Options on a Stock That Pays Known Dividends

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.

Pricing Discrete Barrier Options (continued)

• Suppose each period has a duration of Δt and the

 > 1 monitored times are

t₀ = 0, t₁, t₂, . . . , t_ = T.

• It is unlikely that all monitored times coincide with the end of a period on the tree, or Δt divides t_i 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 .

* j* j* j* j

* j* j* j

 : j

R z

*

R z

*

R z

*

R z

*

R z

*

t

-

^Δt1 ^Δt-¹ ^Δt-¹ ^Δt2-

t 2σ√

Δt1



2σ√ Δt2



Pricing Discrete Barrier Options (concluded)

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

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. 305) 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).

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)S_t), 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. 304).

• The binomial-trinomial tree can avoid this problem in most cases.

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/Δt periods between time 0 and the^Δ ex-dividend date.^a

• 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/Δt)σ√Δt} ≥ D.

– Or,

Δt ≥

 tσ ln(S/D)

₂ .

aOr simply assume m is an integer input and Δt =^Δ t/m.

Options on a Stock That Pays Known Dividends (continued)

• Build a CRR 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 S^u and S^u⁻¹.

• The remaining nodes’ successor nodes will have prices S^u⁻³, S^u⁻⁵, S^u⁻⁷, . . . ,

same as the CRR tree.

A Stair Tree

D

0

1

2

3

4

S'' S

D Su²− D S−

D Sd²−

Options on a Stock That Pays Known Dividends (continued)

• 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 Δt^ = Δt on p. 703.

– And even with the displacements ±2σ√

Δt (as on p.

731).

Options on a Stock That Pays Known Dividends (concluded)

• Hence the construction can be completed.

• From time t + Δt 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 (B82506025, R86526008, D8852600) & Lyuu (2004); Dai (B82506025, R86526008, D8852600) (2009).

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

aH. Wu (R96723058) (2009).

bC. Huang (R97922073) (2010).

cY. Huang (R97922081) (2010).

dC. Chou (R97944012) (2010); C. Chen (R98922127) (2011).

eDai (B82506025, R86526008, D8852600), Lyuu, & C. Wang

General Properties of Trees

• 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(Δt) and O(√

Δt), respectively.

• Note that √

Δt  Δt.

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

Δt if the variance is to be matched.^b

aChiu (R98723059) (2012); C. H. Wu (R99922149) (2012).

bLyuu & C. Wang (F95922018) (2009, 2011); Lyuu & Wen (D94922003) (2012).

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 (1976) jump-diffusion model is our focus.

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.

Merton’s Jump-Diffusion Model (continued)

• Let S_t be the stock price at time t.

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

dS_t

S_t = (r − λ¯k) dt + σ dW_t + k dq_t. (96)

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

Merton’s Jump-Diffusion Model (continued)

• The jump event is governed by a compound Poisson process q_t 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) = e^{Δ γ+δ2/2} − 1.

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

Merton’s Jump-Diffusion Model (continued)

• The solution to Eq. (96) on p. 738 is

S_t = S₀e^{(r−λ¯k−σ2/2) t+σWt}U (n(t)), (97) where

U (n(t)) =

n(t)

i=0

(1 + k_i) .

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

– k₀ = 0.

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

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.

Tree for Merton’s Jump-Diffusion Model

• 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 (B82506025, R86526008, D8852600), C. Wang (F95922018), Lyuu,

& Y. Liu (2010).

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

• Take the logarithm of Eq. (97) on p. 740:

M_t = ln^Δ

S_t S₀



= X_t + Y_t, (98) where

X_t =^Δ



r − λ¯k − σ² 2



t + σW_t, (99)

Y_t =^Δ

n(t) i=0

ln (1 + k_i) . (100)

• It decomposes the S₀-logarithmic return of S_t into the diffusion component X_t and the jump component Y_t.

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

• Motivated by decomposition (98) on p. 743, the tree construction divides each period into a diffusion phase followed by a jump phase.

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

• So X_t makes an up move to X_t + σ√

Δt with probability p_u or a down move to X_t − σ√

Δt with probability p_d.

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

• According to BOPM,

p_u = e^μΔt − d u − d , p_d = 1 − p_u, except that μ = r − λ¯k here.

• The diffusion component gives rise to diffusion nodes.

• They are spaced at 2σ√

Δt apart such as the white nodes A, B, C, D, E, F, and G on p. 746.

( − Δ1) t Δ^t ( + Δ1) t

q1

q₋1

pu

pd

q0

2 2

h= γ +δ

2σ Δt

White nodes are diffusion nodes.

Gray nodes are jump nodes. In the diffusion phase, the solid black lines denote the binomial structure of BOPM; the dashed lines denote the trinomial structure. Here m is set to one for simplicity. Only the double-circled nodes will remain af- ter the construction. Note that a and b are diffusion nodes because no jump occurs in the jump phase.

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

• In the jump phase, Y_t+Δt 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 =^Δ

γ² + δ² apart.

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

• The gray nodes at time Δt on p. 746 are jump nodes.

• The size of the tree is O(n^2.5).

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

α_iS_i(τ ) − X, 0 

, where α_i is the percentage of asset i.

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

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

Multivariate Contingent Claims (concluded)

Name Payoff

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

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

Basket average option max((S1(τ) + · · · + S_k(τ))/k − X, 0) Multi-strike option max(S1(τ) − X1, . . . , S_k(τ) − X_k, 0)

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

Madonna option max(

(S1(τ) − X1)² + · · · + (S_k(τ) − X_k)² − X, 0)

aLyuu & Teng (R91723054) (2011).

Correlated Trinomial Model

• Two risky assets S₁ and S₂ follow dS_i

S_i = r dt + σ_i dW_i in a risk-neutral economy, i = 1, 2.

• Let

M_i =^Δ e^rΔt,

V_i =^Δ M_i²(e^σi2Δt − 1).

– S_iM_i is the mean of S_i at time Δt.

– S_i²V_i the variance of S_i at time Δt.

Correlated Trinomial Model (continued)

• The value of S₁S₂ at time Δt has a joint lognormal distribution with mean S₁S₂M₁M₂e^ρσ1σ2Δt, 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 Δt from now, there are 5 distinct outcomes.

Correlated Trinomial Model (continued)

• The five-point probability distribution of the asset prices is

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

p₅ S₁ S₂

• As usual, impose u_id_i = 1.

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.

Correlated Trinomial Model (concluded)

• Let R = M^Δ ₁M₂e^ρσ1σ2Δt.

• 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 appear on p. 246 of the textbook.

Correlated Trinomial Model Simplified

• Let μ^_i = r^Δ − σ_i²/2 and u_i = e^{Δ λσi}

√Δt for i = 1, 2.

• The following simpler scheme is good enough:

p1 = 1

4

 1 λ2 +

√Δt λ

μ1

σ1 + μ2 σ2

 + ρ

λ2

 ,

p2 = 1

4

 1 λ2 +

√Δt λ

μ1

σ1 − μ2 σ2



− ρ λ2

 ,

p3 = 1

4

 1 λ2 +

√Δt λ



−μ1

σ1 − μ2 σ2

 + ρ

λ2

 ,

p4 = 1

4

 1 λ2 +

√Δt λ



−μ1

σ1 + μ2 σ2



− ρ λ2

 ,

p5 = 1 − 1 λ2 .

aMadan, Milne, & Shefrin (1989).

Correlated Trinomial Model Simplified (continued)

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

−1 + λ√ Δt 

 μ^₁ σ1

+ μ^₂ σ2

 ≤ ρ ≤ 1 − λ√ Δt 

 μ^₁ σ1

− μ^₂ σ2

,(101)

1 ≤ λ (102)

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

−1 + O(√

Δt) < ρ < 1 − O(√

Δt), such as the above one.^a

aW. Kao (R98922093) (2011); W. Kao (R98922093), Lyuu, & Wen (D94922003) (2014).

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 Y. Chang (B89704039, R93922034), Hsu (R7526001, D89922012),

& Lyuu (2006); W. Kao (R98922093), Lyuu, & Wen (D94922003) (2014) for solutions.

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

cHull & White (1990); Dai (B82506025, R86526008, D8852600), Lyuu,

& C. Wang (F95922018) (2012).

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



. (103)

• Assume c is an unknown constant independent of n.

– Convergence is basically monotonic and smooth.

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 . (104)

• This estimate should converge faster than 1/n.^a

• The Richardson extrapolation uses n₂ = 2n₁.

aIt is identical to the forward rate formula (21) on p. 140!

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. (103) on p. 758.

• But the plots on p. 288 (redrawn on next page)

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

• Extrapolation is inapplicable at this stage.

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

Improving BOPM with Extrapolation (concluded)

• Take the at-the-money option in the left plot on p. 761.

• The sequence with odd n turns out to be monotonic and smooth (see the left plot on p. 763).^a

• Apply extrapolation (104) on p. 759 with n₂ = n₁ + 2, where n₁ is odd.

• Result is shown in the right plot on p. 763.

• The convergence rate is amazing.

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

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

Numerical Methods

All science is dominated by the idea of approximation.

— Bertrand Russell

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. 767).

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

0 0.05 0.1 0.15 0.2 0.25 80

85 90 95 100 105 110 115

Example: Poisson’s Equation

• It is ∂²θ/∂x² + ∂²θ/∂y² = −ρ(x, y), which describes the electrostatic field.

• Replace second derivatives with finite differences through central difference.

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

• The finite difference form is

−ρ(x_i, y_j) = θ(x_i+1, y_j) − 2θ(x_i, y_j) + θ(x_i−1, y_j) (Δx)²

+θ(x_i, y_j+1) − 2θ(x_i, y_j) + θ(x_i, y_j−1)

(Δy)² .

Example: Poisson’s Equation (concluded)

• In the above, Δx = x^Δ _i − x_i−1 and Δy = y^Δ _j − y_j−1 for i, j = 1, 2, . . . .

• When the grid points are evenly spaced in both axes so that Δx = Δy = h, the difference equation becomes

−h²ρ(x_i, y_j) = θ(x_i+1, y_j) + θ(x_i−1, y_j) +θ(x_i, y_j+1) + θ(x_i, y_j−1) − 4θ(x_i, y_j).

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

• From now on, θ_i,j will denote the finite-difference approximation to the exact θ(x_i, y_j).

Explicit Methods

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

• Use evenly spaced grid points (x_i, t_j) with distances

Δx and Δt, where Δx = x^Δ _i+1 − x_i and Δt = t^Δ _j+1 − t_j.

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

∂θ(x, t)

∂t



t=t_j

= θ(x, t_j+1) − θ(x, t_j)

Δt + · · · , (105)

∂²θ(x, t)

∂x²



x=x_i

= θ(x_i+1, t) − 2θ(x_i, t) + θ(x_i−1, t)

(Δx)² + · · · .(106)

Explicit Methods (continued)

• Next, assemble Eqs. (105) and (106) into a single equation at (x_i, t_j).

• 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. (106), we might as well use x_i for x in Eq. (105).

• Two choices are possible for t in Eq. (106).

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

θ_i,j+1 − θ_i,j

Δt = D θ_i+1,j − 2θ_i,j + θ_i−1,j

(Δx)² .

(107)

Explicit Methods (continued)

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

• Rearrange Eq. (107) on p. 771 as

θi,j+1 = DΔt

(Δx)² θi+1,j +



1 − 2DΔt (Δx)²



θi,j + DΔt

(Δx)² θi−1,j.

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

Stencils

t_j t_j x_i

x_i x_i

t_j t_j x_i

x_i x_i

= >

Explicit Methods (concluded)

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

θ_i,1, i = 1, 2, . . . .

• And then

θ_i,2, i = 1, 2, . . . .

• And so on.

Stability

• The explicit method is numerically unstable unless Δt ≤ (Δx)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 Δx would imply quadrupling (Δt)⁻¹, resulting in a running time 8 times as much.

Explicit Method and Trinomial Tree

• Recall that

θi,j+1 = DΔt

(Δx)² θi+1,j +



1 − 2DΔt (Δx)²



θi,j + DΔt

(Δx)² θi−1,j.

• When the stability condition is satisfied, the three coefficients for θ_i+1,j, θ_i,j, and θ_i−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!

Explicit Method and Trinomial Tree (concluded)

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

• The explicit finite-difference equation is also identical to backward induction on a binomial tree.^a

– Let the binomial tree take 2 steps each of length Δt/2.

– It is now a trinomial tree.

aHilliard (2014).

Implicit Methods

• Suppose we use t = t_j+1 in Eq. (106) on p. 770 instead.

• The finite-difference equation becomes θ_i,j+1 − θ_i,j

Δt = D θ_i+1,j+1 − 2θ_i,j+1 + θ_i−1,j+1

(Δx)² .

(108)

• The stencil involves θ_i,j, θ_i,j+1, θ_i+1,j+1, and θ_i−1,j+1.

• This method is implicit:

– The value of any one of the three quantities at t_j+1 cannot be calculated unless the other two are known.

– See exhibit (b) on p. 773.

Implicit Methods (continued)

• Equation (108) can be rearranged as

θ_i−1,j+1 − (2 + γ) θ_i,j+1 + θ_i+1,j+1 = −γθ_i,j, where γ = (Δx)^{Δ 2}/(DΔt).

• This equation is unconditionally stable.

• Suppose the boundary conditions are given at x = x₀ and x = x_{N +1}.

• After θ_i,j has been calculated for i = 1, 2, . . . , N , the values of θ_i,j+1 at time t_j+1 can be computed as the solution to the following tridiagonal linear system,

Implicit Methods (continued)

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

a 1 0 · · · · · · · · · 0

1 a 1 0 · · · · · · 0

0 1 a 1 0 · · · 0

.. .

...

...

...

...

... .. . ..

.

... ... ... ... ... .. .

0 · · · · · · 0 1 a 1

0 · · · · · · · · · 0 1 a

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎣

θ1,j+1 θ2,j+1 θ3,j+1

.. . .. . .. . θN,j+1

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦

=

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎣

−γθ1,j − θ0,j+1

−γθ2,j

−γθ3,j .. . .. .

−γθN−1,j

−γθN,j − θN+1,j+1

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦ ,

where a =^Δ −2 − γ.

Implicit Methods (concluded)

• Tridiagonal systems can be solved in O(N) time and O(N ) space.

– Never invert a matrix to solve a tridiagonal system.

• The matrix above is nonsingular when γ ≥ 0.

– A square matrix is nonsingular if its inverse exists.

Crank-Nicolson Method

• Take the average of explicit method (107) on p. 771 and implicit method (108) on p. 778:

θi,j+1 − θi,j Δt

= 1

2



D θi+1,j − 2θi,j + θi−1,j

(Δx)2 + D θi+1,j+1 − 2θi,j+1 + θi−1,j+1 (Δx)2

 .

• After rearrangement,

γθi,j+1 − θi+1,j+1 − 2θi,j+1 + θi−1,j+1

2 = γθi,j + θi+1,j − 2θi,j + θi−1,j

2 .

• This is an unconditionally stable implicit method with excellent rates of convergence.

Stencil

t

t

x

x

x

Numerically Solving the Black-Scholes PDE (82) on p.

629

• See text.

• Brennan and Schwartz (1978) analyze the stability of the implicit method.

Monte Carlo Simulation

• Monte Carlo simulation is a sampling scheme.

• In many important applications within finance and without, Monte Carlo is one of the few feasible tools.

• When the time evolution of a stochastic process is not easy to describe analytically, Monte Carlo may very well be the only strategy that succeeds consistently.

aA top 10 algorithm (Dongarra & Sullivan, 2000).

The Big Idea

• Assume X₁, X₂, . . . , X_n have a joint distribution.

• θ = E[ g(X^Δ ₁, X₂, . . . , X_n) ] for some function g is desired.

• We generate

x⁽ⁱ⁾₁ , x⁽ⁱ⁾₂ , . . . , x⁽ⁱ⁾_n



, 1 ≤ i ≤ N

independently with the same joint distribution as (X₁, X₂, . . . , X_n).

• Set

Y_i = g^Δ

x⁽ⁱ⁾₁ , x⁽ⁱ⁾₂ , . . . , x⁽ⁱ⁾_n

 .

The Big Idea (concluded)

• Y₁, Y₂, . . . , Y_N are independent and identically distributed random variables.

• Each Y_i has the same distribution as Y = g(X^Δ ₁, X₂, . . . , X_n).

• Since the average of these N random variables, Y , satisfies E[ Y ] = θ, it can be used to estimate θ.

• The strong law of large numbers says that this procedure converges almost surely.

• The number of replications (or independent trials), N, is called the sample size.

Accuracy

• The Monte Carlo estimate and true value may differ owing to two reasons:

1. Sampling variation.

2. The discreteness of the sample paths.^a

• The first can be controlled by the number of replications.

• The second can be controlled by the number of observations along the sample path.

aThis may not be an issue if the financial derivative only requires discrete sampling along the time dimension, such as the discrete barrier option.

Accuracy and Number of Replications

• The statistical error of the sample mean Y of the random variable Y grows as 1/√

N . – Because Var[ Y ] = Var[ Y ]/N .

• In fact, this convergence rate is asymptotically optimal.^a

• So the variance of the estimator Y can be reduced by a factor of 1/N by doing N times as much work.

• This is amazing because the same order of convergence holds independently of the dimension n.

aThe Berry-Esseen theorem.

Accuracy and Number of Replications (concluded)

• In contrast, classic numerical integration schemes have an error bound of O(N^−c/n) for some constant c > 0.

– n is the dimension.

• The required number of evaluations thus grows

exponentially in n to achieve a given level of accuracy.

– The curse of dimensionality.

• The Monte Carlo method is more efficient than

alternative procedures for multivariate derivatives when n is large.

Monte Carlo Option Pricing

• For the pricing of European options on a

dividend-paying stock, we may proceed as follows.

• Assume

dS

S = μ dt + σ dW.

• Stock prices S₁, S₂, S₃, . . . at times Δt, 2Δt, 3Δt, . . . can be generated via

S_i+1 = S_ie^{(μ−σ2/2) Δt+σ}

√Δt ξ, ξ ∼ N(0, 1).

(109)

Monte Carlo Option Pricing (continued)

• If we discretize dS/S = μ dt + σ dW directly, we will obtain

S_i+1 = S_i + S_iμ Δt + S_iσ√

Δt ξ.

• But this is locally normally distributed, not lognormally, hence biased.^a

• In practice, this is not expected to be a major problem as long as Δt is sufficiently small.

aContributed by Mr. Tai, Hui-Chin (R97723028) on April 22, 2009.

Monte Carlo Option Pricing (continued)

• Non-dividend-paying stock prices in a risk-neutral

economy can be generated by setting μ = r and Δt = T .

1: C := 0; {Accumulated terminal option value.}

2: for i = 1, 2, 3, . . . , N do

3: P := S × e^{(r−σ2/2) T +σ√T ξ}, ξ ∼ N(0, 1);

4: C := C + max(P − X, 0);

5: end for

6: return Ce^−rT/N ;

Monte Carlo Option Pricing (concluded)

• Pricing Asian options is also easy.

1: C := 0;

2: for i = 1, 2, 3, . . . , N do

3: P := S; M := S;

4: for j = 1, 2, 3, . . . , n do

5: P := P × e^(r−σ2/2)(T /n)+σ√

T /n ξ;

6: M := M + P ;

7: end for

8: C := C + max(M/(n + 1) − X, 0);

9: end for

10: return Ce^−rT/N ;

How about American Options?

• Standard Monte Carlo simulation is inappropriate for American options because of early exercise (why?).

– Given a sample path S₀, S₁, . . . , S_n, how to decide which S_i is an early-exercise point?

– What is the option price at each S_i if the option is not exercised?

• It is difficult to determine the early-exercise point based on one single path.

• But Monte Carlo simulation can be modified to price American options with small biases (pp. 847ff).^a

aLongstaff & Schwartz (2001).