed.

• From time t + ∆t onward, the standard binomial tree will be used until the maturity date or the next

ex-dividend date when the procedure can be repeated.

• The resulting tree is called the stair tree.a

aDai (R86526008, D8852600) and Lyuu (2004); Dai (R86526008) (2009).

(28)

Other Applications of Binomial-Trinomial Trees

• Pricing guaranteed minimum withdrawal benefits.a

• Option pricing with stochastic volatilities.b

• Efficient Parisian option pricing.c

• Option pricing with time-varying volatilities and time-varying barriers.d

• Defaultable bond pricing.e

aWu (R96723058) (2009).

bHuang (R97922073) (2010).

cHuang (R97922081) (2010).

dChou (R97944012) (2010) and Chen (R98922127) (2011).

eDai (R86526008, D8852600), Lyuu, and Wang (F95922018) (2009, 2010, 2014).

(29)

General Properties of Trees

a

• Consider the Ito process,

dX = a(X, t) dt + σ dW, where a(X, t) = O(1) and σ is a constant.

• The mean and volatility of the next move’s size are O(∆t) and O(√

∆t), respectively.

• Note that

∆t ≫ ∆t.

• The tree spacing must be in the order of σ√

∆t if the variance is to be matched.b

aChiu (R98723059) (2012) and Wu (R99922149) (2012).

bLyuu and Wang (F95922018) (2009, 2011) and Lyuu and Wen (D94922003) (2012).

(30)

General Properties of Trees (concluded)

• It can also be proved that either B is a tree node or both A and C are tree nodes.

-

 ∆t

? 6

Θ(σ

∆t)

X

A

B

C

(31)

Merton’s Jump-Diffusion Model

• Empirically, stock returns tend to have fat tails,

inconsistent with the Black-Scholes model’s assumptions.

• Stochastic volatility and jump processes have been proposed to address this problem.

• Merton’s jump-diffusion model is our focus.a

aMerton (1976).

(32)

Merton’s Jump-Diffusion Model (continued)

• This model superimposes a jump component on a diffusion component.

• The diffusion component is the familiar geometric Brownian motion.

• The jump component is composed of lognormal jumps driven by a Poisson process.

– It models the sudden changes in the stock price

because of the arrival of important new information.

(33)

Merton’s Jump-Diffusion Model (continued)

• Let St be the stock price at time t.

• The risk-neutral jump-diffusion process for the stock price follows

dSt

St = (r − λ¯k) dt + σ dWt + k dqt. (77)

• Above, σ denotes the volatility of the diffusion component.

(34)

Merton’s Jump-Diffusion Model (continued)

• The jump event is governed by a compound Poisson process qt with intensity λ, where k denotes the magnitude of the random jump.

– The distribution of k obeys ln(1 + k) ∼ N (

γ, δ2) with mean ¯k ≡ E (k) = eγ+δ2/2 − 1.

• The model with λ = 0 reduces to the Black-Scholes model.

(35)

Merton’s Jump-Diffusion Model (continued)

• The solution to Eq. (77) on p. 704 is

St = S0e(r−λ¯k−σ2/2) t+σWtU (n(t)), (78) where

U (n(t)) =

n(t)

i=0

(1 + ki) .

– ki is the magnitude of the ith jump with ln(1 + ki) ∼ N(γ, δ2).

– k0 = 0.

– n(t) is a Poisson process with intensity λ.

(36)

Merton’s Jump-Diffusion Model (concluded)

• Recall that n(t) denotes the number of jumps that occur up to time t.

• As k > −1, stock prices will stay positive.

• The geometric Brownian motion, the lognormal jumps, and the Poisson process are assumed to be independent.

(37)

Tree for Merton’s Jump-Diffusion Model

a

• Define the S-logarithmic return of the stock price S as ln(S/S).

• Define the logarithmic distance between stock prices S and S as

| ln(S) − ln(S) | = | ln(S/S)|.

aDai (R86526008, D8852600), Wang (F95922018), Lyuu, and Liu (2010).

(38)

Tree for Merton’s Jump-Diffusion Model (continued)

• Take the logarithm of Eq. (78) on p. 706:

Mt ≡ ln

(St S0

)

= Xt + Yt, (79) where

Xt (

r − λ¯k − σ2/2)

t + σWt, (80) Yt

n(t) i=0

ln (1 + ki) . (81)

• It decomposes the S0-logarithmic return of St into the diffusion component Xt and the jump component Yt.

(39)

Tree for Merton’s Jump-Diffusion Model (continued)

• Motivated by decomposition (79) on p. 709, the tree construction divides each period into a diffusion phase followed by a jump phase.

• In the diffusion phase, Xt is approximated by the BOPM.

• Hence Xt can make an up move to Xt + σ√

∆t with probability pu or a down move to Xt − σ√

∆t with probability pd.

(40)

Tree for Merton’s Jump-Diffusion Model (continued)

• According to BOPM,

pu = eµ∆t − d u − d , pd = 1 − pu, except that µ = r − λ¯k here.

• The diffusion component gives rise to diffusion nodes.

• They are spaced at 2σ√

∆t apart such as the white nodes A, B, C, D, E, F, and G on p. 712.

(41)

(− ∆1) t t (+ ∆1) t

q1

q1

pu

pd

q0

2 2

h= γ +δ

2σ t

White nodes are diffusion nodes.

Gray nodes are jump nodes. In the diffusion phase, the solid black lines denote the binomial struc- ture of BOPM, whereas the dashed lines denote the trinomial struc- ture. Here m is set to one here for simplicity. Only the double- circled nodes will remain after the construction. Note that a and b are diffusion nodes because no jump oc- curs (j = 0 in Eq. (82)) in the jump phase.

(42)

Tree for Merton’s Jump-Diffusion Model (continued)

• In the jump phase, Yt+∆t is approximated by moves

from each diffusion node to 2m jump nodes that match the first 2m moments of the lognormal jump.

• The m jump nodes above the diffusion node are spaced at h apart.

• The same holds for the m jump nodes below the diffusion node.

• The gray nodes at time ℓ∆t on p. 712 are jump nodes.

(43)

Tree for Merton’s Jump-Diffusion Model (continued)

• Overall, the random variable Mt is approximated as follows.

• Following Mt, the S0-logarithmic returns of the 2 × (2m + 1) nodes at time t + ∆t are

Mt+∆t = Mt + cσ√

∆t + jh, c = ±1, j = 0, ±1, ±2, . . . , ±m.

(82) – c ∈ {−1, 1} denotes the up or down move of the

stock price driven by the diffusion component.

– j denotes the number of jumps above or below the diffusion nodes, driven by the jump component.

(44)

Tree for Merton’s Jump-Diffusion Model (continued)

• The magnitude of the basic jump size is set to h =

γ2 + δ2 . (83)

• In summary, the stock prices at the end of the period are Suejh, Sdejh,−m ≤ j ≤ m.

• Only these 2 × (2m + 1) nodes at the end of the jump phase will be retained for the tree; the nodes at the end of the diffusion phase will not.

(45)

Tree for Merton’s Jump-Diffusion Model (continued)

• The probabilities remain to be calculated.

• Let µi be the ith moment of Yt+∆t − Yt.

• The probabilities for the jump phase are qj, −m ≤ j ≤ m.

(46)

Tree for Merton’s Jump-Diffusion Model (continued)

• They must match the first 2m moments of the jump component, i.e.,

m j=−m

(jh)iqj

= µi

≡ E



n(∆t)

w=0

ln(1 + kw)

i

 , i = 1, 2, . . . , 2m. (84)

(47)

Tree for Merton’s Jump-Diffusion Model (continued)

• For sufficiently small ∆t, the approximation µi ≈ κi is accurate as µ1 = κ1 and µi = κi + O((∆t)2),

i = 2, 3, . . . , wherea

κ1 = λ∆t(γ),

κ2 = λ∆t

( γ2

+ δ2 ) ,

κ3 = λ∆t

( γ3

+ 3γδ2 ) ,

κ4 = λ∆t

( γ4

+ 6γ2 δ2

+ 3δ4 ) ,

κ5 = λ∆t

( γ5

+ 10γ3 δ2

+ 15γδ4 ) ,

κ6 = λ∆t

( γ6

+ 15γ4 δ2

+ 45γ2 δ4

+ 15δ6 ) ,

κ7 = λ∆t

( γ7

+ 21γ5 δ2

+ 105γ3 δ4

+ 105γδ6 ) , ..

.

aStuart and Ord (1994).

(48)

Tree for Merton’s Jump-Diffusion Model (continued)

• Finally,

m j=−m

qj = 1, (85)

where qj ≥ 0.

• Solve the 2m + 1 equations (84)–(85) to obtain the probabilities qj.

(49)

Tree for Merton’s Jump-Diffusion Model (continued)

• The diffusion component and the jump component are independent.

• So the probability of moving from Mt to Mt + σ√

∆t + jh is puqj and that to Mt − σ√

∆t + jh is pdqj.

• In the ensuing diffusion phase, the trinomial branches on p. 674 (the dashed lines) are used to link the jump nodes (like the gray nodes at time ℓ∆t) with the diffusion

nodes (like nodes C, D, E, F, and G).

(50)

Tree for Merton’s Jump-Diffusion Model (concluded)

• The diffusion nodes like nodes a and b at time ℓ∆t, however, continue to follow the structure of BOPM.

• The procedure is repeated until the maturity date.

• The size of the tree is O(n2.5).

(51)

Multivariate Contingent Claims

• They depend on two or more underlying assets.

• The basket call on m assets has the terminal payoff

max

( m

i=1

αiSi(τ ) − X, 0 )

, where αi is the percentage of asset i.

• Basket options are essentially options on a portfolio of stocks or index options.

• Option on the best of two risky assets and cash has a terminal payoff of max(S1(τ ), S2(τ ), X).

(52)

Multivariate Contingent Claims (concluded)

a

Name Payoff

Exchange option max(S1(τ ) − S2(τ ), 0) Better-off option max(S1(τ ), . . . , Sk(τ ), 0) Worst-off option min(S1(τ ), . . . , Sk(τ ), 0)

Binary maximum option I{ max(S1(τ ), . . . , Sk(τ )) > X } Maximum option max(max(S1(τ ), . . . , Sk(τ )) − X, 0) Minimum option max(min(S1(τ ), . . . , Sk(τ )) − X, 0) Spread option max(S1(τ ) − S2(τ ) − X, 0)

Basket average option max((S1(τ ), . . . , Sk(τ ))/k − X, 0) Multi-strike option max(S1(τ ) − X1, . . . , Sk(τ ) − Xk, 0)

Pyramid rainbow option max(| S1(τ ) − X1 | + · · · + | Sk(τ ) − Xk | − X, 0)

Madonna option max(

(S1(τ ) − X1)2 + · · · + (Sk(τ ) − Xk)2 − X, 0)

aLyuu and Teng (R91723054) (2011).

(53)

Correlated Trinomial Model

a

• Two risky assets S1 and S2 follow dSi

Si = r dt + σi dWi in a risk-neutral economy, i = 1, 2.

• Let

Mi ≡ er∆t,

Vi ≡ Mi2(eσi2∆t − 1).

– SiMi is the mean of Si at time ∆t.

– Si2Vi the variance of Si at time ∆t.

aBoyle, Evnine, and Gibbs (1989).

(54)

Correlated Trinomial Model (continued)

• The value of S1S2 at time ∆t has a joint lognormal distribution with mean S1S2M1M2eρσ1σ2∆t, where ρ is the correlation between dW1 and dW2.

• Next match the 1st and 2nd moments of the

approximating discrete distribution to those of the continuous counterpart.

• At time ∆t from now, there are five distinct outcomes.

(55)

Correlated Trinomial Model (continued)

• The five-point probability distribution of the asset prices is (as usual, we impose uidi = 1)

Probability Asset 1 Asset 2 p1 S1u1 S2u2 p2 S1u1 S2d2 p3 S1d1 S2d2 p4 S1d1 S2u2

p5 S1 S2

(56)

Correlated Trinomial Model (continued)

• The probabilities must sum to one, and the means must be matched:

1 = p1 + p2 + p3 + p4 + p5,

S1M1 = (p1 + p2) S1u1 + p5S1 + (p3 + p4) S1d1, S2M2 = (p1 + p4) S2u2 + p5S2 + (p2 + p3) S2d2.

(57)

Correlated Trinomial Model (concluded)

• Let R ≡ M1M2eρσ1σ2∆t.

• Match the variances and covariance:

S12V1 = (p1 + p2)((S1u1)2 − (S1M1)2) + p5(S12 − (S1M1)2) +(p3 + p4)((S1d1)2 − (S1M1)2),

S22V2 = (p1 + p4)((S2u2)2 − (S2M2)2) + p5(S22 − (S2M2)2) +(p2 + p3)((S2d2)2 − (S2M2)2),

S1S2R = (p1u1u2 + p2u1d2 + p3d1d2 + p4d1u2 + p5) S1S2.

• The solutions are complex (see text).

(58)

Correlated Trinomial Model Simplified

a

• Let µi ≡ r − σi2/2 and ui ≡ eλσi∆t for i = 1, 2.

• The following simpler scheme is good enough:

p1 = 1 4

[ 1 λ2

+

∆t

λ

(µ′1

σ1

+ µ′2 σ2

) +

ρ λ2

] ,

p2 = 1

4 [ 1

λ2 +

∆t

λ

(µ′1

σ1 µ′2 σ2

)

ρ λ2

] ,

p3 = 1

4 [ 1

λ2 +

∆t

λ (

µ′1

σ1 µ′2 σ2

) + ρ

λ2 ]

,

p4 = 1

4 [ 1

λ2 +

∆t

λ (

µ′1 σ1

+ µ′2 σ2

)

ρ λ2

] ,

p5 = 1 1 λ2

.

aMadan, Milne, and Shefrin (1989).

(59)

Correlated Trinomial Model Simplified (continued)

• All of the probabilities lie between 0 and 1 if and only if

−1 + λ

∆t µ1

σ1 + µ2 σ2

≤ ρ ≤ 1 − λ

∆t µ1

σ1 µ2 σ2

,(86)

1 ≤ λ (87)

• We call a multivariate tree (correlation-) optimal if it guarantees valid probabilities as long as

−1 + O(√

∆t) < ρ < 1 − O(√

∆t), such as the above one.a

aKao (R98922093) (2011) and Kao (R98922093), Lyuu, and Wen (D94922003) (2014).

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Correlated Trinomial Model Simplified (concluded)

• But this model cannot price 2-asset 2-barrier options accurately.a

• Few multivariate trees are both optimal and able to handle multiple barriers.b

• An alternative is to use orthogonalization.c

aSee Chang (B89704039, R93922034), Hsu (R7526001, D89922012), and Lyuu (2006) and Kao (R98922093), Lyuu and Wen (D94922003) (2014) for solutions.

bSee Kao (R98922093), Lyuu, and Wen (D94922003) (2014) for one.

cHull and White (1990) and Dai (R86526008, D8852600), Lyuu, and Wang (F95922018) (2012).

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Extrapolation

• It is a method to speed up numerical convergence.

• Say f(n) converges to an unknown limit f at rate of 1/n:

f (n) = f + c

n + o (1

n )

. (88)

• Assume c is an unknown constant independent of n.

– Convergence is basically monotonic and smooth.

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Extrapolation (concluded)

• From two approximations f(n1) and f (n2) and ignoring the smaller terms,

f (n1) = f + c n1 , f (n2) = f + c

n2 .

• A better approximation to the desired f is f = n1f (n1) − n2f (n2)

n1 − n2 . (89)

• This estimate should converge faster than 1/n.

• The Richardson extrapolation uses n2 = 2n1.

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Improving BOPM with Extrapolation

• Consider standard European options.

• Denote the option value under BOPM using n time periods by f (n).

• It is known that BOPM convergences at the rate of 1/n, consistent with Eq. (88) on p. 732.

• But the plots on p. 279 (redrawn on next page)

demonstrate that convergence to the true option value oscillates with n.

• Extrapolation is inapplicable at this stage.

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5 10 15 20 25 30 35 n

11.5 12 12.5 13

Call value

0 10 20 30 40 50 60 n

15.1 15.2 15.3 15.4 15.5

Call value

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Improving BOPM with Extrapolation (concluded)

• Take the at-the-money option in the left plot on p. 735.

• The sequence with odd n turns out to be monotonic and smooth (see the left plot on p. 737).a

• Apply extrapolation (89) on p. 733 with n2 = n1 + 2, where n1 is odd.

• Result is shown in the right plot on p. 737.

• The convergence rate is amazing.

• See Exercise 9.3.8 of the text (p. 111) for ideas in the general case.

aThis can be proved; see Chang and Palmer (2007).

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5 10 15 20 25 30 35 n

12.2 12.4 12.6 12.8 13 13.2 13.4

Call value

5 10 15 20 25 30 35 n

12.11 12.12 12.13 12.14 12.15 12.16 12.17

Call value

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Numerical Methods

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All science is dominated by the idea of approximation.

— Bertrand Russell

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Finite-Difference Methods

• Place a grid of points on the space over which the desired function takes value.

• Then approximate the function value at each of these points (p. 741).

• Solve the equation numerically by introducing difference equations in place of derivatives.

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0 0.05 0.1 0.15 0.2 0.25 80

85 90 95 100 105 110 115

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Example: Poisson’s Equation

• It is ∂2θ/∂x2 + ∂2θ/∂y2 = −ρ(x, y).

• Replace second derivatives with finite differences through central difference.

• Introduce evenly spaced grid points with distance of ∆x along the x axis and ∆y along the y axis.

• The finite difference form is

−ρ(xi, yj) = θ(xi+1, yj) − 2θ(xi, yj) + θ(xi−1, yj) (∆x)2

+θ(xi, yj+1) − 2θ(xi, yj) + θ(xi, yj−1)

(∆y)2 .

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Example: Poisson’s Equation (concluded)

• In the above, ∆x ≡ xi − xi−1 and ∆y ≡ yj − yj−1 for i, j = 1, 2, . . . .

• When the grid points are evenly spaced in both axes so that ∆x = ∆y = h, the difference equation becomes

−h2ρ(xi, yj) = θ(xi+1, yj) + θ(xi−1, yj) +θ(xi, yj+1) + θ(xi, yj−1) − 4θ(xi, yj).

• Given boundary values, we can solve for the xis and the yjs within the square [±L, ±L ].

• From now on, θi,j will denote the finite-difference approximation to the exact θ(xi, yj).

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Explicit Methods

• Consider the diffusion equation D(∂2θ/∂x2) − (∂θ/∂t) = 0, D > 0.

• Use evenly spaced grid points (xi, tj) with distances

∆x and ∆t, where ∆x ≡ xi+1 − xi and ∆t ≡ tj+1 − tj.

• Employ central difference for the second derivative and forward difference for the time derivative to obtain

∂θ(x, t)

∂t

t=tj

= θ(x, tj+1) − θ(x, tj)

∆t + · · · , (90)

2θ(x, t)

∂x2

x=xi

= θ(xi+1, t) − 2θ(xi, t) + θ(xi−1, t)

(∆x)2 + · · · . (91)

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Explicit Methods (continued)

• Next, assemble Eqs. (90) and (91) into a single equation at (xi, tj).

• But we need to decide how to evaluate x in the first equation and t in the second.

• Since central difference around xi is used in Eq. (91), we might as well use xi for x in Eq. (90).

• Two choices are possible for t in Eq. (91).

• The first choice uses t = tj to yield the following finite-difference equation,

θi,j+1 − θi,j

∆t = D θi+1,j − 2θi,j + θi−1,j

(∆x)2 . (92)

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Explicit Methods (continued)

• The stencil of grid points involves four values, θi,j+1, θi,j, θi+1,j, and θi−1,j.

• Rearrange Eq. (92) on p. 745 as

θi,j+1 = D∆t

(∆x)2 θi+1,j + (

1 2D∆t (∆x)2

)

θi,j + D∆t

(∆x)2 θi−1,j.

• We can calculate θi,j+1 from θi,j, θi+1,j, θi−1,j, at the previous time tj (see exhibit (a) on next page).

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Stencils

tj tj 1 xi 1

xi 1 xi

tj tj 1 xi 1

xi 1 xi

(a) (b)

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Explicit Methods (concluded)

• Starting from the initial conditions at t0, that is, θi,0 = θ(xi, t0), i = 1, 2, . . . , we calculate

θi,1, i = 1, 2, . . . .

• And then

θi,2, i = 1, 2, . . . .

• And so on.

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Stability

• The explicit method is numerically unstable unless

∆t ≤ (∆x)2/(2D).

– A numerical method is unstable if the solution is highly sensitive to changes in initial conditions.

• The stability condition may lead to high running times and memory requirements.

• For instance, halving ∆x would imply quadrupling (∆t)−1, resulting in a running time 8 times as much.

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Explicit Method and Trinomial Tree

• Recall that

θi,j+1 = D∆t

(∆x)2 θi+1,j + (

1 2D∆t (∆x)2

)

θi,j + D∆t

(∆x)2 θi−1,j.

• When the stability condition is satisfied, the three coefficients for θi+1,j, θi,j, and θi−1,j all lie between zero and one and sum to one.

• They can be interpreted as probabilities.

• So the finite-difference equation becomes identical to backward induction on trinomial trees!

• The freedom in choosing ∆x corresponds to similar freedom in the construction of trinomial trees.

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