Barrier Options

Other Examples

• Subordinated debts as bull call spreads.

• Warrants as calls.

• Callable bonds as American calls with 2 strike prices.

• Convertible bonds.

Barrier Options

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

• A knock-out 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 finance student told me on March 26, 2004, that she did not understand why I covered barrier options until she started working

H

Time Price

S Barrier hit

Barrier Options (concluded)

• A knock-in 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 Formula for Down-and-In Calls

• Assume X ≥ H.

• The value of a European down-and-in 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(H2/(SX))+(r−q+σ²/2) τ σ√

τ .

– λ ≡ (r − q + σ²/2)/σ².

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

A Formula for Up-and-In Puts

• Assume X ≤ H.

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

Xe^−rτ

µH S

¶_2λ−2

N (−x + σ√

τ ) − Se^−qτ

µH S

¶_2λ

N (−x).

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

aMerton (1973).

Are American Options Barrier Options?

• 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.

• On the other hand, the barrier in a barrier option is given a priori.

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

Interesting Observations

• Assume H < X.

• Replace S in the pricing formula for the down-and-in call, Eq. (28) on p. 322, with H²/S.

• Equation (28) becomes Eq. (25) on p. 274 when r − q = σ²/2.

• Equation (28) becomes S/H times Eq. (25) on p. 274 when r − q = 0.

• Why?

Binomial Tree Algorithms

• Barrier options can be priced by binomial tree algorithms.

• Below is for the down-and-out 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.

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.

• In fact, the binomial tree is O(1/√

n) convergent.^a

• Solutions will be presented later.

aLin (R95221010) (2008).

100 150 200 250 300 350 400

#Periods 3

3.5 4 4.5 5 5.5

Down-and-in call value

Daily Monitoring

• Almost all 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 saves time or space?^a

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

A Heptanomial Tree (6 Periods Per 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. 274 apply with the dividend yield equal to ˆr:

C = Se^−ˆrτN (x) − Xe^−rτN (x − σ√

τ ), (29) P = Xe^−rτN (−x + σ√

τ ) − Se^−ˆrτN (−x).

(29⁰) – Above,

x ≡ ln(S/X) + (r − ˆr + σ²/2) 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

Path-Dependent Derivatives

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

• S₀ 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.

Path-Dependent Derivatives (continued)

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

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

max

à 1

n + 1

Xn i=0

S_i − X, 0

! .

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

Ã

X − 1

n + 1

Xn i=0

S_i, 0

! .

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.

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).

Path-Dependent 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 fixed-strike 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 average-strike options.

Average-Rate Options

• Average-rate options are notoriously hard to price.

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

• A straightforward algorithm enumerates the 2ⁿ price paths for an n-period 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.^b

aA “sufficient statistic,” if you will.

bWhat if the average is a moving average?

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 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

³

(S₀S₁ · · · S_n)^1/(n+1) − X, 0

´ , max

³

X − (S₀S₁ · · · S_n)^1/(n+1), 0

´ .

Pricing Some Path-Dependent Options (concluded)

• The limiting analytical solutions are the Black-Scholes formulas.

– With the volatility set to σ_a ≡ σ/√ 3 .

– With the dividend yield set to q_a ≡ (r + q + σ²/6)/2.

• The formula is therefore

C = Se^−qaτN (x) − Xe^−rτN (x − σ_a√

τ ), (30) P = Xe^−rτN (−x + σ_a√

τ ) − Se^−qaτN (−x),

(30⁰) – x ≡ ^ln(S/X)+(^r−q_√ ^a+σa²/2)^τ .

An Approximate Formula for Asian Calls

C = e^−rτ

"

S τ

Z _τ

0

e^µt+σ2t/2N

Ã−γ + (σt/τ )(τ − t/2) pτ /3

! dt

−XN

à −γ pτ /3

! # , where

• µ ≡ r − σ²/2.

• γ is the unique value that satisfies S

τ

Z _τ

0

e3γσt(τ −t/2)/τ²+µt+σ²[ t−(3t²/τ³)(τ −t/2)²]/2 dt = X.

Approximation Algorithm for Asian Options

• Based on the BOPM.

• Consider a node at time j with the underlying asset price equal to S₀u^j−idⁱ.

• Name such a node N (j, i).

• The running sum P_j

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

S₀(1 +

z }|j {

u + u² + · · · + u^j−i + u^j−id + · · · + u^j−idⁱ)

= S₀ 1 − u^j−i+1

+ S₀u^j−id 1 − dⁱ .

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

S₀(1 +

z }|j {

d + d² + · · · + dⁱ + dⁱu + · · · + dⁱu^j−i)

= S₀ 1 − dⁱ⁺¹

1 − d + S₀dⁱu 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^jp

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)

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).

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 S₀u^j+1−idⁱ, we seek the option value corresponding to the running average

A_u ≡ (j + 1) a + S₀u^j+1−idⁱ

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_{`</sub>(j + 1, i) ≤ A<sub>u</sub> ≤ A<sub>`+1}(j + 1, i).

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

A_u = xA_{`</sub>(j + 1, i) + (1 − x) A<sub>`+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_{`</sub>(j + 1, i) + (1 − x) C<sub>`+1}(j + 1, i).

– C_{`</sub>(t, s) denotes the option value at node N (t, s) with running average A<sub>`}(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(kn²).

– There are O(n²) nodes.

– Each node has O(k) buckets.

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 (see text).

• How about the Greeks?^a

aThanks to a lively class discussion on March 31, 2004.

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. 361 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

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

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

60 80 100 120 140 n 0.325

0.33 0.335 0.34 0.345 0.35

Asian option value

aDai (R86526008, D8852600) 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 (R86526008, D8852600), Huang (F83506075), and Lyuu (2002).

bDai (R86526008, D8852600) and Lyuu (2002, 2004).

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).

• In practice, log-linear 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(n²)-time algorithm with an error bound of O(1/n) exists.^b

aDai (R86526008, D8852600), Huang (F83506075), and Lyuu (2002).

bHsu (R7526001) 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

i 15 j

0 20 40

k

0 5

10

i 15

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

aHsu (R7526001) and Lyuu (2004); Zhang (2001,2003); Chen (R92723061) and Lyuu (2006).

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

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

Forwards, Futures, Futures Options, Swaps

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

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 (all measured in years).

• The spot price S, the spot price at maturity is S_T.

• The delivery price is X.

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.

Forward Contracts

• Forward contracts are for the delivery 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.

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

• The processor knows the cost of corn in advance.

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.

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.

Interest Rate Parity

F

S = e^r`−rf. (31)

• 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

Interest Rate Parity (concluded)

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

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

• If F e^rf/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^rf/S < e^r`, an arbitrage profit can result from

borrowing money in the foreign market and lending it in