Barrier Options
^{a}• Their payoﬀ depends on whether the underlying asset’s price reaches a certain price level H throughout its life.
• A knockout (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 knockout option is sometimes called a downandout option if H < S.
• A put knockout option is sometimes called an upandout 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. YuhDauh Lyuu, National Taiwan Universityc Page 395
H
Time Price
S Barrier hit
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 396
Barrier Options (concluded)
• A knockin (KI) option comes into existence if a certain barrier is reached.
• A downandin option is a call knockin option that comes into existence only when the barrier is reached and H < S.
• An upandin is a put knockin 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. YuhDauh Lyuu, National Taiwan Universityc Page 397
A Formula for DownandIn Calls
^{a}• Assume X ≥ H.
• The value of a European downandin call on a stock paying a dividend yield of q is
Se^{−qτ}
H S
_{2λ}
N(x) − Xe^{−rτ}
H S
_{2λ−2}
N(x − σ√ τ),
(56)
– x =^{Δ} ^{ln(H}^{2}/(SX))+(r−q+σ^{2}/2) τ σ√
τ .
– λ = (r − q + σ^{Δ} ^{2}/2)/σ^{2}.
• A European downandout call can be priced via the inout parity (see text).
aMerton (1973). See Exercise 17.1.6 of the textbook for a proof.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 398
A Formula for UpandIn Puts
^{a}• Assume X ≤ H.
• The value of a European upandin put is
Xe^{−rτ}
H S
_{2λ−2}
N(−x + σ√
τ) − Se^{−qτ}
H S
_{2λ}
N(−x).
• Again, a European upandout put can be priced via the inout parity.
aMerton (1973).
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 399
Barrier Options: Popularity
• Knockout options were issued in the U.S. in 1967.^{a}
• Knockin puts are the most popular barrier options.^{b}
• Knockout puts are the second most popular barrier options.^{c}
• Knockout calls are the most popular among barrier call options.^{d}
aCox & Rubinstein (1985).
bBennett (2014).
cBennett (2014).
dBennett (2014).
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 400
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, JuiChung (D97723002) on March 25, 2009.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 401
Are American Options Barrier Options? (concluded)
• In contrast, the barrier in a barrier option is ﬁxed by a contract.^{a}
– The option remains Europeanstyle, 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, SinHuei (Amber) (P00922005) on March 31, 2021.
cContributed by Mr. Lu, YuMing (R06723032, D08922008) on March 31, 2021.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 402
Interesting Observations
• Assume H < X.
• Replace S in the Merton pricing formula Eq. (44) on p.
333 for the call with H^{2}/S.
– Equation (56) on p. 398 for the downandin call becomes Eq. (44) when r − q = σ^{2}/2.
– Equation (56) becomes S/H times Eq. (44) when r − q = 0.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 403
Interesting Observations (concluded)
• Replace S in the pricing formula for the downandin call, Eq. (56), with H^{2}/S.
– Equation (56) becomes Eq. (44) when r − q = σ^{2}/2.
– Equation (56) becomes H/S times Eq. (44) when r − q = 0.^{a}
• Why?
aContributed by Mr. Chou, MingHsin (R02723073) on April 24, 2014.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 404
Binomial Tree Algorithms
• Barrier options can be priced by binomial tree algorithms.
• Below is for the downandout option.
0 H
• Pricing downandin options is subtler.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 405
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.
Backwardinduction: C = (0.5 × C_{u} + 0.5 × C_{d})/1.25.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 406
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 closeby 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. YuhDauh Lyuu, National Taiwan Universityc Page 407
Binomial Tree Algorithms (concluded)
^{a}aLyuu (1998).
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 408
Other Types of Barrier Options
^{a}• Partial barrier options.
• Forwardstarting 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. YuhDauh Lyuu, National Taiwan Universityc Page 409
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, TzuChun (R94922003) and others on April 12, 2006.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 410
A Heptanomial Tree (6 Periods Per Day)

_{1 day}
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 411
Discrete Monitoring vs. Continuous Monitoring
• Discrete barriers are more expensive for knockout options than continuous ones.
• But discrete barriers are less expensive for knockin 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. YuhDauh Lyuu, National Taiwan Universityc Page 412
Data! data! data!
— Arthur Conan Doyle (1892), The Adventures of Sherlock Holmes
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 413
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 countervalue currency).
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 414
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. YuhDauh Lyuu, National Taiwan Universityc Page 415
Time Series of the Daily Euro–USD Exchange Rate
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 416
Distribution of the Daily Euro–USD Exchange Rate
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 417
Time Series of the Minutely Euro–USD Exchange Rate
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 418
Distribution of the Minutely Euro–USD Exchange Rate
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 419
Time Series of the Daily GBP–USD Exchange Rate
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 420
Distribution of the Daily GBP–USD Exchange Rate
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 421
Distribution of the Minutely GBP–USD Exchange Rate
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 422
Distribution of the GBP–USD Exchange Rate (after the Collapse of Lehman Brothers and before Brexit)
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 423
Distribution of the Daily JPY–USD Exchange Rate
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 424
Distribution of the Daily JPY–USD Exchange Rate (concluded)
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 425
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. YuhDauh Lyuu, National Taiwan Universityc Page 426
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. YuhDauh Lyuu, National Taiwan Universityc Page 427
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 JPYUSD exchange rate drops below 0.0088.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 428
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. YuhDauh Lyuu, National Taiwan Universityc Page 429
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^{−ˆrτ}N (x) − Xe^{−rτ}N (x − σ√
τ ), (57) P = Xe^{−rτ}N (−x + σ√
τ ) − Se^{−ˆrτ}N (−x).
(57^{}) – Above,
x =^{Δ} ln(S/X) + (r − ˆr + σ^{2}/2) τ σ√
τ .
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 430
Distribution of the Logarithmic Euro–USD Exchange Rate
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 431
Distribution of the Logarithmic GBP–USD Exchange Rate
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 432
Distribution of the Logarithmic JPY–USD Exchange Rate
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 433
Bar the roads!
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. YuhDauh Lyuu, National Taiwan Universityc Page 434
PathDependent Derivatives
• Let S_{0}, S_{1}, . . . , S_{n} denote the prices of the underlying asset over the life of the option.
• S_{0} is the known price at time zero.
• S_{n} is the price at expiration.
• The standard European call has a terminal value depending only on the last price, max(S_{n} − 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. YuhDauh Lyuu, National Taiwan Universityc Page 435
PathDependent Derivatives (continued)
• Some derivatives are pathdependent in that their terminal payoﬀ depends explicitly on the path.
• The (arithmetic) averagerate call has this terminal value:
max
1 n + 1
n i=0
S_{i} − X, 0
.
• The averagerate put’s terminal value is given by
max
X − 1
n + 1
n i=0
S_{i}, 0
.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 436
PathDependent Derivatives (continued)
• Averagerate 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. YuhDauh Lyuu, National Taiwan Universityc Page 437
PathDependent Derivatives (continued)
• A lookback call option on the minimum has a terminal payoﬀ of
S_{n} − min
0≤i≤nS_{i}.
• A lookback put on the maximum has a terminal payoﬀ of
0≤i≤nmax S_{i} − S_{n}.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 438
PathDependent Derivatives (concluded)
• The ﬁxedstrike lookback option^{a} provides a payoﬀ of – max(max0≤i≤n S_{i} − X, 0) for the call.
– max(X − min_{0≤i≤n} S_{i}, 0) for the put.
• Lookback calls and puts on the average (instead of a constant X) are called averagestrike options.
aAlso called forward lookback option.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 439
AverageRate Options
• Averagerate options are notoriously hard to price.
• The binomial tree for the averages does not combine (see next page).
• A naive algorithm enumerates the 2^{n} paths for an nperiod binomial tree and then averages the payoﬀs.
• But the complexity is exponential.^{a}
• The Monte Carlo method^{b} and approximation algorithms are some of the alternatives left.
aDai (B82506025, R86526008, D8852600) & Lyuu (2007) reduce it to 2^{O(}^{√}^{n )}.
bSee pp. 858ﬀ.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 440
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. YuhDauh Lyuu, National Taiwan Universityc Page 441
States and Their Transitions
• The tuple
(i, S, P )
captures the state^{a} 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. YuhDauh Lyuu, National Taiwan Universityc Page 442
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 exponentialtime algorithm.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 443
Pricing Some PathDependent Options
• Not all pathdependent derivatives are hard to price.
– Barrier options are easy to price.
• When averaging is done geometrically, the option payoﬀs are
max
(S_{0}S_{1} · · · S_{n})^{1/(n+1)} − X, 0 , max
X − (S_{0}S_{1} · · · S_{n})^{1/(n+1)}, 0
.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 444
Pricing Some PathDependent Options (concluded)
• The limiting analytical solutions are the BlackScholes formulas:^{a}
C = Se^{−q}^{a}^{τ}N(x) − Xe^{−rτ}N(x − σa
√τ), (58) P = Xe^{−rτ}N(−x + σa
√τ) − Se^{−q}^{a}^{τ}N(−x),
(58^{})
– With the volatility set to σ_{a} = σ/^{Δ} √ 3 .
– With the dividend yield set to q_{a} = (r + q + σ^{Δ} ^{2}/6)/2.
– x =^{Δ} ^{ln(S/X)+}(r−q^{a}+σa^{2}/2)τ σa√
τ .
aAngus (1999).
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 445
An Approximate Formula for Asian Calls
^{a}C = e^{−rτ}
S
τ
_{τ}
0
e^{μt+σ}^{2}^{t/2}N
−γ + (σt/τ)(τ − t/2) τ /3
dt
−XN
−γ τ /3
, where
• μ = r − σ^{Δ} ^{2}/2.
• γ is the unique value that satisﬁes S
τ
_{τ}
0
e3γσt(τ −t/2)/τ^{2}+μt+σ^{2}[ t−(3t^{2}/τ^{3})(τ −t/2)^{2}]/2 dt = X.
aRogers & Shi (1995); Thompson (1999); K. Chen (R92723061) (2005); K. Chen (R92723061) & Lyuu (2006).
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 446
Approximation Algorithm for Asian Options
• Based on the BOPM.
• Consider a node at time j with the underlying asset price equal to S_{0}u^{j−i}d^{i}.
• Name such a node N(j, i).
• The running sum _{j}
m=0 S_{m} at this node has a maximum value of
S_{0}(1 +
j
u + u^{2} + · · · + u^{j−i} + u^{j−i}d + · · · + u^{j−i}d^{i})
= S_{0} 1 − u^{j−i+1}
1 − u + S_{0}u^{j−i}d 1 − d^{i} 1 − d .
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 447
Path with maximum running average
Path with minimum running average
N
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 448
Approximation Algorithm for Asian Options (continued)
• Divide this value by j + 1 and call it A_{max}(j, i).
• Similarly, the running sum has a minimum value of
S_{0}(1 +
j
d + d^{2} + · · · + d^{i} + d^{i}u + · · · + d^{i}u^{j−i})
= S_{0} 1 − d^{i+1}
1 − d + S_{0}d^{i}u 1 − u^{j−i} 1 − u .
• Divide this value by j + 1 and call it A_{min}(j, i).
• A_{min} and A_{max} are running averages.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 449
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
∼ 2^{j}
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. YuhDauh Lyuu, National Taiwan Universityc Page 450
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 451
Approximation Algorithm for Asian Options (continued)
• But all averages must lie between A_{min}(j, i) and A_{max}(j, i).
• Pick k + 1 equally spaced values in this range and treat them as the true and only running averages:
A_{m}(j, i) =^{Δ}
k − m k
A_{min}(j, i) +
m k
A_{max}(j, i) for m = 0, 1, . . . , k.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 452
m A_{min}(j,i)
A_{max}(j,i) A_{m}(j,i)
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 453
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 loglinear interpolation.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 454
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 S_{0}u^{j+1−i}d^{i}, we seek the
option value corresponding to the new running average A_{u} =^{Δ} (j + 1) a + S_{0}u^{j+1−i}d^{i}
j + 2 .
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 455
Approximation Algorithm for Asian Options (continued)
• But A_{u} 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) ≤ A_{u} < A_{+1}(j + 1, i).
• In “most” cases, the fastest way to nail is via
=
A_{u} − A_{min}(j + 1, i)
[ A_{max}(j + 1, i) − A_{min}(j + 1, i) ]/k
.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 456
0 ... m ...
k
.. 0
.
+ 1 ...
k
.. 0
. ^{}
^{} + 1 ...
k
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 457
Approximation Algorithm for Asian Options (continued)
• But watch out for the rare case where A_{u} = A_{}(j + 1, i) for some .
• Also watch out for the case where A_{u} = A_{max}(j, i).
• Finally, watch out for the degenerate case where A_{0}(j + 1, i) = · · · = A_{k}(j + 1, i).
– It happens to the two extreme paths.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 458
Approximation Algorithm for Asian Options (continued)
• Express A_{u} as a linearly interpolated value of the two running averages,
A_{u} = xA_{}(j + 1, i) + (1 − x) A_{+1}(j + 1, i), 0 < x ≤ 1.
• Obtain the approximate option value given the running average A_{u} via
C_{u} = xC^{Δ} _{}(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. YuhDauh Lyuu, National Taiwan Universityc Page 459
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 C_{d}.
• Finally obtain the option value as
[ pC_{u} + (1 − p) C_{d} ] e^{−rΔt}.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 460
Approximation Algorithm for Asian Options (continued)
• No interpolation is needed for the calculations at time step n − 1.^{a}
– The running time is O(kn^{2}).
∗ There are O(n^{2}) nodes.
∗ Each node has O(k) buckets.
– The option values are simply (for calls):
C_{u} = max(A_{u} − X, 0), C_{d} = max(A_{d} − X, 0).
– That saves O(nk) calculations.
aNot n. Contributed by Mr. Chen, ShihHang (R02723031) on April 9, 2014.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 461
Approximation Algorithm for Asian Options (concluded)
• Arithmetic averagerate options were assumed to be newly issued: no historical average to deal with.
• This problem can be easily addressed.^{a}
• How about the Greeks?^{b}
aSee Exercise 11.7.4 of the textbook.
bThanks to lively class discussions on March 31, 2004, and April 9, 2014.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 462
A Numerical Example
• Consider a European arithmetic averagerate 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.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 463
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
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 464
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.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 465
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.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 466
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.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 467
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.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 468
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.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 469
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).
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 470
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 algorithm^{c} that does away with interpolation with a running time of
2^{O(}^{√}^{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).
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 471
Remarks on Asian Option Pricing (continued)
• There is an O(kn^{2})time algorithm with an error bound of O(Xn/k) from the naive O(2^{n})time binomial tree algorithm in the case of European Asian options.^{a}
– k can be varied for tradeoﬀ between time and accuracy.
– If we pick k = O(n^{2}), then the error is O(1/n), and the running time is O(n^{4}).
aAingworth, Motwani (1962–2009), & Oldham (2000).
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 472
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(n^{1.5}), then the error is O(1/n), and the running time is O(n^{3.5}).
• Under “reasonable assumptions,” an O(n^{2})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).
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 473
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
k^{ij}
0 5
10 15
20
i
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 474
A Grand Comparison
^{a}aHsu (R7526001, D89922012) & Lyuu (2004); J. E. Zhang (2001,2003);
K. Chen (R92723061) & Lyuu (2006).
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 475
X σ r Exact AA2 AA3 HsuLyuu ChenLyuu 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
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 476
A Grand Comparison (concluded)
X σ r Exact AA2 AA3 HsuLyuu ChenLyuu
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
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 477
Forwards, Futures, Futures Options, Swaps
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 478
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. YuhDauh Lyuu, National Taiwan Universityc Page 479
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 S_{T}.
• The delivery price is X.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 480
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.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 481
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.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 482
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.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 483
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).
• r_{f} 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.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 484
Interest Rate Parity
^{a}F
S = e^{r}^{}^{−r}^{f}. (59)
• A holder of the local currency can do either of:
– Lend the money in the domestic market to receive e^{r}^{} 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 (ManciniGriﬀoli & Ranaldo, 2013).
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 485
Interest Rate Parity (concluded)
• No money changes hand in entering into a forward contract.
• One unit of local currency will hence become F e^{r}^{f}/S one year from now in the 2nd case.
• If F e^{r}^{f}/S > e^{r}^{}, an arbitrage proﬁt can result from
borrowing money in the domestic market and lending it in the foreign market.
• If F e^{r}^{f}/S < e^{r}^{}, an arbitrage proﬁt can result from
borrowing money in the foreign market and lending it in the domestic market.
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 486
Forward Price
• The payoﬀ of a forward contract at maturity is S_{T} − X.
– Contrast that with call’s payoﬀ max(S_{T} − 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. YuhDauh Lyuu, National Taiwan Universityc Page 487
Forward Price (continued)
6 
0 1 2 3 · · · n
ST − X
2022 Prof. YuhDauh Lyuu, National Taiwan Universityc Page 488