Barrier Options
^{a}• Their payoff depends on whether the underlying asset’s price reaches a certain price level H.
• A knockout 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 student told me on March 26, 2004, that she did not un derstand why I covered barrier options until she started working in a bank. She was working for Lehman Brothers in HK as of April, 2006.
H
Time Price
S Barrier hit
Barrier Options (concluded)
• A knockin 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 kinds of barrier options.
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}
µH S
¶_{2λ}
N (x) − Xe^{−rτ}
µH S
¶_{2λ−2}
N (x − σ√ τ ),
(28)
– x ≡ ^{ln(H}^{2}/(SX))+(r−q+σ^{2}/2) τ σ√
τ .
– λ ≡ (r − q + σ^{2}/2)/σ^{2}.
• A European downandout call can be priced via the inout parity (see text).
A Formula for UpandIn Puts
^{a}• Assume X ≤ H.
• The value of a European upandin put is
Xe^{−rτ}
µH S
¶_{2λ−2}
N (−x + σ√
τ ) − Se^{−qτ}
µH S
¶_{2λ}
N (−x).
• A European upandout put can be priced via the inout parity.
aMerton (1973).
Are American Options Barrier Options?
^{a}• It looks like American options are barrier options with the exercise boundary treated as the barrier and the payoff as the rebate.
• One salient difference is that the exercise boundary must be derived during backward induction, whereas the
barrier in a barrier option is given a priori.
aContributed by Mr. JuiChung Yang (D97723002) on March 25, 2009.
Interesting Observations
• Assume H < X.
• Replace S in the pricing formula for the downandin call, Eq. (28) on p. 315, with H^{2}/S.
• Equation (28) becomes Eq. (25) on p. 267 when r − q = σ^{2}/2.
• Equation (28) becomes S/H times Eq. (25) on p. 267 when r − q = 0.
• Why?
Binomial Tree Algorithms
• Barrier options can be priced by binomial tree algorithms.
• Below is for the downandout option.
0 H
H 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 X
0.0
S = 8, X = 6, H = 4, R = 1.25, u = 2, and d = 0.5.
Backwardinduction: C = (0.5 × C + 0.5 × C )/1.25.
Binomial Tree Algorithms (concluded)
• But convergence is erratic because H is not at a price level on the tree (see plot on next page).
– The barrier has to be adjusted to be at a price level.
– The “effective barrier” changes as n increases.
• Solutions will be presented later.
100 150 200 250 300 350 400
#Periods 3
3.5 4 4.5 5 5.5
Downandin call value
Daily Monitoring
• Almost all barrier options monitor the barrier only for the daily closing prices.
• In that case, 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 saves time or space?^{a}
aContributed by Ms. Chen, TzuChun (R94922003) and others on
A Heptanomial Tree (6 Periods Per Day)

¾ _{1 day}
Foreign Currencies
• S denotes the spot exchange rate in domestic/foreign terms.
• σ denotes the volatility of the exchange rate.
• r denotes the domestic interest rate.
• ˆr denotes the foreign interest rate.
• 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.
Foreign Exchange Options
• 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.
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 price of $.0088 and an exercise month in March 2000.
• This gives the company the right to sell 100,000,000 Japanese yen for 100,000,000 × .0088 = 880,000 U.S.
dollars.
Foreign Exchange Options (concluded)
• The formulas derived for stock index options in Eqs. (25) on p. 267 apply with the dividend yield equal to ˆr:
C = Se^{−ˆ}^{rτ}N (x) − Xe^{−rτ}N (x − σ√
τ ), (29) P = Xe^{−rτ}N (−x + σ√
τ ) − Se^{−ˆ}^{rτ}N (−x).
(29^{0}) – Above,
x ≡ ln(S/X) + (r − ˆr + σ^{2}/2) τ σ√
τ .
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
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.
PathDependent Derivatives (continued)
• Some derivatives are pathdependent in that their terminal payoff depends critically on the path.
• The (arithmetic) averagerate call has this terminal value:
max
Ã 1
n + 1
Xn i=0
S_{i} − X, 0
! .
• The averagerate put’s terminal value is given by max
Ã
X − 1
n + 1
Xn i=0
S_{i}, 0
! .
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.
aAs of the late 1990s, the outstanding volume was in the range of 5–10 billion U.S. dollars according to Nielsen and Sandmann (2003).
PathDependent Derivatives (concluded)
• A lookback call option on the minimum has a terminal payoff of S_{n} − min_{0≤i≤n} S_{i}.
• A lookback put on the maximum has a terminal payoff of max_{0≤i≤n} S_{i} − S_{n}.
• The fixedstrike lookback option provides a payoff of – max(max_{0≤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.
AverageRate Options
• Averagerate options are notoriously hard to price.
• The binomial tree for the averages does not combine.
• A straightforward algorithm enumerates the 2^{n} price paths for an nperiod binomial tree and then averages the payoffs.
• But the complexity is exponential.
• As a result, the Monte Carlo method and approximation algorithms are some of the alternatives left.
S
Su
Sd
Suu
Sud
Sdu
Sdd
p
1−^{ p}
+ + −
=
PD[ 6 6X 6XX ;
&_{XX}
+ + −
=
PD[ 6 6X 6XG ;
&_{XG}
+ + −
=
PD[ 6 6G 6GX ;
&_{GX}
+ + −
=
PD[ 6 6G 6GG ;
&_{GG}
( )
U XG
X XX
H
&
S S&
& = + −
( )
U GG
G GX
H
&
S S&
& = + −
( )
U G
X
H
&
S S&
&= + − p
1−^{ p} p
1−^{ p} p
1−^{ p}
p
1−^{ p} p
1−^{ p}
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.
aIt is a sufficient statistic.
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
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
³
(S_{0}S_{1} · · · S_{n})^{1/(n+1)} − X, 0
´ , max
³
X − (S_{0}S_{1} · · · S_{n})^{1/(n+1)}, 0
´ .
Pricing Some PathDependent Options (concluded)
• The limiting analytical solutions are the BlackScholes formulas.
– With the volatility set to σ_{a} ≡ σ/√ 3 .
– With the dividend yield set to q_{a} ≡ (r + q + σ^{2}/6)/2.
• The formula is therefore
C = Se^{−q}^{a}^{τ}N (x) − Xe^{−rτ}N (x − σ_{a}√
τ ), (30) P = Xe^{−rτ}N (−x + σ_{a}√
τ ) − Se^{−q}^{a}^{τ}N (−x),
(30^{0}) – x ≡ ^{ln(S/X)+}(^{r−q}_{√} ^{a}^{+σ}a^{2}/2)^{τ} .
An Approximate Formula for Asian Calls
^{a}C = e^{−rτ}
"
S τ
Z _{τ}
0
e^{µt+σ}^{2}^{t/2}N
Ã−γ + (σt/τ )(τ − t/2) pτ /3
! dt
−XN
Ã −γ pτ /3
! # , where
• µ ≡ r − σ^{2}/2.
• γ is the unique value that satisfies S
τ
Z _{τ}
0
e3γσt(τ −t/2)/τ^{2}+µt+σ^{2}[ t−(3t^{2}/τ^{3})(τ −t/2)^{2}]/2 dt = X.
aRogers and Shi (1995); Thompson (1999); Chen (2005); Chen and
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 P_{j}
m=0 S_{m} at this node has a maximum value of
S_{0}(1 +
z }j {
u + u^{2} + · · · + u^{j−i} + u^{j−i}d + · · · + u^{j−i}d^{i})
= S_{0} 1 − u^{j−i+1}
+ S_{0}u^{j−i}d 1 − d^{i} .
Path with maximum running average
Path with minimum running average
N
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 +
z }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.
Approximation Algorithm for Asian Options (continued)
• The possible running averages at N (j, i) are far too many: ¡_{j}
i
¢.
– For example, ¡ _{j}
j/2
¢ ≈ 2^{j}p
2/(πj) .
• But all 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) ≡
µk − m k
¶
A_{min}(j, i) + ³ m k
´
A_{max}(j, i) for m = 0, 1, . . . , k.
m A_{min}(j,i)
A_{max}(j,i) A_{m}(j,i)
Approximation Algorithm for Asian Options (continued)
• Such “bucketing” introduces errors, but it works reasonably well in practice.^{a}
• A better alternative picks values whose logarithms are equally spaced.
• Still other alternatives are possible.
• Generally, k must scale with at least n to show convergence.^{b}
aHull and White (1993).
bDai, Huang, and Lyuu (2002).
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 running average
A_{u} ≡ (j + 1) a + S_{0}u^{j+1−i}d^{i}
j + 2 .
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).
• 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.
0 ... m ...
k
¾
.. 0
. `
` + 1 ...
k
.. 0
. `^{0}
`^{0} + 1 ...
k
Approximation Algorithm for Asian Options (continued)
• Obtain the approximate option value given the running average A_{u} via
C_{u} ≡ xC_{`}(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).
• This interpolation introduces the second source of error.
Approximation Algorithm for Asian Options (continued)
• 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^{−r∆t}.
• The running time is O(kn^{2}).
– There are O(n^{2}) nodes.
– Each node has O(k) buckets.
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 (see text).
• How about the Greeks?^{a}
aThanks to a lively class discussion on March 31, 2004.
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. 354 is 48.925.
• The maximum running average at node A in the same figure is 51.149.
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
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.
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.
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.
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.
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
Convergence Behavior of the Approximation Algorithm
^{a}60 80 100 120 140 n 0.325
0.33 0.335 0.34 0.345 0.35
Asian option value
aDai and Lyuu (2002).
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.
• There is a convergent approximation algorithm that does away with interpolation with a provable running time of 2^{O(}^{√}^{n )}.^{b}
aDai, Huang, and Lyuu (2002).
bDai and Lyuu (2002, 2004).
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}).
• In practice, loglinear interpolation works better.
aAingworth, Motwani, and Oldham (2000).
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, Huang, and Lyuu (2002).
bHsu and Lyuu (2004).
Remarks on Asian Option Pricing (concluded)
• The basic idea is a nonuniform allocation of running averages instead of a uniform k.
• It strikes a 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
A Grand Comparison
^{a}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
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