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Barrier Options

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Barrier Options

a

• Their payoﬀ depends on whether the underlying asset’s price reaches a certain price level H throughout its life.

• A knock-out (KO) option is an ordinary European

option which ceases to exist if the barrier H is reached by the price of its underlying asset.

• A call knock-out option is sometimes called a down-and-out option if H < S.

• A put knock-out option is sometimes called an up-and-out option when H > S.

aA former MBA student in ﬁnance told me on March 26, 2004, that she did not understand why I covered barrier options until she started working in a bank. She was working for Lehman Brothers in Hong Kong as of April, 2006.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 395

(2)

S Barrier hit

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 396

(3)

Barrier Options (concluded)

• A knock-in (KI) option comes into existence if a certain barrier is reached.

• A down-and-in option is a call knock-in option that comes into existence only when the barrier is reached and H < S.

• An up-and-in is a put knock-in option that comes into existence only when the barrier is reached and H > S.

• Formulas exist for all the possible barrier options mentioned above.a

aHaug (2006).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 397

(4)

A Formula for Down-and-In Calls

a

• Assume X ≥ H.

• The value of a European down-and-in call on a stock paying a dividend yield of q is

Se−qτ

H S



N(x) − Xe−rτ

H S

2λ−2

N(x − σ τ),

(56)

– x =Δ ln(H2/(SX))+(r−q+σ2/2) τ σ

τ .

– λ = (r − q + σΔ 2/2)/σ2.

• A European down-and-out call can be priced via the in-out parity (see text).

aMerton (1973). See Exercise 17.1.6 of the textbook for a proof.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 398

(5)

A Formula for Up-and-In Puts

a

• Assume X ≤ H.

• The value of a European up-and-in put is

Xe−rτ

H S

2λ−2

N(−x + σ

τ) − Se−qτ

H S



N(−x).

• Again, a European up-and-out put can be priced via the in-out parity.

aMerton (1973).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 399

(6)

Barrier Options: Popularity

• Knock-out options were issued in the U.S. in 1967.a

• Knock-in puts are the most popular barrier options.b

• Knock-out puts are the second most popular barrier options.c

• Knock-out calls are the most popular among barrier call options.d

aCox & Rubinstein (1985).

bBennett (2014).

cBennett (2014).

dBennett (2014).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 400

(7)

Are American Options Barrier Options?

a

• American options are barrier options with the exercise boundary as the barrier and the payoﬀ as the rebate?

• One salient diﬀerence is that the exercise boundary must be found by backward induction.

• It cannot be speciﬁed in an arbitrary way.

aContributed by Mr. Yang, Jui-Chung (D97723002) on March 25, 2009.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 401

(8)

Are American Options Barrier Options? (concluded)

• In contrast, the barrier in a barrier option is ﬁxed by a contract.a

– The option remains European-style, without the right to early exercise.b

• One can also have American barrier options.

– Need to specify whether one can exercise the option early if the stock price has not touched the barrier.c

aCox & Rubinstein (1985).

bContributed by Ms. Chen, Sin-Huei (Amber) (P00922005) on March 31, 2021.

cContributed by Mr. Lu, Yu-Ming (R06723032, D08922008) on March 31, 2021.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 402

(9)

Interesting Observations

• Assume H < X.

• Replace S in the Merton pricing formula Eq. (44) on p.

333 for the call with H2/S.

– Equation (56) on p. 398 for the down-and-in call becomes Eq. (44) when r − q = σ2/2.

– Equation (56) becomes S/H times Eq. (44) when r − q = 0.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 403

(10)

Interesting Observations (concluded)

• Replace S in the pricing formula for the down-and-in call, Eq. (56), with H2/S.

– Equation (56) becomes Eq. (44) when r − q = σ2/2.

– Equation (56) becomes H/S times Eq. (44) when r − q = 0.a

• Why?

aContributed by Mr. Chou, Ming-Hsin (R02723073) on April 24, 2014.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 404

(11)

Binomial Tree Algorithms

• Barrier options can be priced by binomial tree algorithms.

• Below is for the down-and-out option.

0 H

• Pricing down-and-in options is subtler.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 405

(12)

8

16

4

32

8

2

64

16

4

1

4.992

12.48

1.6

27.2

4.0

0

58

10

0

0 0.0

S = 8, X = 6, H = 4, R = 1.25, u = 2, and d = 0.5.

Backward-induction: C = (0.5 × Cu + 0.5 × Cd)/1.25.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 406

(13)

Binomial Tree Algorithms (continued)

• But convergence is erratic because H is not at a price level on the tree (see plot on next page).a

– The barrier H is moved lower (or higher, if you so choose) to a close-by node price.

– This “eﬀective barrier” thus changes as n increases.

• In fact, the binomial tree is O(1/√

n) convergent.b

• Solutions will be presented later.

aBoyle & Lau (1994).

bTavella & Randall (2000); J. Lin (R95221010) (2008); J. Lin (R95221010) & Palmer (2013).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 407

(14)

Binomial Tree Algorithms (concluded)

a

aLyuu (1998).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 408

(15)

Other Types of Barrier Options

a

• Partial barrier options.

• Forward-starting barrier options.

• Window barrier options.

• Rolling barrier options.

• Moving barrier options.

aArmtrong (2001); Carr & A. Chou (1997); Davydov & Linetsky (2001/2002); Haug (1998).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 409

(16)

Daily Monitoring

• Many barrier options monitor the barrier only for daily closing prices.

• If so, only nodes at the end of a day need to check for the barrier condition.

• We can even remove intraday nodes to create a multinomial tree.

– A node is then followed by d + 1 nodes if each day is partitioned into d periods.

• Does this save time or space?a

aContributed by Ms. Chen, Tzu-Chun (R94922003) and others on April 12, 2006.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 410

(17)

A Heptanomial Tree (6 Periods Per Day)

-

 1 day

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 411

(18)

Discrete Monitoring vs. Continuous Monitoring

• Discrete barriers are more expensive for knock-out options than continuous ones.

• But discrete barriers are less expensive for knock-in options than continuous ones.

• Discrete barriers are far less popular than continuous ones for individual stocks.a

• They are equally popular for indices.b

aBennett (2014).

bBennett (2014).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 412

(19)

Data! data! data!

— Arthur Conan Doyle (1892), The Adventures of Sherlock Holmes

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 413

(20)

Foreign Currencies

• S denotes the spot exchange rate in domestic/foreign terms.

– By that we mean the number of domestic currencies per unit of foreign currency.a

• σ denotes the volatility of the exchange rate.

• r denotes the domestic interest rate.

• ˆr denotes the foreign interest rate.

aThe market convention is the opposite: A/B = x means one unit of currency A (the reference currency or base currency) is equal to x units of currency B (the counter-value currency).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 414

(21)

Foreign Currencies (concluded)

• A foreign currency is analogous to a stock paying a known dividend yield.

– Foreign currencies pay a “continuous dividend yield”

equal to ˆr in the foreign currency.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 415

(22)

Time Series of the Daily Euro–USD Exchange Rate

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 416

(23)

Distribution of the Daily Euro–USD Exchange Rate

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 417

(24)

Time Series of the Minutely Euro–USD Exchange Rate

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 418

(25)

Distribution of the Minutely Euro–USD Exchange Rate

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 419

(26)

Time Series of the Daily GBP–USD Exchange Rate

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 420

(27)

Distribution of the Daily GBP–USD Exchange Rate

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 421

(28)

Distribution of the Minutely GBP–USD Exchange Rate

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 422

(29)

Distribution of the GBP–USD Exchange Rate (after the Collapse of Lehman Brothers and before Brexit)

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 423

(30)

Distribution of the Daily JPY–USD Exchange Rate

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 424

(31)

Distribution of the Daily JPY–USD Exchange Rate (concluded)

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 425

(32)

Foreign Exchange Options

• In 2000 the total notional volume of foreign exchange options was US\$13 trillion.a

– 38.5% were vanilla calls and puts with a maturity less than one month.

– 52.5% were vanilla calls and puts with a maturity between one and 18 months.

– 4% were barrier options.

– 1.5% were vanilla calls and puts with a maturity more than 18 months.

– 1% were binary options (recall p. 207 or see p. 873).

– 0.7% were Asian options (see p. 438).

aLipton (2002).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 426

(33)

Foreign Exchange Options (continued)

• Foreign exchange options are settled via delivery of the underlying currency.

• A primary use of foreign exchange (or forex) options is to hedge currency risk.

• Consider a U.S. company expecting to receive 100 million Japanese yen in March 2000.

• Those 100 million Japanese yen will be exchanged for U.S. dollars.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 427

(34)

Foreign Exchange Options (continued)

• The contract size for the Japanese yen option is JPY6,250,000.

• The company purchases

100,000,000

6,250,000 = 16

puts on the Japanese yen with a strike of \$.0088/JPY1 and an exercise month in March 2000.

• This put is in the money if the JPY-USD exchange rate drops below 0.0088.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 428

(35)

Foreign Exchange Options (continued)

• These puts provide the company the right to sell 100,000,000 Japanese yen for

100,000,000 × .0088 = 880,000 U.S. dollars.

• Note that these puts are equivalent to the right to buy 880,000 U.S. dollars with 100,000,000 Japanese yen.

– From this angle, they become calls.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 429

(36)

Foreign Exchange Options (concluded)

• Assume the exchange rate S is lognormally distributed.

• The formulas derived for stock index options in Eqs. (44) on p. 333 apply with the dividend yield equal to ˆr:

C = Se−ˆrτN (x) − Xe−rτN (x − σ√

τ ), (57) P = Xe−rτN (−x + σ√

τ ) − Se−ˆrτN (−x).

(57) – Above,

x =Δ ln(S/X) + (r − ˆr + σ2/2) τ σ√

τ .

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 430

(37)

Distribution of the Logarithmic Euro–USD Exchange Rate

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 431

(38)

Distribution of the Logarithmic GBP–USD Exchange Rate

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 432

(39)

Distribution of the Logarithmic JPY–USD Exchange Rate

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 433

(40)

Bar the paths!

Wert thou to ﬂee from here, wert thou to ﬁnd all the roads of the world, the way thou seekst the path to that thou’dst ﬁnd not[.]

— Richard Wagner (1813–1883), Parsifal

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 434

(41)

Path-Dependent Derivatives

• Let S0, S1, . . . , Sn denote the prices of the underlying asset over the life of the option.

• S0 is the known price at time zero.

• Sn is the price at expiration.

• The standard European call has a terminal value depending only on the last price, max(Sn − X, 0).

• Its value thus depends only on the underlying asset’s terminal price regardless of how it gets there.a

aCalled simple claims (Bj¨ork, 2009).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 435

(42)

Path-Dependent Derivatives (continued)

• Some derivatives are path-dependent in that their terminal payoﬀ depends explicitly on the path.

• The (arithmetic) average-rate call has this terminal value:

max

 1 n + 1

n i=0

Si − X, 0

 .

• The average-rate put’s terminal value is given by

max



X − 1

n + 1

n i=0

Si, 0

 .

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 436

(43)

Path-Dependent Derivatives (continued)

• Average-rate options are also called Asian options.

• They are very popular.a

• They are useful hedging tools for ﬁrms that will make a stream of purchases over a time period because the costs are likely to be linked to the average price.

• They are mostly European.

• The averaging clause is also common in convertible bonds and structured notes.

aAs of the late 1990s, the outstanding volume was in the range of 5–10 billion U.S. dollars (Nielsen & Sandmann, 2003).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 437

(44)

Path-Dependent Derivatives (continued)

• A lookback call option on the minimum has a terminal payoﬀ of

Sn − min

0≤i≤nSi.

• A lookback put on the maximum has a terminal payoﬀ of

0≤i≤nmax Si − Sn.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 438

(45)

Path-Dependent Derivatives (concluded)

• The ﬁxed-strike lookback optiona provides a payoﬀ of – max(max0≤i≤n Si − X, 0) for the call.

– max(X − min0≤i≤n Si, 0) for the put.

• Lookback calls and puts on the average (instead of a constant X) are called average-strike options.

aAlso called forward lookback option.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 439

(46)

Average-Rate Options

• Average-rate options are notoriously hard to price.

• The binomial tree for the averages does not combine (see next page).

• A naive algorithm enumerates the 2n paths for an n-period binomial tree and then averages the payoﬀs.

• But the complexity is exponential.a

• The Monte Carlo methodb and approximation algorithms are some of the alternatives left.

aDai (B82506025, R86526008, D8852600) & Lyuu (2007) reduce it to 2O(n ).

bSee pp. 858ﬀ.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 440

(47)

S

Su

Sd

Suu

Sud

Sdu

Sdd

p

1 p

+ +

= 

PD[ 6 6X 6XX ;

&::

+ +

= 

PD[ 6 6X 6XG ;

&:/

+ +

= 

PD[ 6 6G 6GX ;

&/:

+ +

= 

PD[ 6 6G 6GG ;

&//

( )

7 :/

: ::

H

&

S S&

& = + 

( )

7 //

/ /:

H

&

S S&

& = + 

( )

7 /

:

H

&

S S&

&= +  p

1 p p

1 p p

1 p

p

1 p p

1 p

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 441

(48)

States and Their Transitions

• The tuple

(i, S, P )

captures the statea for the Asian option.

– i: the time.

– S: the prevailing stock price.

– P : the running sum.b

aA “suﬃcient statistic,” if you will.

bWhen the average is a moving average, a diﬀerent technique is needed (C. Kao (R89723057) & Lyuu, 2003).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 442

(49)

States and Their Transitions (concluded)

• For the binomial model, the state transition is:

(i + 1, Su, P + Su), for the up move

 (i, S, P )



(i + 1, Sd, P + Sd), for the down move

• This leads to an exponential-time algorithm.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 443

(50)

Pricing Some Path-Dependent Options

• Not all path-dependent derivatives are hard to price.

– Barrier options are easy to price.

• When averaging is done geometrically, the option payoﬀs are

max



(S0S1 · · · Sn)1/(n+1) − X, 0 , max



X − (S0S1 · · · Sn)1/(n+1), 0

 .

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 444

(51)

Pricing Some Path-Dependent Options (concluded)

• The limiting analytical solutions are the Black-Scholes formulas:a

C = Se−qaτN(x) − Xe−rτN(x − σa

τ), (58) P = Xe−rτN(−x + σa

τ) − Se−qaτN(−x),

(58)

– With the volatility set to σa = σ/Δ 3 .

– With the dividend yield set to qa = (r + q + σΔ 2/6)/2.

– x =Δ ln(S/X)+(r−qaa2/2)τ σa

τ .

aAngus (1999).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 445

(52)

An Approximate Formula for Asian Calls

a

C = e−rτ

 S

τ

 τ

0

eμt+σ2t/2N

−γ + (σt/τ)(τ − t/2) τ /3

 dt

−XN

 −γ τ /3

 , where

• μ = r − σΔ 2/2.

• γ is the unique value that satisﬁes S

τ

 τ

0

e3γσt(τ −t/2)/τ2+μt+σ2[ t−(3t23)(τ −t/2)2]/2 dt = X.

aRogers & Shi (1995); Thompson (1999); K. Chen (R92723061) (2005); K. Chen (R92723061) & Lyuu (2006).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 446

(53)

Approximation Algorithm for Asian Options

• Based on the BOPM.

• Consider a node at time j with the underlying asset price equal to S0uj−idi.

• Name such a node N(j, i).

• The running sum j

m=0 Sm at this node has a maximum value of

S0(1 +

j 

u + u2 + · · · + uj−i + uj−id + · · · + uj−idi)

= S0 1 − uj−i+1

1 − u + S0uj−id 1 − di 1 − d .

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 447

(54)

Path with maximum running average

Path with minimum running average

N

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 448

(55)

Approximation Algorithm for Asian Options (continued)

• Divide this value by j + 1 and call it Amax(j, i).

• Similarly, the running sum has a minimum value of

S0(1 +

j 

d + d2 + · · · + di + diu + · · · + diuj−i)

= S0 1 − di+1

1 − d + S0diu 1 − uj−i 1 − u .

• Divide this value by j + 1 and call it Amin(j, i).

• Amin and Amax are running averages.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 449

(56)

Approximation Algorithm for Asian Options (continued)

• The number of paths to N(j, i) are far too many: j

i

. – For example,

 j j/2



∼ 2j

2/(πj) .

• The number of distinct running averages for the nodes at any given time step n seems to be bimodal for n big enough.a

– In the plot on the next page, u = 5/4 and d = 4/5.

aContributed by Mr. Liu, Jun (R99944027) on April 15, 2014.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 450

(57)

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 451

(58)

Approximation Algorithm for Asian Options (continued)

• But all averages must lie between Amin(j, i) and Amax(j, i).

• Pick k + 1 equally spaced values in this range and treat them as the true and only running averages:

Am(j, i) =Δ

k − m k



Amin(j, i) +

m k



Amax(j, i) for m = 0, 1, . . . , k.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 452

(59)

m Amin(j,i)

Amax(j,i) Am(j,i)

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 453

(60)

Approximation Algorithm for Asian Options (continued)

• Such “bucketing” or “binning ” introduces errors, but it works reasonably well in practice.a

• A better alternative picks values whose logarithms are equally spaced.b

• Still other alternatives are possible, considering the distribution of averages on p. 452.

aHull & White (1993); Ritchken, Sankarasubramanian, & Vijh (1993).

bCalled log-linear interpolation.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 454

(61)

Approximation Algorithm for Asian Options (continued)

• Backward induction calculates the option values at each node for the k + 1 running averages.

• Suppose the current node is N(j, i) and the running average is a.

• Assume the next node is N(j + 1, i), after an up move.

• As the asset price there is S0uj+1−idi, we seek the

option value corresponding to the new running average Au =Δ (j + 1) a + S0uj+1−idi

j + 2 .

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 455

(62)

Approximation Algorithm for Asian Options (continued)

• But Au is not likely to be one of the k + 1 running averages at N (j + 1, i)!

• Find the 2 running averages that bracket it:

A(j + 1, i) ≤ Au < A+1(j + 1, i).

• In “most” cases, the fastest way to nail  is via

 =

 Au − Amin(j + 1, i)

[ Amax(j + 1, i) − Amin(j + 1, i) ]/k

 .

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 456

(63)

0 ... m ...

k



.. 0

. 

 + 1 ...

k

.. 0

. 

 + 1 ...

k

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 457

(64)

Approximation Algorithm for Asian Options (continued)

• But watch out for the rare case where Au = A(j + 1, i) for some .

• Also watch out for the case where Au = Amax(j, i).

• Finally, watch out for the degenerate case where A0(j + 1, i) = · · · = Ak(j + 1, i).

– It happens to the two extreme paths.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 458

(65)

Approximation Algorithm for Asian Options (continued)

• Express Au as a linearly interpolated value of the two running averages,

Au = xA(j + 1, i) + (1 − x) A+1(j + 1, i), 0 < x ≤ 1.

• Obtain the approximate option value given the running average Au via

Cu = xCΔ (j + 1, i) + (1 − x) C+1(j + 1, i).

– C(t, s) denotes the option value at node N (t, s) with running average A(t, s).

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 459

(66)

Approximation Algorithm for Asian Options (continued)

• This interpolation introduces the second source of error.

– Alternatives to linear interpolation exist.

• The same steps are repeated for the down node

N (j + 1, i + 1) to obtain another approximate option value Cd.

• Finally obtain the option value as

[ pCu + (1 − p) Cd ] e−rΔt.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 460

(67)

Approximation Algorithm for Asian Options (continued)

• No interpolation is needed for the calculations at time step n − 1.a

– The running time is O(kn2).

∗ There are O(n2) nodes.

∗ Each node has O(k) buckets.

– The option values are simply (for calls):

Cu = max(Au − X, 0), Cd = max(Ad − X, 0).

– That saves O(nk) calculations.

aNot n. Contributed by Mr. Chen, Shih-Hang (R02723031) on April 9, 2014.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 461

(68)

Approximation Algorithm for Asian Options (concluded)

• Arithmetic average-rate options were assumed to be newly issued: no historical average to deal with.

• This problem can be easily addressed.a

aSee Exercise 11.7.4 of the textbook.

bThanks to lively class discussions on March 31, 2004, and April 9, 2014.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 462

(69)

A Numerical Example

• Consider a European arithmetic average-rate call with strike price 50.

• Assume zero interest rate in order to dispense with discounting.

• The minimum running average at node A in the ﬁgure on p. 465 is 48.925.

• The maximum running average at node A in the same ﬁgure is 51.149.

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51.168

49.500 50.612 51.723

48.944

53.506

48.979 50.056

48.388

46.827 52.356

50

53.447

46.775

0.0269

50.056 51.206

47.903 50.056 0.2956

0.5782 0.8617

50.056

1.206 0.056

2.356 3.506

49.666 48.925

50.408 51.149

0.000 0.000

0.000 0.056 p = 0.483

u = 1.069 d = 0.936

A

B

C

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A Numerical Example (continued)

• Each node picks k = 3 for 4 equally spaced running averages.

• The same calculations are done for node A’s successor nodes B and C.

• Suppose node A is 2 periods from the root node.

• Consider the up move from node A with running average 49.666.

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A Numerical Example (continued)

• Because the stock price at node B is 53.447, the new running average will be

3 × 49.666 + 53.447

4 ≈ 50.612.

• With 50.612 lying between 50.056 and 51.206 at node B, we solve

50.612 = x × 50.056 + (1 − x) × 51.206 to obtain x ≈ 0.517.

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A Numerical Example (continued)

• The option value corresponding to running average 50.056 at node B is 0.056.

• The option values corresponding to running average 51.206 at node B is 1.206.

• Their contribution to the option value corresponding to running average 49.666 at node A is weighted linearly as

x × 0.056 + (1 − x) × 1.206 ≈ 0.611.

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A Numerical Example (continued)

• Now consider the down move from node A with running average 49.666.

• Because the stock price at node C is 46.775, the new running average will be

3 × 49.666 + 46.775

4 ≈ 48.944.

• With 48.944 lying between 47.903 and 48.979 at node C, we solve

48.944 = x × 47.903 + (1 − x) × 48.979 to obtain x ≈ 0.033.

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A Numerical Example (concluded)

• The option values corresponding to running averages 47.903 and 48.979 at node C are both 0.0.

• Their contribution to the option value corresponding to running average 49.666 at node A is 0.0.

• Finally, the option value corresponding to running average 49.666 at node A equals

p × 0.611 + (1 − p) × 0.0 ≈ 0.2956, where p = 0.483.

• The remaining three option values at node A can be computed similarly.

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Convergence Behavior of the Approximation Algorithm with k = 50000

a

60 80 100 120 140 n 0.325

0.33 0.335 0.34 0.345 0.35

Asian option value

aDai (B82506025, R86526008, D8852600) & Lyuu (2002).

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Remarks on Asian Option Pricing

• Asian option pricing is an active research area.

• The above algorithm overestimates the “true” value.a

• To guarantee convergence, k needs to grow with n at least.b

• There is a convergent approximation algorithmc that does away with interpolation with a running time of

2O(n ).

aDai (B82506025, R86526008, D8852600), G. Huang (F83506075), &

Lyuu (2002).

bDai (B82506025, R86526008, D8852600), G. Huang (F83506075), &

Lyuu (2002).

cDai (B82506025, R86526008, D8852600) & Lyuu (2002, 2004).

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Remarks on Asian Option Pricing (continued)

• There is an O(kn2)-time algorithm with an error bound of O(Xn/k) from the naive O(2n)-time binomial tree algorithm in the case of European Asian options.a

– k can be varied for trade-oﬀ between time and accuracy.

– If we pick k = O(n2), then the error is O(1/n), and the running time is O(n4).

aAingworth, Motwani (1962–2009), & Oldham (2000).

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Remarks on Asian Option Pricing (continued)

• Another approximation algorithm reduces the error to O(X√

n/k).a

– It varies the number of buckets per node.

– If we pick k = O(n), the error is O(n−0.5).

– If we pick k = O(n1.5), then the error is O(1/n), and the running time is O(n3.5).

• Under “reasonable assumptions,” an O(n2)-time algorithm with an error bound of O(1/n) exists.b

aDai (B82506025, R86526008, D8852600), G. Huang (F83506075), &

Lyuu (2002).

bHsu (R7526001, D89922012) & Lyuu (2004).

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Remarks on Asian Option Pricing (concluded)

• The basic idea is a nonuniform allocation of running averages instead of a uniform k.

• It strikes a tight balance between error and complexity.

Uniform allocation

0 5

10 15

20

i j

0 20 40

k

0 5

10 15

20

i

Nonuniform allocation

0 5

10 15

20

i j

0 100 200 300 400

kij

0 5

10 15

20

i

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A Grand Comparison

a

aHsu (R7526001, D89922012) & Lyuu (2004); J. E. Zhang (2001,2003);

K. Chen (R92723061) & Lyuu (2006).

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X σ r Exact AA2 AA3 Hsu-Lyuu Chen-Lyuu 95 0.05 0.05 7.1777275 7.1777244 7.1777279 7.178812 7.177726

100 2.7161745 2.7161755 2.7161744 2.715613 2.716168

105 0.3372614 0.3372601 0.3372614 0.338863 0.337231

95 0.09 8.8088392 8.8088441 8.8088397 8.808717 8.808839

100 4.3082350 4.3082253 4.3082331 4.309247 4.308231

105 0.9583841 0.9583838 0.9583841 0.960068 0.958331

95 0.15 11.0940944 11.0940964 11.0940943 11.093903 11.094094

100 6.7943550 6.7943510 6.7943553 6.795678 6.794354

105 2.7444531 2.7444538 2.7444531 2.743798 2.744406

90 0.10 0.05 11.9510927 11.9509331 11.9510871 11.951610 11.951076

100 3.6413864 3.6414032 3.6413875 3.642325 3.641344

110 0.3312030 0.3312563 0.3311968 0.331348 0.331074

90 0.09 13.3851974 13.3851165 13.3852048 13.385563 13.385190

100 4.9151167 4.9151388 4.9151177 4.914254 4.915075

110 0.6302713 0.6302538 0.6302717 0.629843 0.630064

90 0.15 15.3987687 15.3988062 15.3987860 15.398885 15.398767

100 7.0277081 7.0276544 7.0277022 7.027385 7.027678

110 1.4136149 1.4136013 1.4136161 1.414953 1.413286

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A Grand Comparison (concluded)

X σ r Exact AA2 AA3 Hsu-Lyuu Chen-Lyuu

90 0.20 0.05 12.5959916 12.5957894 12.5959304 12.596052 12.595602

100 5.7630881 5.7631987 5.7631187 5.763664 5.762708

110 1.9898945 1.9894855 1.9899382 1.989962 1.989242

90 0.09 13.8314996 13.8307782 13.8313482 13.831604 13.831220

100 6.7773481 6.7775756 6.7773833 6.777748 6.776999

110 2.5462209 2.5459150 2.5462598 2.546397 2.545459

90 0.15 15.6417575 15.6401370 15.6414533 15.641911 15.641598

100 8.4088330 8.4091957 8.4088744 8.408966 8.408519

110 3.5556100 3.5554997 3.5556415 3.556094 3.554687

90 0.30 0.05 13.9538233 13.9555691 13.9540973 13.953937 13.952421

100 7.9456288 7.9459286 7.9458549 7.945918 7.944357

110 4.0717942 4.0702869 4.0720881 4.071945 4.070115

90 0.09 14.9839595 14.9854235 14.9841522 14.984037 14.982782

100 8.8287588 8.8294164 8.8289978 8.829033 8.827548

110 4.6967089 4.6956764 4.6969698 4.696895 4.694902

90 0.15 16.5129113 16.5133090 16.5128376 16.512963 16.512024 100 10.2098305 10.2110681 10.2101058 10.210039 10.208724

110 5.7301225 5.7296982 5.7303567 5.730357 5.728161

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Forwards, Futures, Futures Options, Swaps

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Summon the nations to come to the trial.

Which of their gods can predict the future?

— Isaiah 43:9 The sure fun of the evening outweighed the uncertain treasure[.]

— Mark Twain (1835–1910), The Adventures of Tom Sawyer

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 479

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Terms

• r will denote the riskless interest rate.

• The current time is t.

• The maturity date is T .

• The remaining time to maturity is τ = T − t (years).Δ

• The spot price is S.

• The spot price at maturity is ST.

• The delivery price is X.

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Terms (concluded)

• The forward or futures price is F for a newly written contract.

• The value of the contract is f.

• A price with a subscript t usually refers to the price at time t.

• Continuous compounding will be assumed.

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Forward Contracts

• Long forward contracts are for the purchase of the

underlying asset for a certain delivery price on a speciﬁc time.

– Foreign currencies, bonds, corn, etc.

• Ideal for hedging purposes.

• A farmer enters into a forward contract with a food

processor to deliver 100,000 bushels of corn for \$2.5 per bushel on September 27, 1995.a

• The farmer is assured of a buyer at an acceptable price.

• The processor knows the cost of corn in advance.

aThe farmer assumes a short position.

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Forward Contracts (concluded)

• A forward agreement limits both risk and rewards.

– If the spot price of corn rises on the delivery date, the farmer will miss the opportunity of extra proﬁts.

– If the price declines, the processor will be paying more than it would.

• Either side has an incentive to default.

• Other problems: The food processor may go bankrupt, the farmer can go bust, the farmer might not be able to harvest 100,000 bushels of corn because of bad weather, the cost of growing corn may skyrocket, etc.

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Spot and Forward Exchange Rates

• Let S denote the spot exchange rate.

• Let F denote the forward exchange rate one year from now (both in domestic/foreign terms).

• rf denotes the annual interest rate of the foreign currency.

• r denotes the annual interest rate of the local currency.

• Arbitrage opportunities will arise unless these four numbers satisfy an equation.

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Interest Rate Parity

a

F

S = er−rf. (59)

• A holder of the local currency can do either of:

– Lend the money in the domestic market to receive er one year from now.

– Convert local currency for foreign currency, lend for 1 year in foreign market, and convert foreign currency into local currency at the ﬁxed forward exchange rate, F , by selling forward foreign currency now.

aKeynes (1923). John Maynard Keynes (1883–1946) was one of the greatest economists in history. The parity broke down in late 2008 (Mancini-Griﬀoli & Ranaldo, 2013).

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Interest Rate Parity (concluded)

• No money changes hand in entering into a forward contract.

• One unit of local currency will hence become F erf/S one year from now in the 2nd case.

• If F erf/S > er, an arbitrage proﬁt can result from

borrowing money in the domestic market and lending it in the foreign market.

• If F erf/S < er, an arbitrage proﬁt can result from

borrowing money in the foreign market and lending it in the domestic market.

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Forward Price

• The payoﬀ of a forward contract at maturity is ST − X.

– Contrast that with call’s payoﬀ max(ST − X, 0).

• Forward contracts do not involve any initial cash ﬂow.

• The forward price F is the delivery price X which makes the forward contract zero valued.

– That is,

f = 0 when X = F.

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 487

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Forward Price (continued)

6 -

0 1 2 3 · · · n

ST − X

2022 Prof. Yuh-Dauh Lyuu, National Taiwan Universityc Page 488

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