Correlated Trinomial Model

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 (or index options).^a

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

aExcept that membership and weights do not change for basket op- tions (Bennett, 2014).

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.

aBoyle, Evnine, & Gibbs (1989).

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 often 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

,(116)

1 ≤ λ. (117)

• 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 (continued)

• 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 an exception.

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

Wang (F95922018), & Lyuu (2013).

Correlated Trinomial Model Simplified (concluded)

• Suppose we allow each asset’s volatility to be a function of time.^a

• There are k assets.

• Can you build an optimal multivariate tree that can handle two barriers on each asset in time O(n^k+1)?^b

aRecall p. 315.

bSee Y. Zhang (R05922052) (2019) for a complete solution.

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



. (118)

• 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 . (119)

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

• The Richardson extrapolation uses n2 = 2n₁.

aIt is identical to the forward rate formula (22) on p. 150!

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,^a consistent with Eq. (118) on p. 830.

• The plots on p. 306 (redrawn on next page) show that convergence to the true option value oscillates with n.

• Extrapolation is inapplicable at this stage.

aL. Chang & Palmer (2007); F. Diener & M. Diener (2004).

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. 833.

• The sequence with odd n turns out to be monotonic and smooth (see the left plot on p. 835).^a

• Apply extrapolation (119) on p. 831 with n2 = n₁ + 2, where n₁ is odd.

• Result is shown in the right plot on p. 835.

• The convergence rate is amazing.

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

aThis can be proved (L. Chang & Palmer, 2007; F. Diener & M.

Diener, 2004).

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. 839).

• 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 , y ).

Explicit Methods

• Consider the diffusion equation^a 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=tj

=

θ(x, t_j+1 ) − θ(x, t_j )

Δt + · · · , (120)

∂²θ(x, t)

∂x²



x=xi

=

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

(Δx)² + · · · .(121)

aIt is a parabolic partial differential equation.

Explicit Methods (continued)

• Next, assemble Eqs. (120) and (121) 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. (121), we might as well use x_i for x in Eq. (120).

• Two choices are possible for t in Eq. (121).

• 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 .

(122)

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. (122) on p. 843 as

θi,j+1 = DΔt

(Δx)2 θi+1,j +



1 − 2DΔt (Δx)2



θi,j + DΔt

(Δx)2 θi−1,j . (123)

• 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 Eq. (123) on p. 844:

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. (121) on p. 842 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 .

(124)

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

• This method is now 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. 845.

Implicit Methods (continued)

• Equation (124) 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 (122) on p. 843 and implicit method (124) on p. 850:

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 (94) on p.

685

• 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).

• Output Y = (1/N )^Δ _N

i=1 Y_i, where 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 time, 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.

• 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 for n 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), (125) by Eq. (87) on p. 619.

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.

– 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.^a

aLongstaff & Schwartz (2001). See pp. 931ff.

Obtaining Profit and Loss of Delta Hedge

• Profit and loss of delta hedge should be calculated under the real-world probability measure.^b

• So stock prices should be sampled from dS

S = μ dt + σ dW.

• Suppose backward induction on a tree under the risk-neutral measure is performed for the delta.^c

aContributed by Mr. Lu, Zheng-Liang (D00922011) on August 12, 2021.

bRecall p. 711.

cBecause, say, no closed-form formulas are available for the delta.

Obtaining Profit and Loss of Delta Hedge (concluded)

• Note that one needs a delta per stock price.

• So Nn trees are needed for the distribution of the profit and loss from N paths with n + 1 stock prices per path.

• These are a lot of trees!

• How to do it efficiently?^a

aHint: Eq. (43) on p. 299.

Delta and Common Random Numbers

• In estimating delta, it is natural to start with the finite-difference estimate

e^−rτ E[ P (S + ) ] − E[ P (S − ) ]

2 .

– P (x) is the terminal payoff of the derivative security when the underlying asset’s initial price equals x.

• Use simulation to estimate E[ P (S + ) ] first.

• Use another simulation to estimate E[ P (S − ) ].

• Finally, apply the formula to approximate the delta.

• This is also called the bump-and-revalue method.

Delta and Common Random Numbers (concluded)

• This method is not recommended because of its high variance.

• A much better approach is to use common random numbers to lower the variance:

e^−rτ E

P (S + ) − P (S − ) 2

 .

• Here, the same random numbers are used for P (S + ) and P (S − ).

• This holds for gamma and cross gamma.^a

aFor multivariate derivatives.

Problems with the Bump-and-Revalue Method

• Consider the binary option with payoff

⎧⎨

⎩

1, if S(T ) > X, 0, otherwise.

• Then

P (S+)−P (S−) =

⎧⎨

⎩

1, if S + > X and S − < X, 0, otherwise.

• So the finite-difference estimate per run for the (undiscounted) delta is 0 or O(1/).

• This means high variance.

Problems with the Bump-and-Revalue Method (concluded)

• The price of the binary option equals e^−rτN (x − σ√

τ ).

– It equals minus the derivative of the European call with respect to X.

– It also equals Xτ times the rho of a European call (p.

362).

• Its delta is

N^ (x − σ√ τ ) Sσ√

τ .

Gamma

• The finite-difference formula for gamma is e^−rτ E

P (S + ) − 2 × P (S) + P (S − ) ²

 .

• For a correlation option with multiple underlying assets, the finite-difference formula for the cross gamma

∂²P (S₁, S₂, . . . )/(∂S₁∂S₂) is:

e^−rτ E

 P (S1 + 1, S2 + 2) − P (S1 − 1, S2 + 2) 412

−P (S1 + 1, S2 − 2) +P (S1 − 1, S2 − 2)  .

Gamma (continued)

• Choosing an of the right magnitude can be challenging.

– If is too large, inaccurate Greeks result.

– If is too small, unstable Greeks result.

• This phenomenon is sometimes called the curse of differentiation.^a

aA¨ıt-Sahalia & Lo (1998); Bondarenko (2003).

Gamma (continued)

• In general, suppose (in some sense)

∂ⁱ

∂θⁱe^−rτE[ P (S) ] = e^−rτE

∂ⁱP (S)

∂θⁱ

holds for all i > 0, where θ is a parameter of interest.^a – A common requirement is Lipschitz continuity.^b

• Then Greeks become integrals.

• As a result, we avoid , finite differences, and resimulation.

aThe ∂ⁱP (S)/∂θⁱ within E[ · ] may not be partial differentiation in the classic sense.

bBroadie & Glasserman (1996).

Gamma (continued)

• This is indeed possible for a broad class of payoff functions.^a

– Roughly speaking, any payoff function that is equal to a sum of products of differentiable functions and indicator functions with the right kind of support.

– For example, the payoff of a call is

max(S(T ) − X, 0) = (S(T ) − X)I{ S(T )−X≥0 }. – The results are too technical to cover here (see next

page).

aTeng (R91723054) (2004); Lyuu & Teng (R91723054) (2011).

Gamma (continued)

• Suppose h(θ, x) ∈ H with pdf f(x) for x and g_j(θ, x) ∈ G for j ∈ B, a finite set of natural numbers.

• Then

∂

∂θ



h(θ, x) 

j∈B1{gj (θ,x)>0}(x) f (x) dx

=



hθ (θ, x) 

j∈B1{gj (θ,x)>0}(x) f (x) dx

+ 

l∈ B

⎡

⎢⎣h(θ, x)Jl(θ, x) 

j∈B\l1{gj (θ, x)>0}(x) f (x)

⎤

⎥⎦

x=χl (θ) ,

where

Jl(θ, x) = sign

∂gl(θ, x)

∂xk

 ∂gl(θ, x)/∂θ

∂gl(θ, x)/∂x for l ∈ B.

Gamma (concluded)

• Similar results have been derived for Levy processes.^a

• Formulas are also recently obtained for credit derivatives.^b

• In queueing networks, this is called infinitesimal perturbation analysis (IPA).^c

aLyuu, Teng (R91723054), & S. Wang (2013).

bLyuu, Teng (R91723054), Tseng, & S. Wang (2014, 2019).

cCao (1985); Y. C. Ho & Cao (1985).

Biases in Pricing Continuously Monitored Options with Monte Carlo

• We are asked to price a continuously monitored up-and-out call with barrier H.

• The Monte Carlo method samples the stock price at n discrete time points t₁, t₂, . . . , t_n.

• A sample path

S(t₀), S(t₁), . . . , S(t_n) is produced.

– Here, t₀ = 0 is the current time, and t_n = T is the expiration time of the option.

Biases in Pricing Continuously Monitored Options with Monte Carlo (continued)

• If all of the sampled prices are below the barrier, this sample path pays max(S(t_n) − X, 0).

• Repeat these steps and average the payoffs for a Monte Carlo estimate.

1: C := 0;

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

3: P := S; hit := 0;

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

5: P := P × e^(r−σ2/2) (T /n)+σ√

(T /n) ξ; {By Eq. (125) on p.

863.}

6: if P ≥ H then

7: hit := 1;

8: break;

9: end if

10: end for

11: if hit = 0 then

12: C := C + max(P − X, 0);

13: end if

14: end for

15: return Ce^−rT/N ;

Biases in Pricing Continuously Monitored Options with Monte Carlo (continued)

• This estimate is biased.^a

– Suppose none of the sampled prices on a sample path equals or exceeds the barrier H.

– It remains possible for the continuous sample path that passes through them to hit the barrier between sampled time points (see plot on next page).

– Hence the knock-out probability is underestimated.

aShevchenko (2003).

H

Biases in Pricing Continuously Monitored Options with Monte Carlo (concluded)

• The bias can be lowered by increasing the number of observations along the sample path.

– For trees, the knock-out probability may decrease as the number of time steps is increased.

• However, even daily sampling may not suffice.

• The computational cost also rises as a result.

Brownian Bridge Approach to Pricing Barrier Options

• We desire an unbiased estimate which can be calculated efficiently.

• The above-mentioned payoff should be multiplied by the probability p that a continuous sample path does not hit the barrier conditional on the sampled prices.

– Formally,

p = Prob[ S(t) < H, 0^Δ ≤ t ≤ T | S(t0), S(t₁), . . . , S(t_n) ].

• This methodology is called the Brownian bridge approach.

Brownian Bridge Approach to Pricing Barrier Options (continued)

• As a barrier is not hit over a time interval if and only if the maximum stock price over that period is at most H,

p = Prob

0≤t≤Tmax S(t) < H | S(t0), S(t₁), . . . , S(t_n) 

.

• Luckily, the conditional distribution of the maximum over a time interval given the beginning and ending stock prices is known.

Brownian Bridge Approach to Pricing Barrier Options (continued)

Lemma 22 Assume S follows dS/S = μ dt + σ dW and define^a ζ(x) = exp^Δ



− 2 ln(x/S(t)) ln(x/S(t + Δt)) σ²Δt

 . (1) If H > max(S(t), S(t + Δt)), then

Prob



t≤u≤t+Δtmax S(u) < H 

 S(t),S(t + Δt)

= 1 − ζ(H).

(2) If h < min(S(t), S(t + Δt)), then Prob



t≤u≤t+Δtmin S(u) > h

 S(t),S(t + Δt)

= 1 − ζ(h).

aHere, Δt is an arbitrary positive real number.

Brownian Bridge Approach to Pricing Barrier Options (continued)

• Lemma 22 gives the probability that the barrier is not hit in a time interval, given the starting and ending stock prices.

• For our up-and-out^a call, choose n = 1.

• As a result,

p =

⎧⎨

⎩

1 − exp

−2 ln(H/S(0)) ln(H/S(T )) σ²T

, if H > max(S(0), S(T )),

0, otherwise.

aSo S(0) < H by definition.

Brownian Bridge Approach to Pricing Barrier Options (continued)

The following algorithm works for up-and-out and down-and-out calls.

1: C := 0;

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

3: P := S × e^{(r−q−σ2/2) T +σ}

√T ξ( );

4: if (S < H and P < H) or (S > H and P > H) then

5: C := C+max(P −X, 0)×



1 − exp

−2 ln(H/S)×ln(H/P ) σ²T

;

6: end if

7: end for

8: return Ce^−rT/N ;

Brownian Bridge Approach to Pricing Barrier Options (concluded)

• The idea can be generalized.

• For example, we can handle more complex barrier options.

• Consider an up-and-out call with barrier H_i for the time interval (t_i, t_i+1 ], 0 ≤ i < m.

• This option contains m barriers.

• Multiply the probabilities for the m time intervals to obtain the desired probability adjustment term.

Pricing Barrier Options without Brownian Bridge

• Let T_h denote the amount of time for a process X_t to hit h for the first time.

• It is called the first passage time or the first hitting time.

• Suppose X_t is a (μ, σ) Brownian motion:

dX_t = μ dt + σ dW_t, t ≥ 0.