Tracking Error Revisited

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Tracking Error Revisited

• Define the dollar gamma as S2Γ.

• The change in value of a delta-hedged long option position after a duration of ∆t is proportional to the dollar gamma.

• It is about

(1/2)S2Γ[ (∆S/S)2 − σ2∆t ].

– (∆S/S)2 is called the daily realized variance.

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Tracking Error Revisited (continued)

• Let the rebalancing times be t1, t2, . . . , tn.

• Let ∆Si = Si+1 − Si.

• The total tracking error at expiration is about

n−1X

i=0

er(T −ti)Si2Γi 2

" µ

∆Si Si

2

− σ2∆t

# ,

• The tracking error is path dependent.

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Tracking Error Revisited (concluded)

a

• The tracking error ²n over n rebalancing acts (such as 251,235 on p. 534) has about the same probability of being positive as being negative.

• Subject to certain regularity conditions, the root-mean-square tracking error p

E[ ²2n ] is O(1/√

n ).b

• The root-mean-square tracking error increases with σ at first and then decreases.

aBertsimas, Kogan, and Lo (2000).

bSee also Grannan and Swindle (1996).

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Delta-Gamma Hedge

• Delta hedge is based on the first-order approximation to changes in the derivative price, ∆f , due to changes in the stock price, ∆S.

• When ∆S is not small, the second-order term, gamma Γ ≡ ∂2f /∂S2, helps (theoretically).

• A delta-gamma hedge is a delta hedge that maintains zero portfolio gamma, or gamma neutrality.

• To meet this extra condition, one more security needs to be brought in.

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Delta-Gamma Hedge (concluded)

• Suppose we want to hedge short calls as before.

• A hedging call f2 is brought in.

• To set up a delta-gamma hedge, we solve

−N × f + n1 × S + n2 × f2 − B = 0 (self-financing),

−N × ∆ + n1 + n2 × ∆2 − 0 = 0 (delta neutrality),

−N × Γ + 0 + n2 × Γ2 − 0 = 0 (gamma neutrality),

for n1, n2, and B.

– The gammas of the stock and bond are 0.

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Other Hedges

• If volatility changes, delta-gamma hedge may not work well.

• An enhancement is the delta-gamma-vega hedge, which also maintains vega zero portfolio vega.

• To accomplish this, one more security has to be brought into the process.

• In practice, delta-vega hedge, which may not maintain gamma neutrality, performs better than delta hedge.

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Trees

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I love a tree more than a man.

— Ludwig van Beethoven (1770–1827) And though the holes were rather small, they had to count them all.

— The Beatles, A Day in the Life (1967)

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The Combinatorial Method

• The combinatorial method can often cut the running time by an order of magnitude.

• The basic paradigm is to count the number of admissible paths that lead from the root to any terminal node.

• We first used this method in the linear-time algorithm for standard European option pricing on p. 234.

– In general, it cannot apply to American options.

• We will now apply it to price barrier options.

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The Reflection Principle

a

• Imagine a particle at position (0, −a) on the integral lattice that is to reach (n, −b).

• Without loss of generality, assume a > 0 and b ≥ 0.

• This particle’s movement:

(i, j) *(i + 1, j + 1) up move S → Su j(i + 1, j − 1) down move S → Sd

• How many paths touch the x axis?

aAndr´e (1887).

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(0, a) (n, b) (0, a)

J

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The Reflection Principle (continued)

• For a path from (0, −a) to (n, −b) that touches the x axis, let J denote the first point this happens.

• Reflect the portion of the path from (0, −a) to J.

• A path from (0, a) to (n, −b) is constructed.

• It also hits the x axis at J for the first time.

• The one-to-one mapping shows the number of paths from (0, −a) to (n, −b) that touch the x axis equals the number of paths from (0, a) to (n, −b).

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The Reflection Principle (concluded)

• A path of this kind has (n + b + a)/2 down moves and (n − b − a)/2 up moves.

• Hence there are

µ n

n+a+b 2

(57) such paths for even n + a + b.

– Convention: ¡n

k

¢ = 0 for k < 0 or k > n.

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Pricing Barrier Options (Lyuu, 1998)

• Focus on the down-and-in call with barrier H < X.

• Assume H < S without loss of generality.

• Define

a

»ln (X/ (Sdn)) ln(u/d)

¼

=

»ln(X/S)

∆t + n 2

¼ , h

¹ln (H/ (Sdn)) ln(u/d)

º

=

¹ln(H/S)

∆t + n 2

º .

– h is such that ˜H ≡ Suhdn−h is the terminal price that is closest to, but does not exceed H.

– a is such that ˜X ≡ Suadn−a is the terminal price that is closest to, but is not exceeded by X.

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Pricing Barrier Options (continued)

• The true barrier is replaced by the effective barrier ˜H in the binomial model.

• A process with n moves hence ends up in the money if and only if the number of up moves is at least a.

• The price Sukdn−k is at a distance of 2k from the lowest possible price Sdn on the binomial tree.

Sukdn−k = Sd−kdn−k = Sdn−2k. (58)

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0 0

2 h n

2 a

S

0 0 0 0

2 j

X~ Su da n a Su dj n j

H~ Su dh n h

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Pricing Barrier Options (continued)

• The number of paths from S to the terminal price Sujdn−j is ¡n

j

¢, each with probability pj(1 − p)n−j.

• With reference to p. 550, the reflection principle can be applied with a = n − 2h and b = 2j − 2h in Eq. (57) on p. 547 by treating the S line as the x axis.

• Therefore,

µ n

n+(n−2h)+(2j−2h) 2

=

µ n

n − 2h + j

paths hit ˜H in the process for h ≤ n/2.

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Pricing Barrier Options (concluded)

• The terminal price Sujdn−j is reached by a path that hits the effective barrier with probability

µ n

n − 2h + j

pj(1 − p)n−j.

• The option value equals

P2h

j=a

¡ n

n−2h+j

¢pj(1 − p)n−j ¡

Sujdn−j − X¢

Rn . (59)

– R ≡ erτ /n is the riskless return per period.

• It implies a linear-time algorithm.

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Convergence of BOPM

• Equation (59) results in the sawtooth-like convergence shown on p. 321.

• The reasons are not hard to see.

• The true barrier most likely does not equal the effective barrier.

• The same holds for the strike price and the effective strike price.

• The issue of the strike price is less critical.

• But the issue of the barrier is not negligible.

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Convergence of BOPM (continued)

• Convergence is actually good if we limit n to certain values—191, for example.

• These values make the true barrier coincide with or occur just above one of the stock price levels, that is, H ≈ Sdj = Se−jσ

τ /n for some integer j.

• The preferred n’s are thus n =

$ τ

(ln(S/H)/(jσ))2

%

, j = 1, 2, 3, . . .

• There is only one minor technicality left.

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Convergence of BOPM (continued)

• We picked the effective barrier to be one of the n + 1 possible terminal stock prices.

• However, the effective barrier above, Sdj, corresponds to a terminal stock price only when n − j is even.a

• To close this gap, we decrement n by one, if necessary, to make n − j an even number.

aThis is because j = n − 2k for some k by Eq. (58) on p. 549. Of course we could have adopted the form Sdj (−n ≤ j ≤ n) for the effective barrier.

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Convergence of BOPM (concluded)

• The preferred n’s are now n =



` if ` − j is even

` − 1 otherwise , j = 1, 2, 3, . . . , where

` ≡

$ τ

(ln(S/H)/(jσ))2

% .

• Evaluate pricing formula (59) on p. 552 only with the n’s above.

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0 500 1000 1500 2000 2500 3000 3500

#Periods 5.5

5.55 5.6 5.65 5.7

Down-and-in call value

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Practical Implications

• Now that barrier options can be efficiently priced, we can afford to pick very large n’s (p. 559).

• This has profound consequences.

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n Combinatorial method Value Time (milliseconds)

21 5.507548 0.30

84 5.597597 0.90

191 5.635415 2.00

342 5.655812 3.60

533 5.652253 5.60

768 5.654609 8.00

1047 5.658622 11.10

1368 5.659711 15.00

1731 5.659416 19.40

2138 5.660511 24.70

2587 5.660592 30.20

3078 5.660099 36.70

3613 5.660498 43.70

4190 5.660388 44.10

4809 5.659955 51.60

5472 5.660122 68.70

6177 5.659981 76.70

6926 5.660263 86.90

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Practical Implications (concluded)

• Pricing is prohibitively time consuming when S ≈ H because n ∼ 1/ ln2(S/H).

• This observation is indeed true of standard quadratic-time binomial tree algorithms.

• But it no longer applies to linear-time algorithms (p. 561).

• In fact, this model is O(1/n) convergent.a

aLin (2008).

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Barrier at 95.0 Barrier at 99.5 Barrier at 99.9

n Value Time n Value Time n Value Time

..

. 795 7.47761 8 19979 8.11304 253

2743 2.56095 31.1 3184 7.47626 38 79920 8.11297 1013 3040 2.56065 35.5 7163 7.47682 88 179819 8.11300 2200 3351 2.56098 40.1 12736 7.47661 166 319680 8.11299 4100 3678 2.56055 43.8 19899 7.47676 253 499499 8.11299 6300 4021 2.56152 48.1 28656 7.47667 368 719280 8.11299 8500

True 2.5615 7.4767 8.1130

(All times in milliseconds.)

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Trinomial Tree

• Set up a trinomial approximation to the geometric Brownian motion dS/S = r dt + σ dW .a

• The three stock prices at time ∆t are S, Su, and Sd, where ud = 1.

• Impose the matching of mean and that of variance:

1 = pu + pm + pd,

SM (puu + pm + (pd/u)) S,

S2V pu(Su − SM )2 + pm(S − SM )2 + pd(Sd − SM )2.

aBoyle (1988).

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• Above,

M ≡ er∆t,

V ≡ M2(eσ2∆t − 1), by Eqs. (18) on p. 149.

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* - j

pu pm pd

Su S Sd S

-

¾

∆t

* - j

* - j

* - j

* - j

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Trinomial Tree (continued)

• Use linear algebra to verify that pu = u¡

V + M2 − M¢

− (M − 1) (u − 1) (u2 − 1) , pd = u2 ¡

V + M2 − M¢

− u3(M − 1) (u − 1) (u2 − 1) . – In practice, must make sure the probabilities lie

between 0 and 1.

• Countless variations.

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Trinomial Tree (concluded)

• Use u = eλσ∆t, where λ ≥ 1 is a tunable parameter.

• Then

pu 1

2 +

¡r + σ2¢ √

∆t

2λσ ,

pd 1

2

¡r − 2σ2¢ √

∆t

2λσ .

• A nice choice for λ is p

π/2 .a

aOmberg (1988).

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Barrier Options Revisited

• BOPM introduces a specification error by replacing the barrier with a nonidentical effective barrier.

• The trinomial model solves the problem by adjusting λ so that the barrier is hit exactly.a

• It takes

h = ln(S/H) λσ√

∆t

consecutive down moves to go from S to H if h is an integer, which is easy to achieve by adjusting λ.

– This is because Se−hλσ∆t = H.

aRitchken (1995).

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Barrier Options Revisited (continued)

• Typically, we find the smallest λ ≥ 1 such that h is an integer.

• That is, we find the largest integer j ≥ 1 that satisfies

ln(S/H)

∆t ≥ 1 and then let

λ = ln(S/H) jσ√

∆t .

– Such a λ may not exist for very small n’s.

– This is not hard to check.

• This done, one of the layers of the trinomial tree coincides with the barrier.

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Barrier Options Revisited (concluded)

• The following probabilities may be used, pu = 1

2 + µ0

∆t 2λσ , pm = 1 − 1

λ2, pd = 1

2 µ0

∆t 2λσ . – µ0 ≡ r − σ2/2.

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0 50 100 150 200

#Periods 5.61

5.62 5.63 5.64 5.65 5.66

Down-and-in call value

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Algorithms Comparison

a

• So which algorithm is better, binomial or trinomial?

• Algorithms are often compared based on the n value at which they converge.

– The one with the smallest n wins.

• So giraffes are faster than cheetahs because they take fewer strides to travel the same distance!

• Performance must be based on actual running times.

aLyuu (1998).

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Algorithms Comparison (concluded)

• Pages 321 and 570 show the trinomial model converges at a smaller n than BOPM.

• It is in this sense when people say trinomial models converge faster than binomial ones.

• But is the trinomial model better then?

• The linear-time binomial tree algorithm actually

performs better than the trinomial one (see next page expanded from p. 559).

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n Combinatorial method Trinomial tree algorithm

Value Time Value Time

21 5.507548 0.30

84 5.597597 0.90 5.634936 35.0 191 5.635415 2.00 5.655082 185.0 342 5.655812 3.60 5.658590 590.0 533 5.652253 5.60 5.659692 1440.0 768 5.654609 8.00 5.660137 3080.0 1047 5.658622 11.10 5.660338 5700.0 1368 5.659711 15.00 5.660432 9500.0 1731 5.659416 19.40 5.660474 15400.0 2138 5.660511 24.70 5.660491 23400.0 2587 5.660592 30.20 5.660493 34800.0 3078 5.660099 36.70 5.660488 48800.0 3613 5.660498 43.70 5.660478 67500.0 4190 5.660388 44.10 5.660466 92000.0 4809 5.659955 51.60 5.660454 130000.0 5472 5.660122 68.70

6177 5.659981 76.70

(All times in milliseconds.)

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Double-Barrier Options

• Double-barrier options are barrier options with two barriers L < H.

• Assume L < S < H.

• The binomial model produces oscillating option values (see plot next page).a

aChao (1999); Dai and Lyuu (2005);

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20 40 60 80 100 8

10 12 14 16

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Double-Barrier Knock-Out Options

• We knew how to pick the λ so that one of the layers of the trinomial tree coincides with one barrier, say H.

• This choice, however, does not guarantee that the other barrier, L, is also hit.

• One way to handle this problem is to lower the layer of the tree just above L to coincide with L.a

– More general ways to make the trinomial model hit both barriers are available.b

aRitchken (1995).

bHsu and Lyuu (2006). Dai and Lyuu (2006) combine binomial and trinomial trees to derive an O(n)-time algorithm for double-barrier op-

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H

L S

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Double-Barrier Knock-Out Options (continued)

• The probabilities of the nodes on the layer above L must be adjusted.

• Let ` be the positive integer such that Sd`+1 < L < Sd`.

• Hence the layer of the tree just above L has price Sd`.

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Double-Barrier Knock-Out Options (concluded)

• Define γ > 1 as the number satisfying L = Sd`−1e−γλσ∆t. – The prices between the barriers are

L, Sd`−1, . . . , Sd2, Sd, S, Su, Su2, . . . , Suh−1, Suh = H.

• The probabilities for the nodes with price equal to Sd`−1 are

p0u = b + aγ

1 + γ , p0d = b − a

γ + γ2 , and p0m = 1 − p0u − p0d, where a ≡ µ0

∆t/(λσ) and b ≡ 1/λ2.

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Convergence: Binomial vs. Trinomial

30 32.5 35 37.5 40 42.5 45 n

1.6 1.8 2 2.2 2.4 2.6

Optionvalue

Trinomial Binomial

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Multivariate Contingent Claims

• They depend on two or more underlying assets.

• The basket call on m assets has the terminal payoff max(Pm

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).

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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).

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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.

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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

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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.

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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).

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Correlated Trinomial Model Simplified

a

• Let µ0i ≡ 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

õ01

σ1 + µ02 σ2

! + ρ

λ2

# ,

p2 = 1

4

"

1 λ2 +

∆t

λ

õ01

σ1 µ02 σ2

!

ρ λ2

# ,

p3 = 1

4

"

1 λ2 +

∆t

λ Ã

µ01

σ1 µ02 σ2

! + ρ

λ2

# ,

p4 = 1

4

"

1 λ2 +

∆t

λ Ã

µ01

σ1 + µ02 σ2

!

ρ λ2

# ,

p5 = 1 − 1 λ2.

• It cannot price 2-asset 2-barrier options accurately.b

aMadan, Milne, and Shefrin (1989).

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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 µ1

n

. (60)

• 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 by 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 . (61)

• This estimate should converge faster than 1/n.

• The Richardson extrapolation uses n = 2n .

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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. (60) on p. 588.

• But the plots on p. 249 (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. 591.

• The sequence with odd n turns out to be monotonic and smooth (see the left plot on p. 593).a

• Apply extrapolation (61) on p. 589 with n2 = n1 + 2, where n1 is odd.

• Result is shown in the right plot on p. 593.

• The convergence rate is amazing.

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

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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. 597).

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

(63)

0 0.05 0.1 0.15 0.2 0.25 80

85 90 95 100 105 110 115

(64)

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

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Explicit Methods

• Consider the diffusion equation D(∂2θ/∂x2) − (∂θ/∂t) = 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 + · · · , (62)

2θ(x, t)

∂x2

¯¯

¯¯

x=xi

= θ(xi+1, t) − 2θ(xi, t) + θ(xi−1, t)

(∆x)2 + · · · . (63)

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Explicit Methods (continued)

• Next, assemble Eqs. (62) and (63) 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. (63), we might as well use xi for x in Eq. (62).

• Two choices are possible for t in Eq. (63).

• 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 . (64)

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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. (64) on p. 601 as

θi,j+1 = D∆t

(∆x)2 θi+1,j + µ

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 figure (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.

(71)

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 eight times as much.

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Explicit Method and Trinomial Tree

• Rearrange Eq. (64) on p. 601 as

θi,j+1 = D∆t

(∆x)2 θi+1,j + µ

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

Figure

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References

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