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(1)

Pricing Discrete Barrier Options

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

• They are more common than the continuously monitored versions.

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

• Here is why.

• Suppose each period has a duration of ∆t and the

` > 1 monitored times are t0 = 0, t1, t2, . . . , t` = T .

(2)

Pricing Discrete Barrier Options (continued)

• It is extremely unlikely that all monitored times

coincide with the end of a period on the tree, meaning

∆t divides ti for all i.

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

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

• See p. 615.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 614

(3)

* j* j* j* j

* j* j* j µ

: j

R z

*

R z

*

R z

*

R z

*

R z

*

t0

-

¾∆t01 ¾∆t-1 ¾∆t-1 ¾∆t02-

t1

∆t1 (

∆t2 )

(4)

Pricing Discrete Barrier Options (concluded)

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

• If the ith binomial-trinomial tree has ni periods, the size of the whole tree is

O

à ` X

i=1

ni

!2

 .

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 616

(5)

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 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).

(6)

Options on a Stock That Pays Known Dividends (continued)

• The most realistic model assumes the stock price decreases by the amount of the dividend paid at the ex-dividend date.

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

• But this model results in binomial trees that grow exponentially.

• The binomial-trinomial tree can often avoid the

exponential explosion for the known-dividends case.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 618

(7)

Options on a Stock That Pays Known Dividends (continued)

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

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

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

Se−(t/∆t)σ∆t ≥ D.

– Equivalently,

∆t ≥

· ln(S/D)

¸2 .

(8)

Options on a Stock That Pays Known Dividends (continued)

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

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

• Assume the top node’s price equals S0.

– As usual, its two successor nodes will have prices S0u and S0u−1.

• The remaining nodes’ successor nodes will have prices S0u−3, S0u−5, S0u−7, . . . ,

same as the binomial tree.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 620

(9)

Options on a Stock That Pays Known Dividends (concluded)

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

• Note that the binomial-trinomial structure remains valid in the special case when ∆t0 = ∆t on p. 600.

• Hence the construction can be completed.

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

ex-dividend date when the procedure can be repeated.

• The resulting tree is called the stair tree.a

aDai (R86526008, D8852600) and Lyuu (2004).

(10)

Multivariate Contingent Claims

• They depend on two or more underlying assets.

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

à m X

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).

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 622

(11)

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).

(12)

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.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 624

(13)

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

(14)

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.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 626

(15)

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).

(16)

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).

bSee Chang, Hsu, and Lyuu (2006) for a solution.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 628

(17)

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

. (70)

• Assume c is an unknown constant independent of n.

– Convergence is basically monotonic and smooth.

(18)

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 . (71)

• This estimate should converge faster than 1/n.

• The Richardson extrapolation uses n2 = 2n1.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 630

(19)

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. (70) on p. 629.

• But the plots on p. 253 (redrawn on next page)

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

• Extrapolation is inapplicable at this stage.

(20)

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

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 632

(21)

Improving BOPM with Extrapolation (concluded)

• Take the at-the-money option in the left plot on p. 632.

• The sequence with odd n turns out to be monotonic and smooth (see the left plot on p. 634).a

• Apply extrapolation (71) on p. 630 with n2 = n1 + 2, where n1 is odd.

• Result is shown in the right plot on p. 634.

• The convergence rate is amazing.

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

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

(22)

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

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 634

(23)

Numerical Methods

(24)

All science is dominated by the idea of approximation.

— Bertrand Russell

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 636

(25)

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. 638).

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

(26)

0 0.05 0.1 0.15 0.2 0.25 80

85 90 95 100 105 110 115

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 638

(27)

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 .

(28)

Example: Poisson’s Equation (concluded)

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

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

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

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

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

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 640

(29)

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 + · · · , (72)

2θ(x, t)

∂x2

¯¯

¯¯

x=xi

= θ(xi+1, t) − 2θ(xi, t) + θ(xi−1, t)

(∆x)2 + · · · . (73)

(30)

Explicit Methods (continued)

• Next, assemble Eqs. (72) and (73) 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. (73), we might as well use xi for x in Eq. (72).

• Two choices are possible for t in Eq. (73).

• 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 . (74)

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 642

(31)

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. (74) on p. 642 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 exhibit (a) on next page).

(32)

Stencils

tj tj 1 xi 1

xi 1 xi

tj tj 1 xi 1

xi 1 xi

(a) (b)

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 644

(33)

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.

(34)

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.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 646

(35)

Explicit Method and Trinomial Tree

• Rearrange Eq. (74) on p. 642 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 freedom in the construction of trinomial trees.

(36)

Implicit Methods

• Suppose we use t = tj+1 in Eq. (73) on p. 641 instead.

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

∆t = D θi+1,j+1 − 2θi,j+1 + θi−1,j+1

(∆x)2 .

(75)

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

• This method is 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. 644.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 648

(37)

Implicit Methods (continued)

• Equation (75) can be rearranged as

θi−1,j+1 − (2 + γ) θi,j+1 + θi+1,j+1 = −γθi,j, where γ ≡ (∆x)2/(D∆t).

• 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,

(38)

Implicit Methods (continued)

a 1 0 · · · · · · 0

1 a 1 0 · · · 0

0 1 a 1 · · · 0

.. .

.. .

.. .

.. .

.. .

.. . ..

.

.. .

.. .

.. .

.. .

.. . 0 · · · · · · 1 a 1 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 ≡ −2 − γ.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 650

(39)

Implicit Methods (concluded)

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

• The matrix above is nonsingular when γ ≥ 0.

– A square matrix is nonsingular if its inverse exists.

(40)

Crank-Nicolson Method

• Take the average of explicit method (74) on p. 642 and implicit method (75) on p. 648:

θi,j+1 − θi,j

∆t

= 1

2 Ã

D θi+1,j − 2θi,j + θi−1,j

(∆x)2 + D θi+1,j+1 − 2θi,j+1 + θi−1,j+1 (∆x)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.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 652

(41)

Stencil

t

j

t

j+1

x

i

x

i+1

x

i+1

(42)

Numerically Solving the Black-Scholes PDE

• See text.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 654

(43)

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 according to Dongarra and Sullivan (2000).

(44)

The Big Idea

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

• θ ≡ E[ g(X1, X2, . . . , Xn) ] for some function g is desired.

• We generate

³

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

´

, 1 ≤ i ≤ N

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

• Set

Yi ≡ g

³

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

´ .

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 656

(45)

The Big Idea (concluded)

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

• Each Yi has the same distribution as Y ≡ g(X1, 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.

(46)

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 derivative only requires discrete sampling along the time dimension.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 658

(47)

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 by the Berry-Esseen theorem.

• 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.

(48)

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, for example, is more efficient than alternative procedures for securities depending on more than one asset, the multivariate derivatives.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 660

(49)

Monte Carlo Option Pricing

• For the pricing of European options on a

dividend-paying stock, we may proceed as follows.

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

Si+1 = Sie(µ−σ2/2) ∆t+σ∆t ξ, ξ ∼ N (0, 1)

(76) when dS/S = µ dt + σ dW .

(50)

Monte Carlo Option Pricing (continued)

• If we discretize dS/S = µ dt + σ dW , we will obtain Si+1 = Si + (µ − σ2/2) ∆t + σ√

∆t 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 ∆t is sufficiently small.

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

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 662

(51)

Monte Carlo Option Pricing (concluded)

• Non-dividend-paying stock prices in a risk-neutral economy can be generated by setting µ = r.

1: C := 0;

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

3: P := S × e(r−σ2/2) T +σT ξ;

4: C := C + max(P − X, 0);

5: end for

6: return Ce−rT/m;

• Pricing Asian options is easy (see text).

(52)

How about American Options?

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

• 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 (p. 709ff).a

aLongstaff and Schwartz (2001).

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 664

(53)

Delta and Common Random Numbers

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

e−rτ E[ P (S + ²) ] − 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 + ²) ] first.

• Use another simulation to estimate E[ P (S − ²) ].

• Finally, apply the formula to approximate the delta.

(54)

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 + ²) − P (S − ²)

¸ .

• Here, the same random numbers are used for P (S + ²) and P (S − ²).

• This holds for gamma and cross gammas (for multivariate derivatives).

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 666

(55)

Gamma

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

· P (S + ²) − 2 × P (S) + P (S − ²)

²2

¸ .

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

2P (S1, S2, . . . )/(∂S1∂S2) is:

e−rτ E

· P (S1 + ²1, S2 + ²2) − P (S1 − ²1, S2 + ²2)

1²2

−P (S1 + ²1, S2 − ²2) + P (S1 − ²1, S2 − ²2) ¸ .

(56)

Gamma (concluded)

• Choosing an ² of the right magnitude can be challenging.

– If ² is too large, inaccurate Greeks result.

– If ² is too small, unstable Greeks result.

• This phenomenon is sometimes called the curse of differentiation.

• Need formulas for Greeks which are integrals (thus avoiding ², finite differences, and resimulation).a

aLyuu and Teng (R91723054) (2008).

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 668

(57)

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.

(58)

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).

• Repeating these steps and averaging the payoffs yield a Monte Carlo estimate.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 670

(59)

1: C := 0;

2: for i = 1, 2, 3, . . . , m 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;

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/m;

(60)

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

• This estimate is biased.

– 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).

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 672

(61)

H

(62)

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

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

• However, even daily sampling may not suffice.

• The computational cost also rises as a result.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 674

(63)

Brownian Bridge Approach to Pricing Barrier Options

• We desire an unbiased estimate efficiently.

• So 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.

• This methodology is called the Brownian bridge approach.

• Formally, we have

p ≡ Prob[ S(t) < H, 0 ≤ t ≤ T | S(t0), S(t1), . . . , S(tn) ].

(64)

Brownian Bridge Approach to Pricing Barrier Options (continued)

• As a barrier is hit over a time interval if and only if the maximum stock price over that period is at least 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.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 676

(65)

Brownian Bridge Approach to Pricing Barrier Options (continued)

Lemma 19 Assume S follows dS/S = µ dt + σ dW and define ζ(x) ≡ exp

·

2 ln(x/S(t)) ln(x/S(t + ∆t)) σ2∆t

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

Prob

·

t≤u≤t+∆tmax S(u) < H

¯¯

¯¯ S(t), S(t + ∆t)

¸

= 1 − ζ(H).

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

·

t≤u≤t+∆tmin S(u) > h

¯¯

¯¯ S(t), S(t + ∆t)

¸

= 1 − ζ(h).

(66)

Brownian Bridge Approach to Pricing Barrier Options (continued)

• Lemma 19 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 call, choose n = 1.

• As a result,

p =

1 − exp h

2 ln(H/S(0)) ln(H/S(T )) σ2T

i

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

0, otherwise.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 678

(67)

Brownian Bridge Approach to Pricing Barrier Options (continued)

1: C := 0;

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

3: P := S × e(r−q−σ2/2) T +σT ξ( );

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

5: C := C+max(P −X, 0)×

n

1 − exp h

2 ln(H/S)×ln(H/P ) σ2T

io

;

6: end if

7: end for

8: return Ce−rT/m;

(68)

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 < n.

• This option thus contains n barriers.

• It is a simple matter of multiplying the probabilities for the n time intervals properly to obtain the desired

probability adjustment term.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 680

(69)

Variance Reduction

• The statistical efficiency of Monte Carlo simulation can be measured by the variance of its output.

• If this variance can be lowered without changing the expected value, fewer replications are needed.

• Methods that improve efficiency in this manner are called variance-reduction techniques.

• Such techniques become practical when the added costs are outweighed by the reduction in sampling.

(70)

Variance Reduction: Antithetic Variates

• We are interested in estimating E[ g(X1, X2, . . . , Xn) ], where X1, X2, . . . , Xn are independent.

• Let Y1 and Y2 be random variables with the same distribution as g(X1, X2, . . . , Xn).

• Then

Var

· Y1 + Y2 2

¸

= Var[ Y1 ]

2 + Cov[ Y1, Y2 ]

2 .

– Var[ Y1 ]/2 is the variance of the Monte Carlo method with two (independent) replications.

• The variance Var[ (Y1 + Y2)/2 ] is smaller than

Var[ Y1 ]/2 when Y1 and Y2 are negatively correlated.

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 682

(71)

Variance Reduction: Antithetic Variates (continued)

• For each simulated sample path X, a second one is obtained by reusing the random numbers on which the first path is based.

• This yields a second sample path Y .

• Two estimates are then obtained: One based on X and the other on Y .

• If N independent sample paths are generated, the antithetic-variates estimator averages over 2N

estimates.

(72)

Variance Reduction: Antithetic Variates (continued)

• Consider process dX = at dt + bt

dt ξ.

• Let g be a function of n samples X1, X2, . . . , Xn on the sample path.

• We are interested in E[ g(X1, X2, . . . , Xn) ].

• Suppose one simulation run has realizations

ξ1, ξ2, . . . , ξn for the normally distributed fluctuation term ξ.

• This generates samples x1, x2, . . . , xn.

• The estimate is then g(x), where x ≡ (x1, x2 . . . , xn).

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 684

(73)

Variance Reduction: Antithetic Variates (concluded)

• The antithetic-variates method does not sample n more numbers from ξ for the second estimate g(x0).

• Instead, generate the sample path x0 ≡ (x01, x02 . . . , x0n) from −ξ1, −ξ2, . . . , −ξn.

• Compute g(x0).

• Output (g(x) + g(x0))/2.

• Repeat the above steps for as many times as required by accuracy.

(74)

Variance Reduction: Conditioning

• We are interested in estimating E[ X ].

• Suppose here is a random variable Z such that

E[ X | Z = z ] can be efficiently and precisely computed.

• E[ X ] = E[ E[ X | Z ] ] by the law of iterated conditional expectations.

• Hence the random variable E[ X | Z ] is also an unbiased estimator of E[ X ].

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 686

(75)

Variance Reduction: Conditioning (concluded)

• As

Var[ E[ X | Z ] ] ≤ Var[ X ],

E[ X | Z ] has a smaller variance than observing X directly.

• First obtain a random observation z on Z.

• Then calculate E[ X | Z = z ] as our estimate.

– There is no need to resort to simulation in computing E[ X | Z = z ].

• The procedure can be repeated a few times to reduce the variance.

(76)

Control Variates

• Use the analytic solution of a similar yet simpler problem to improve the solution.

• Suppose we want to estimate E[ X ] and there exists a random variable Y with a known mean µ ≡ E[ Y ].

• Then W ≡ X + β(Y − µ) can serve as a “controlled”

estimator of E[ X ] for any constant β.

– However β is chosen, W remains an unbiased estimator of E[ X ] as

E[ W ] = E[ X ] + βE[ Y − µ ] = E[ X ].

°2010 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 688

(77)

Control Variates (continued)

• Note that

Var[ W ] = Var[ X ] + β2 Var[ Y ] + 2β Cov[ X, Y ],

(77)

• Hence W is less variable than X if and only if

β2 Var[ Y ] + 2β Cov[ X, Y ] < 0. (78)

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