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# Correlated Trinomial Model

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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

 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).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).

(2)

### 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)

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

aLyuu & Teng (R91723054) (2011).

(3)

### 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 =Δ erΔt,

Vi =Δ Mi2(eσi2Δt − 1).

– SiMi is the mean of Si at time Δt.

– Si2Vi the variance of Si at time Δt.

aBoyle, Evnine, & Gibbs (1989).

(4)

### Correlated Trinomial Model (continued)

• The value of S1S2 at time Δt has a joint lognormal distribution with mean S1S2M1M2eρσ1σ2Δt, 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.

(5)

### 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 uidi = 1.

(6)

### 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.

(7)

### Correlated Trinomial Model (concluded)

• Let R = MΔ 1M2eρσ1σ2Δt.

• 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.

(8)

### Correlated Trinomial Model Simpliﬁed

a

• Let μi = rΔ − σi2/2 and ui = eΔ λσi

Δt for i = 1, 2.

• The following simpler scheme is often good enough:

p1 = 1

4

 1 λ2

+

Δt λ

μ1 σ1

+ μ2 σ2

 + ρ

λ2

 ,

p2 = 1

4

 1 λ2

+

Δt λ

μ1

σ1 μ2 σ2



ρ λ2

 ,

p3 = 1

4

 1 λ2

+

Δt λ



μ1

σ1 μ2 σ2

 + ρ

λ2

 ,

p4 = 1

4

 1 λ2

+

Δt λ



μ1 σ1

+ μ2 σ2



ρ λ2

 ,

p5 = 1 − 1 λ2

.

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### 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).

(10)

### 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).

(11)

### 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(nk+1)?b

aRecall p. 315.

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

(12)

### 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.

(13)

### 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!

(14)

### 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).

(15)

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

(16)

### 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).

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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

(18)

## Numerical Methods

(19)

All science is dominated by the idea of approximation.

— Bertrand Russell

(20)

### 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.

(21)

0 0.05 0.1 0.15 0.2 0.25 80

85 90 95 100 105 110 115

(22)

### 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(xi, yj) = θ(xi+1, yj) − 2θ(xi, yj) + θ(xi−1, yj) (Δx)2

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

(Δy)2 .

(23)

### Example: Poisson’s Equation (concluded)

• In the above, Δx = xΔ i − xi−1 and Δy = yΔ j − yj−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(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(x , y ).

(24)

### Explicit Methods

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

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

Δx and Δt, where Δx = xΔ i+1 − xi and Δt = tΔ j+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 )

Δt + · · · , (120)

2θ(x, t)

∂x2



x=xi

=

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

x)2 + · · · .(121)

aIt is a parabolic partial diﬀerential equation.

(25)

### Explicit Methods (continued)

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

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

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

θi,j+1 − θi,j

Δt = D θi+1,j − 2θi,j + θi−1,j

(Δx)2 .

(122)

(26)

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

(27)

tj tj  xi

xi  xi

tj tj  xi

xi  xi

= >

(28)

### 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.

(29)

### 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.

(30)

### 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 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!

(31)

### 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).

(32)

### Implicit Methods

• Suppose we use t = tj+1 in Eq. (121) on p. 842 instead.

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

Δt = D θi+1,j+1 − 2θi,j+1 + θi−1,j+1

(Δx)2 .

(124)

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

• This method is now implicit:

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

– See exhibit (b) on p. 845.

(33)

### Implicit Methods (continued)

• Equation (124) can be rearranged as

θi−1,j+1 − (2 + γ) θi,j+1 + θi+1,j+1 = −γθi,j, where γ = (Δx)Δ 2/(DΔt).

• This equation is unconditionally stable.

• Suppose the boundary conditions are given at x = x0

and x = xN +1.

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

(34)

### 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.

(35)

### 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.

(36)

### 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.

(37)

j

j+1

i

i+1

i+1

(38)

### 685

• See text.

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

(39)

### Monte Carlo Simulation

a

• 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).

(40)

### The Big Idea

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

• θ = E[ g(XΔ 1, X2, . . . , Xn) ] 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, . . . , Xn).

• Output Y = (1/N )Δ N

i=1 Yi, where Yi = gΔ



x(i)1 , x(i)2 , . . . , x(i)n

 .

(41)

### The Big Idea (concluded)

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

• Each Yi has the same distribution as Y = g(XΔ 1, X2, . . . , Xn).

• 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.

(42)

### 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.

(43)

### 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.

(44)

### 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.

(45)

### 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

Si+1

= Sie(μ−σ2/2) Δt+σ

Δt ξ, ξ ∼ N(0, 1), (125) by Eq. (87) on p. 619.

(46)

### Monte Carlo Option Pricing (continued)

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

Si+1 = Si + Siμ Δt + Siσ√

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.

(47)

### 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 +σT ξ, ξ ∼ N(0, 1);

4: C := C + max(P − X, 0);

5: end for

6: return Ce−rT /N ;

(48)

### 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 ;

(49)

• Standard Monte Carlo simulation is inappropriate for American options because of early exercise.

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

– What is the option price at each Si 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.

(50)

### Obtaining Proﬁt and Loss of Delta Hedge

a

• 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.

(51)

### 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.

(52)

### 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.

(53)

### 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.

(54)

### 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.

(55)

### 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 .

(56)

### 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) 412

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

(57)

### 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).

(58)

### 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.

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### 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).

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### Gamma (continued)

• Suppose h(θ, x) ∈ H with pdf f(x) for x and gj(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.

(61)

### 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).

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### 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, . . . , tn.

• A sample path

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

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

(63)

### 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(tn) − X, 0).

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

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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 ;

(65)

### 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).

(66)

H

(67)

### 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.

(68)

### 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, 0Δ ≤ t ≤ T | S(t0), S(t1), . . . , S(tn) ].

• This methodology is called the Brownian bridge approach.

(69)

### 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(tn)

.

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

(70)

### Brownian Bridge Approach to Pricing Barrier Options (continued)

Lemma 22 Assume S follows dS/S = μ dt + σ dW and definea ζ(x) = expΔ



− 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.

(71)

### 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-outa 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.

(72)

### 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 ;

(73)

### 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 Hi for the time interval (ti, ti+1 ], 0 ≤ i < m.

• This option contains m barriers.

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

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### Pricing Barrier Options without Brownian Bridge

• Let Th denote the amount of time for a process Xt to hit h for the first time.

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

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

dXt = μ dt + σ dWt, t ≥ 0.

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