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# Brownian Bridge Approach to Pricing Barrier Options

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(1)

### 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.

(2)

### 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.

(3)

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. (120) on p.

853.}

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 ;

(4)

### 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 knock-out probabilities are underestimated.

aShevchenko (2003).

(5)

H

(6)

### 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 probabilities 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.

(7)

### 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.

• This methodology is called the Brownian bridge approach.

• Formally, we have

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

(8)

### 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.

(9)

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

Lemma 21 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.

(10)

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

• Lemma 21 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.

(11)

### 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 ;

(12)

### 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 contains n barriers.

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

(13)

### 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.

(14)

### Pricing Barrier Options without Brownian Bridge (continued)

• The ﬁrst passage time Th follows the inverse Gaussian (IG) distribution with probability density function:a

| h − X(0) | σt3/2

e−(h−X(0)−μx)2/(2σ2x).

• For pricing a barrier option with barrier H by simulation, the density function becomes

| ln(H/S(0)) | σt3/2

e[ln(H/S(0))−(r−σ2/2) x]2/(2σ2x).

aA. N. Borodin & Salminen (1996), with Laplace transform

(15)

### Pricing Barrier Options without Brownian Bridge (concluded)

• Draw an x from this distribution.a

• If x > T , a knock-in option fails to knock in, whereas a knock-out option does not knock out.

• If x ≤ T , the opposite is true.

• If the barrier option survives at maturity T , then draw an S(T ) to calculate its payoﬀ.

• Repeat the above process many times to average the discounted payoﬀ.

aThe IG distribution can be very eﬃciently sampled (Michael, Schu- cany, & Haas, 1976).

(16)

### Brownian Bridge Approach to Pricing Lookback Options

a

• By Lemma 21(1) (p. 876), Fmax(y) =Δ Prob



0≤t≤Tmax S(t) < y | S(0), S(T )



= 1 − exp



−2 ln(y/S(0)) ln(y/S(T )) σ2T

 .

• So Fmax is the conditional distribution function of the maximum stock price.

aEl Babsiri & Noel (1998).

(17)

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

• A random variable with that distribution can be

generated by Fmax−1 (x), where x is uniformly distributed over (0, 1).a

• In other words,

x = 1 − exp



−2 ln(y/S(0)) ln(y/S(T )) σ2T

 .

aThis is called the inverse-transform technique (see p. 259 of the text- book).

(18)

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

• Equivalently,

ln(1 − x)

= −2 ln(y/S(0)) ln(y/S(T )) σ2T

= 2

σ2T { [ ln(y) − ln S(0) ] [ ln(y) − ln S(T ) ] }.

(19)

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

• There are two solutions for ln y.

• But only one is consistent with y ≥ max(S(0), S(T )):

ln y

=

ln(S(0) S(T )) +



ln S(T )S(0)

2

− 2σ2T ln(1 − x)

2 .

(20)

### Brownian Bridge Approach to Pricing Lookback Options (concluded)

The following algorithm works for the lookback put on the maximum.

1: C := 0;

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

3: P := S × e(r−q−σ2/2) T +σ

T ξ( ); {By Eq. (120) on p. 853.}

4: Y := exp

ln(SP )+

(ln PS )2−2σ2T ln[ 1−U(0,1) ] 2



;

5: C := C + (Y − P );

6: end for

7: return Ce−rT/N ;

(21)

### Pricing Lookback Options without Brownian Bridge

• Suppose we do not draw S(T ) in simulation.

• Now, the distribution function of the maximum logarithmic stock price isa

Prob



max

0≤t≤T ln S(t)

S(0) < y



= 1 − N

−y +

r − q − σ22 T σ

T

−e

2y



r−q− σ2 2



σ2 N

−y −

r − q − σ22 T σ

T

⎠ , y ≥ 0.

• The inverse of that is much harder to calculate.

aA. N. Borodin & Salminen (1996).

(22)

### 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.

(23)

### Variance Reduction: Antithetic Variates

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

• 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.

(24)

### 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.

(25)

### 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 = (xΔ 1, x2 . . . , xn).

(26)

### Variance Reduction: Antithetic Variates (concluded)

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

• Instead, generate the sample path x Δ= (x1, x2 . . . , xn) from −ξ1,−ξ2, . . . ,−ξn.

• Compute g(x).

• Output (g(x) + g(x))/2.

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

(27)

### 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 ].

(28)

### 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

(29)

### 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 ].Δ

• Then W = X + β(YΔ − 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 ].

(30)

### Control Variates (continued)

• Note that

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

(121)

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

β2 Var[ Y ] + 2β Cov[ X, Y ] < 0. (122)

(31)

### Control Variates (concluded)

• The success of the scheme clearly depends on both β and the choice of Y .

– American options can be priced by choosing Y to be the otherwise identical European option and μ the Black-Scholes formula.a

– Arithmetic Asian options can be priced by choosing Y to be the otherwise identical geometric Asian option’s price and β = −1.

• This approach is much more eﬀective than the antithetic-variates method.b

aHull & White (1988).

(32)

### Choice of Y

• In general, the choice of Y is ad hoc,a and experiments must be performed to conﬁrm the wisdom of the choice.

• Try to match calls with calls and puts with puts.b

• On many occasions, Y is a discretized version of the derivative that gives μ.

– Discretely monitored geometric Asian option vs. the continuously monitored version.c

• The discrepancy can be large (e.g., lookback options).d

aBut see Dai (B82506025, R86526008, D8852600), C. Chiu (B90201037, R94922072), & Lyuu (2015, 2018).

bContributed by Ms. Teng, Huei-Wen (R91723054) on May 25, 2004.

c

(33)

### Optimal Choice of β

• Equation (121) on p. 897 is minimized when β = −Cov[ X, Y ]/Var[ Y ].

– It is called beta in the book.

• For this speciﬁc β,

Var[ W ] = Var[ X ] Cov[ X, Y ]2

Var[ Y ] = 

1 − ρ2X,Y 

Var[ X ], where ρX,Y is the correlation between X and Y .

(34)

### Optimal Choice of β (continued)

• Note that the variance can never be increased with the optimal choice.

• Furthermore, the stronger X and Y are correlated, the greater the reduction in variance.

• For example, if this correlation is nearly perfect (±1), we could control X almost exactly.

(35)

### Optimal Choice of β (continued)

• Typically, neither Var[ Y ] nor Cov[ X, Y ] is known.

• Therefore, we cannot obtain the maximum reduction in variance.

• We can guess these values and hope that the resulting W does indeed have a smaller variance than X.

• A second possibility is to use the simulated data to estimate these quantities.

– How to do it eﬃciently in terms of time and space?

(36)

### Optimal Choice of β (concluded)

• Observe that −β has the same sign as the correlation between X and Y .

• Hence, if X and Y are positively correlated, β < 0, then X is adjusted downward whenever Y > μ and upward otherwise.

• The opposite is true when X and Y are negatively correlated, in which case β > 0.

• Suppose a suboptimal β +  is used instead.

• The variance increases by only 2Var[ Y ].a

(37)

### A Pitfall

• A potential pitfall is to sample X and Y independently.

• In this case, Cov[ X, Y ] = 0.

• Equation (121) on p. 897 becomes

Var[ W ] = Var[ X ] + β2 Var[ Y ].

• So whatever Y is, the variance is increased!

• Lesson: X and Y must be correlated.

(38)

### Problems with the Monte Carlo Method

• The error bound is only probabilistic.

• The probabilistic error bound of O(1/√

N ) does not beneﬁt from regularity of the integrand function.

• The requirement that the points be independent random samples are wasteful because of clustering.

• In reality, pseudorandom numbers generated by completely deterministic means are used.

• Monte Carlo simulation exhibits a great sensitivity on the seed of the pseudorandom-number generator.

(39)

## Matrix Computation

(40)

To set up a philosophy against physics is rash;

philosophers who have done so have always ended in disaster.

— Bertrand Russell

(41)

### Definitions and Basic Results

• Let A = [ aΔ ij ]1≤i≤m,1≤j≤n, or simply A ∈ Rm×n, denote an m × n matrix.

• It can also be represented as [ a1, a2, . . . , an ] where ai ∈ Rm are vectors.

– Vectors are column vectors unless stated otherwise.

• A is a square matrix when m = n.

• The rank of a matrix is the largest number of linearly independent columns.

(42)

### Definitions and Basic Results (continued)

• A square matrix A is said to be symmetric if AT = A.

• A real n × n matrix

A = [ aΔ ij ]i,j is diagonally dominant if | aii | >

j=i | aij | for 1 ≤ i ≤ n.

– Such matrices are nonsingular.

• The identity matrix is the square matrix I = diag[ 1, 1, . . . , 1 ].Δ

(43)

### Definitions and Basic Results (concluded)

• A matrix has full column rank if its columns are linearly independent.

• A real symmetric matrix A is positive deﬁnite if xTAx =

i,j

aijxixj > 0 for any nonzero vector x.

• A matrix A is positive deﬁnite if and only if there exists a matrix W such that A = WTW and W has full

column rank.

(44)

### Cholesky Decomposition

• Positive deﬁnite matrices can be factored as A = LLT,

called the Cholesky decomposition.

– Above, L is a lower triangular matrix.

(45)

### Generation of Multivariate Distribution

• Let x = [ xΔ 1, x2, . . . , xn ]T be a vector random variable with a positive deﬁnite covariance matrix C.

• As usual, assume E[ x ] = 0.

• This covariance structure can be matched by P y.

– y = [ yΔ 1, y2, . . . , yn ]T is a vector random variable

with a covariance matrix equal to the identity matrix.

– C = P PT is the Cholesky decomposition of C.a

aWhat if C is not positive deﬁnite? See Y. Y. Lai (R93942114) &

Lyuu (2007).

(46)

### Generation of Multivariate Distribution (concluded)

• For example, suppose

C =

⎣ 1 ρ ρ 1

⎦ .

• Then

P =

⎣ 1 0

ρ 

1 − ρ2

as P PT = C.a

aRecall Eq. (28) on p. 179.

(47)

### Generation of Multivariate Normal Distribution

• Suppose we want to generate the multivariate normal distribution with a covariance matrix C = P PT.

– First, generate independent standard normal distributions y1, y2, . . . , yn.

– Then

P [ y1, y2, . . . , yn ]T has the desired distribution.

– These steps can then be repeated.

(48)

### Multivariate Derivatives Pricing

• Generating the multivariate normal distribution is essential for the Monte Carlo pricing of multivariate derivatives (pp. 809ﬀ).

• For example, the rainbow option on k assets has payoﬀ max(max(S1, S2, . . . , Sk) − X, 0)

at maturity.

• The closed-form formula is a multi-dimensional integral.a

aJohnson (1987); C. Y. Chen (D95723006) & Lyuu (2009).

(49)

### Multivariate Derivatives Pricing (concluded)

• Suppose dSj/Sj = r dt + σj dWj, 1 ≤ j ≤ k, where C is the correlation matrix for dW1, dW2, . . . , dWk.

• Let C = P PT.

• Let ξ consist of k independent random variables from N (0, 1).

• Let ξ = P ξ.

• Similar to Eq. (120) on p. 853, for each asset 1 ≤ j ≤ k, Si+1 = Sie(r−σj2/2) Δt+σj

Δt ξj

by Eq. (120) on p. 853.

(50)

### Least-Squares Problems

• The least-squares (LS) problem is concerned with

x∈Rminn  Ax − b , where A ∈ Rm×n, b ∈ Rm, and m ≥ n.

• The LS problem is called regression analysis in statistics and is equivalent to minimizing the mean-square error.

• Often written as

Ax = b.

(51)

### Polynomial Regression

• In polynomial regression, x0 + x1x + · · · + xnxn is used to ﬁt the data { (a1, b1), (a2, b2), . . . , (am, bm)}.

• This leads to the LS problem,

⎢⎢

⎢⎢

⎢⎢

1 a1 a21 · · · an1 1 a2 a22 · · · an2 ... ... ... . .. ... 1 am a2m · · · anm

⎥⎥

⎥⎥

⎥⎥

⎢⎢

⎢⎢

⎢⎢

x0 x1 ... xn

⎥⎥

⎥⎥

⎥⎥

=

⎢⎢

⎢⎢

⎢⎢

b1 b2 ... bm

⎥⎥

⎥⎥

⎥⎥

.

• Consult p. 273 of the textbook for solutions.

(52)

### American Option Pricing by Simulation

• The continuation value of an American option is the conditional expectation of the payoﬀ from keeping the option alive now.

• The option holder must compare the immediate exercise value and the continuation value.

• In standard Monte Carlo simulation, each path is treated independently of other paths.

• But the decision to exercise the option cannot be reached by looking at one path alone.

(53)

### The Least-Squares Monte Carlo Approach

• The continuation value can be estimated from the cross-sectional information in the simulation by using least squares.a

• The result is a function (of the state) for estimating the continuation values.

• Use the function to estimate the continuation value for each path to determine its cash ﬂow.

• This is called the least-squares Monte Carlo (LSM) approach.

aLongstaﬀ & Schwartz (2001).

(54)

### The Least-Squares Monte Carlo Approach (concluded)

• The LSM is provably convergent.a

• The LSM can be easily parallelized.b

– Partition the paths into subproblems and perform LSM on each of them independently.

– The speedup is close to linear (i.e., proportional to the number of cores).

• Surprisingly, accuracy is not aﬀected.

aCl´ement, Lamberton, & Protter (2002); Stentoft (2004).

bK. Huang (B96902079, R00922018) (2013); C. W. Chen (B97902046, R01922005) (2014); C. W. Chen (B97902046, R01922005), K. Huang

(55)

### A Numerical Example

• Consider a 3-year American put on a non-dividend-paying stock.

• The put is exercisable at years 0, 1, 2, and 3.

• The strike price X = 105.

• The annualized riskless rate is r = 5%.

– The annual discount factor hence equals 0.951229.

• The current stock price is 101.

• We use 8 price paths to illustrate the algorithm.

(56)

### A Numerical Example (continued)

Stock price paths

Path Year 0 Year 1 Year 2 Year 3

1 101 97.6424 92.5815 107.5178

2 101 101.2103 105.1763 102.4524 3 101 105.7802 103.6010 124.5115

4 101 96.4411 98.7120 108.3600

5 101 124.2345 101.0564 104.5315

6 101 95.8375 93.7270 99.3788

7 101 108.9554 102.4177 100.9225

(57)

0 0.5 1 1.5 2 2.5 3 95

100 105 110 115 120 125

1

2

3

4 5

6 7

8

(58)

### A Numerical Example (continued)

• We use the basis functions 1, x, x2. – Other basis functions are possible.a

• The plot next page shows the ﬁnal estimated optimal exercise strategy given by LSM.

• We now proceed to tackle our problem.

• The idea is to calculate the cash ﬂow along each path, using information from all paths.

aLaguerre polynomials, Hermite polynomials, Legendre polynomials, Chebyshev polynomials, Gedenbauer polynomials, or Jacobi polynomi-

(59)

0 0.5 1 1.5 2 2.5 3 95

100 105 110 115 120 125

1

2 3

4 5

6 7 8

(60)

### A Numerical Example (continued)

Cash flows at year 3

Path Year 0 Year 1 Year 2 Year 3

1 — — — 0

2 — — — 2.5476

3 — — — 0

4 — — — 0

5 — — — 0.4685

6 — — — 5.6212

7 — — — 4.0775

(61)

### A Numerical Example (continued)

• The cash ﬂows at year 3 are the exercise value if the put is in the money.

• Only 4 paths are in the money: 2, 5, 6, 7.

• Some of the cash ﬂows may not occur if the put is exercised earlier, which we will ﬁnd out later.

• Incidentally, the European counterpart has a value of

0.9512293 × 2.5476 + 0.4685 + 5.6212 + 4.0775 8

= 1.3680.

(62)

### A Numerical Example (continued)

• We move on to year 2.

• For each state that is in the money at year 2, we must decide whether to exercise it.

• There are 6 paths for which the put is in the money: 1, 3, 4, 5, 6, 7 (p. 923).

• Only in-the-money paths will be used in the regression because they are where early exercise is relevant.

– If there were none, move on to year 1.

(63)

### A Numerical Example (continued)

• Let x denote the stock prices at year 2 for those 6 paths.

• Let y denote the corresponding discounted future cash ﬂows (at year 3) if the put is not exercised at year 2.

(64)

### A Numerical Example (continued)

Regression at year 2

Path x y

1 92.5815 0 × 0.951229

2 — —

3 103.6010 0 × 0.951229 4 98.7120 0 × 0.951229 5 101.0564 0.4685 × 0.951229 6 93.7270 5.6212 × 0.951229 7 102.4177 4.0775 × 0.951229

(65)

### A Numerical Example (continued)

• We regress y on 1, x, and x2.

• The result is

f (x) = 22.08 − 0.313114 × x + 0.00106918 × x2.

• f(x) estimates the continuation value conditional on the stock price at year 2.

• We next compare the immediate exercise value and the estimated continuation value.a

aThe f(102.4177) entry on the next page was corrected by Mr. Tu, Yung-Szu (B79503054, R83503086) on May 25, 2017.

(66)

### A Numerical Example (continued)

Optimal early exercise decision at year 2 Path Exercise Continuation 1 12.4185 f (92.5815) = 2.2558

2 — —

3 1.3990 f (103.6010) = 1.1168 4 6.2880 f (98.7120) = 1.5901 5 3.9436 f (101.0564) = 1.3568 6 11.2730 f (93.7270) = 2.1253 7 2.5823 f (102.4177) = 1.2266

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### A Numerical Example (continued)

• Amazingly, the put should be exercised in all 6 paths: 1, 3, 4, 5, 6, 7.

• Now, any positive cash ﬂow at year 3 vanishes for these paths as the put is exercised before year 3 (p. 923).

– They are paths 5, 6, 7.

• The cash ﬂows on p. 927 become the ones on next slide.

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### A Numerical Example (continued)

Cash flows at years 2 & 3

Path Year 0 Year 1 Year 2 Year 3

1 — — 12.4185 0

2 — — 0 2.5476

3 — — 1.3990 0

4 — — 6.2880 0

5 — — 3.9436 0

6 — — 11.2730 0

7 — — 2.5823 0

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### A Numerical Example (continued)

• We move on to year 1.

• For each state that is in the money at year 1, we must decide whether to exercise it.

• There are 5 paths for which the put is in the money: 1, 2, 4, 6, 8 (p. 923).

• Only in-the-money paths will be used in the regression because they are where early exercise is relevant.

– If there were none, move on to year 0.

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### A Numerical Example (continued)

• Let x denote the stock prices at year 1 for those 5 paths.

• Let y denote the corresponding discounted future cash ﬂows if the put is not exercised at year 1.

• From p. 935, we have the following table.

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### A Numerical Example (continued)

Regression at year 1

Path x y

1 97.6424 12.4185 × 0.951229 2 101.2103 2.5476 × 0.9512292

3 — —

4 96.4411 6.2880 × 0.951229

5 — —

6 95.8375 11.2730 × 0.951229

7 — —

8 104.1475 0 × 0.951229

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### A Numerical Example (continued)

• We regress y on 1, x, and x2.

• The result is

f (x) = −420.964 + 9.78113 × x − 0.0551567 × x2.

• f(x) estimates the continuation value conditional on the stock price at year 1.

• We next compare the immediate exercise value and the estimated continuation value.

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### A Numerical Example (continued)

Optimal early exercise decision at year 1

Path Exercise Continuation

1 7.3576 f (97.6424) = 8.2230 2 3.7897 f (101.2103) = 3.9882

3 — —

4 8.5589 f (96.4411) = 9.3329

5 — —

6 9.1625 f (95.8375) = 9.83042

7 — —

8 0.8525 f (104.1475) = −0.551885

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### A Numerical Example (continued)

• The put should be exercised for 1 path only: 8.

– Note that f(104.1475) < 0.

• Now, any positive future cash ﬂow vanishes for this path.

– But there is none.

• The cash ﬂows on p. 935 become the ones on next slide.

• They also conﬁrm the plot on p. 926.

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### A Numerical Example (continued)

Cash flows at years 1, 2, & 3

Path Year 0 Year 1 Year 2 Year 3

1 — 0 12.4185 0

2 — 0 0 2.5476

3 — 0 1.3990 0

4 — 0 6.2880 0

5 — 0 3.9436 0

6 — 0 11.2730 0

7 — 0 2.5823 0

8 — 0.8525 0 0

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### A Numerical Example (continued)

• We move on to year 0.

• The continuation value is, from p 942,

(12.4185 × 0.9512292 + 2.5476 × 0.9512293 +1.3990 × 0.9512292 + 6.2880 × 0.9512292 +3.9436 × 0.9512292 + 11.2730 × 0.9512292 +2.5823 × 0.9512292 + 0.8525 × 0.951229)/8

= 4.66263.

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### A Numerical Example (concluded)

• As this is larger than the immediate exercise value of 105 − 101 = 4,

the put should not be exercised at year 0.

• Hence the put’s value is estimated to be 4.66263.

• Compare this with the European put’s value of 1.3680 (p. 928).

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## Time Series Analysis

(79)

The historian is a prophet in reverse.

— Friedrich von Schlegel (1772–1829)

(80)

### GARCH Option Pricing

a

• Options can be priced when the underlying asset’s return follows a GARCH process.

• Let St denote the asset price at date t.

• Let h2t be the conditional variance of the return over the period [ t, t + 1) given the information at date t.

– “One day” is merely a convenient term for any elapsed time Δt.

aARCH (autoregressive conditional heteroskedastic) is due to Engle (1982), co-winner of the 2003 Nobel Prize in Economic Sciences. GARCH (generalized ARCH) is due to Bollerslev (1986) and Taylor (1986). A Bloomberg quant said to me on Feb 29, 2008, that GARCH is seldom

(81)

### GARCH Option Pricing (continued)

• Adopt the following risk-neutral process for the price dynamics:a

ln St+1

St = r h2t

2 + htt+1, (123) where

h2t+1 = β0 + β1h2t + β2h2t(t+1 − c)2, (124)

t+1 ∼ N(0, 1) given information at date t, r = daily riskless return,

c ≥ 0.

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### GARCH Option Pricing (continued)

• The ﬁve unknown parameters of the model are c, h0, β0, β1, and β2.

• It is postulated that β0, β1, β2 ≥ 0 to make the conditional variance positive.

• There are other inequalities to satisfy (see text).

• The above process is called the nonlinear asymmetric GARCH (or NGARCH) model.

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### GARCH Option Pricing (continued)

• It captures the volatility clustering in asset returns ﬁrst noted by Mandelbrot (1963).a

– When c = 0, a large t+1 results in a large ht+1,

which in turns tends to yield a large ht+2, and so on.

• It also captures the negative correlation between the asset return and changes in its (conditional) volatility.b

– For c > 0, a positive t+1 (good news) tends to decrease ht+1, whereas a negative t+1 (bad news) tends to do the opposite.

a. . . large changes tend to be followed by large changes—of either sign—and small changes tend to be followed by small changes . . . ”

bNoted by Black (1976): Volatility tends to rise in response to “bad news” and fall in response to “good news.”

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### GARCH Option Pricing (continued)

• This is called the leverage eﬀect.

– A falling stock price raises the ﬁxed costs, relatively speaking.a

• With yt Δ

= ln St denoting the logarithmic price, the model becomes

yt+1 = yt + r h2t

2 + htt+1. (125)

• The pair (yt, h2t) completely describes the current state.

aBlack (1992).

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### GARCH Option Pricing (concluded)

• The conditional mean and variance of yt+1 are clearly E[ yt+1 | yt, h2t ] = yt + r h2t

2 , (126) Var[ yt+1 | yt, h2t ] = h2t. (127)

• Finally, given (yt, h2t), the correlation between yt+1 and ht+1 equals

2c

√2 + 4c2 , which is negative for c > 0.

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### GARCH Model: Inferences

• Suppose the parameters c, h0, β0, β1, and β2 are given.

• Then we can recover h1, h2, . . . , hn and 1, 2, . . . , n from the prices

S0, S1, . . . , Sn

under the GARCH model (123) on p. 948.

• This property is useful in statistical inferences.

• Use the Black-Scholes formula with the stock price reduced by the PV of the dividends.. • This essentially decomposes the stock price into a riskless one paying known dividends and

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 ) −

• But Monte Carlo simulation can be modiﬁed to price American options with small biases (pp..

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• But Monte Carlo simulation can be modiﬁed to price American options with small biases..

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