Approximation Algorithm for Asian Options (continued)

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 new 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_(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, i)

[ A_max(j, i) − A_min(j, i) ]/k

 .

0 ... m ...

k



.. 0

. 

 + 1 ...

k

.. 0

. ^

^ + 1 ...

k

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 A₀(j + 1, i) = · · · = A_k(j + 1, i).

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(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^−rΔt.

• The running time is O(kn²).

– There are O(n²) nodes.

– Each node has O(k) buckets.

Approximation Algorithm for Asian Options (continued)

• For the calculations from time step n − 1, no interpolation is needed.^a

– 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.

aContributed by Mr. Chen, Shih-Hang (R02723031) on April 9, 2014.

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

• 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.

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. 416 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 computed similarly.

Convergence Behavior of the Approximation Algorithm with k = 50000

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) 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 at least.

• There is a convergent approximation algorithm that does away with interpolation with a running time of^b

2^{O(√n )}.

aDai (B82506025, R86526008, D8852600), Huang (F83506075), and Lyuu (2002).

bDai (B82506025, 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).

aAingworth, Motwani (1962–2009), 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 (B82506025, R86526008, D8852600), Huang (F83506075), and Lyuu (2002).

bHsu (R7526001, D89922012) 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 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

A Grand Comparison

aHsu (R7526001, D89922012) 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

110 5.7301225 5.7296982 5.7303567 5.730357 5.728161

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 ST.

• 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.^a

• 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. (48)

• 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.

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

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 is the delivery price which makes the forward contract zero valued.

– That is,

f = 0 when X = F.

Forward Price (continued)

6 -

0 1 2 3 · · · n

ST − F

Forward Price (concluded)

• The delivery price cannot change because it is written in the contract.

• But the forward price may change after the contract comes into existence.

• So although the value of a forward contract, f, is 0 at the outset, it will fluctuate thereafter.

– This value is enhanced when the spot price climbs.

– It is depressed when the spot price declines.

• The forward price also varies with the maturity of the contract.

Forward Price: Underlying Pays No Income

Lemma 12 For a forward contract on an underlying asset providing no income,

F = Se^rτ. (49)

• If F > Se^rτ:

– Borrow S dollars for τ years.

– Buy the underlying asset.

– Short the forward contract with delivery price F .

Proof (concluded)

• At maturity:

– Deliver the asset for F .

– Use Se^rτ to repay the loan, leaving an arbitrage profit of

F − Se^rτ > 0.

• If F < Se^rτ, do the opposite.

Example

• r is the annualized 3-month riskless interest rate.

• S is the spot price of the 6-month zero-coupon bond.

• A new 3-month forward contract on a 6-month

zero-coupon bond should command a delivery price of Se^r/4.

• So if r = 6% and S = 970.87, then the delivery price is 970.87 × e^0.06/4 = 985.54.

Contract Value: The Underlying Pays No Income

The value of a forward contract is

f = S − Xe^−rτ. (50)

• Consider a portfolio consisting of:

– One long forward contract;

– Cash amount Xe^−rτ;

– One short position in the underlying asset.

Contract Value: The Underlying Pays No Income (concluded)

• The cash will grow to X at maturity, which can be used to take delivery of the forward contract.

• The delivered asset will then close out the short position.

• Since the value of the portfolio is zero at maturity, its PV must be zero.^a

aRecall p. 200.

Lemma 12 (p. 442) Revisited

• Set f = 0 in Eq. (50) on p. 445.

• Then X = Se^rτ, the forward price.

Forward Price: Underlying Pays Predictable Income

Lemma 13 For a forward contract on an underlying asset providing a predictable income with a PV of I,

F = (S − I) e^rτ. (51)

• If F > (S − I) e^rτ, borrow S dollars for τ years, buy the underlying asset, and short the forward contract with delivery price F .

• At maturity, the asset is delivered for F , and (S − I) e^rτ is used to repay the loan, leaving an arbitrage profit of F − (S − I) e^rτ > 0.

Example

• Consider a 10-month forward contract on a $50 stock.

• The stock pays a dividend of $1 every 3 months.

• The forward price is



50 − e^−r3/4 − e^−r6/2 − e^−3×r9/4

e^r10×(10/12). – r_i is the annualized i-month interest rate.

Underlying Pays a Continuous Dividend Yield of q

• The value of a forward contract at any time prior to T is^a

f = Se^−qτ − Xe^−rτ. (52)

• One consequence of Eq. (52) is that the forward price is

F = Se^{(r−q) τ}. (53)

aSee text for proof.

Futures Contracts vs. Forward Contracts

• They are traded on a central exchange.

• A clearinghouse.

– Credit risk is minimized.

• Futures contracts are standardized instruments.

• Gains and losses are marked to market daily.

– Adjusted at the end of each trading day based on the settlement price.

Size of a Futures Contract

• The amount of the underlying asset to be delivered under the contract.

– 5,000 bushels for the corn futures on the CBT.

– One million U.S. dollars for the Eurodollar futures on the CME.

• A position can be closed out (or offset) by entering into a reversing trade to the original one.

• Most futures contracts are closed out in this way rather than have the underlying asset delivered.

– Forward contracts are meant for delivery.

Daily Settlements

• Price changes in the futures contract are settled daily.

• Hence the spot price rather than the initial futures price is paid on the delivery date.

• Marking to market nullifies any financial incentive for not making delivery.

– A farmer enters into a forward contract to sell a food processor 100,000 bushels of corn at $2.00 per bushel in November.

– Suppose the price of corn rises to $2.5 by November.

Daily Settlements (concluded)

• (continued)

– The farmer has incentive to sell his harvest in the spot market at $2.5.

– With marking to market, the farmer has transferred

$0.5 per bushel from his futures account to that of the food processor by November (see p. 455).

– When the farmer makes delivery, he is paid the spot price, $2.5 per bushel.

– The farmer has little incentive to default.

– The net price remains $2.00 per bushel, the original

Daily Cash Flows

• Let F_i denote the futures price at the end of day i.

• The contract’s cash flow on day i is Fi − Fi−1.

• The net cash flow over the life of the contract is

(F₁ − F₀) + (F₂ − F₁) + · · · + (Fn − Fn−1)

= F_n − F₀ = S_T − F₀.

• A futures contract has the same accumulated payoff S_T − F₀ as a forward contract.

• The actual payoff may vary because of the reinvestment of daily cash flows and how S_T − F₀ is distributed.

Daily Cash Flows (concluded)

6 -

? ?

6

0 1 2 3 · · · n

F1 − F0 F2 − F1 F3 − F2 · · · Fn − Fn−1

Delivery and Hedging

• Delivery ties the futures price to the spot price.

– Futures price is the delivery price that makes the futures contract zero-valued.

• On the delivery date, the settlement price of the futures contract is determined by the spot price.

• Hence, when the delivery period is reached, the futures price should be very close to the spot price.^a

• Changes in futures prices usually track those in spot price, making hedging possible.

aBut since early 2006, futures for corn, wheat and soybeans occasion- ally expired at a price much higher than that day’s spot price.

Forward and Futures Prices

• Surprisingly, futures price equals forward price if interest rates are nonstochastic!

– See text for proof.

• This result “justifies” treating a futures contract as if it were a forward contract, ignoring its marking-to-market feature.

aCox, Ingersoll, and Ross (1981).

Remarks

• When interest rates are stochastic, forward and futures prices are no longer theoretically identical.

– Suppose interest rates are uncertain and futures prices move in the same direction as interest rates.

– Then futures prices will exceed forward prices.

• For short-term contracts, the differences tend to be small.

• Unless stated otherwise, assume forward and futures prices are identical.

Futures Options

• The underlying of a futures option is a futures contract.

• Upon exercise, the option holder takes a position in the futures contract with a futures price equal to the

option’s strike price.

– A call holder acquires a long futures position.

– A put holder acquires a short futures position.

• The futures contract is then marked to market.

• And the futures position of the two parties will be at the prevailing futures price (thus zero-valued).

Futures Options (concluded)

• It works as if the call holder received a futures contract plus cash equivalent to the prevailing futures price Ft

minus the strike price X:

F_t − X.

– This futures contract has zero value.

• It works as if the put holder sold a futures contract for X − Ft

dollars.

Forward Options

• Similar to futures options except that what is delivered is a forward contract with a delivery price equal to the option’s strike price.

– Exercising a call forward option results in a long position in a forward contract.

– Exercising a put forward option results in a short position in a forward contract.

• Exercising a forward option incurs no immediate cash flows.

Example

• Consider a call with strike $100 and an expiration date in September.

• The underlying asset is a forward contract with a delivery date in December.

• Suppose the forward price in July is $110.

• Upon exercise, the call holder receives a forward contract with a delivery price of $100.

• If an offsetting position is then taken in the forward market,^a a $10 profit in December will be assured.

• A call on the futures would realize the $10 profit in July.

aThe counterparty will pay you $110 for the underlying asset.

Some Pricing Relations

• Let delivery take place at time T , the current time be 0, and the option on the futures or forward contract have expiration date t (t ≤ T ).

• Assume a constant, positive interest rate.

• Although forward price equals futures price, a forward option does not have the same value as a futures option.

• The payoffs of calls at time t are, respectively,

futures option = max(F_t − X, 0), (55) forward option = max(F_t − X, 0) e^{−r(T −t)}. (56)

Some Pricing Relations (concluded)

• A European futures option is worth the same as the corresponding European option on the underlying asset if the futures contract has the same maturity as the options.

– Futures price equals spot price at maturity.

– This conclusion is independent of the model for the spot price.

Put-Call Parity

The put-call parity is slightly different from the one in Eq. (26) on p. 208.

Theorem 14 (1) For European options on futures contracts,

C = P − (X − F ) e^−rt.

(2) For European options on forward contracts, C = P − (X − F ) e^−rT.

• See Theorem 12.4.4 of the textbook for proof.

Early Exercise

The early exercise feature is not valuable for forward options.

Theorem 15 American forward options should not be

exercised before expiration as long as the probability of their ending up out of the money is positive.

• See Theorem 12.4.5 of the textbook for proof.

Early exercise may be optimal for American futures options even if the underlying asset generates no payouts.

Theorem 16 American futures options may be exercised optimally before expiration.

Black’s Model

• Formulas for European futures options:

C = F e^−rtN (x) − Xe^−rtN (x − σ√

t), (57) P = Xe^−rtN (−x + σ√

t) − F e^−rtN (−x), where x ≡ ^ln(F/X)+(σ_σ^√_t ^{2/2) t}.

• Formulas (57) are related to those for options on a stock paying a continuous dividend yield.

• They are exactly Eqs. (37) on p. 311 with q set to r and S replaced by F .

a

Black Model (concluded)

• This observation incidentally proves Theorem 16 (p. 467).

• For European forward options, just multiply the above formulas by e^{−r(T −t)}.

– Forward options differ from futures options by a factor of e^{−r(T −t)}.^a

aRecall Eqs. (55)–(56) on p. 464.

Binomial Model for Forward and Futures Options

• Futures price behaves like a stock paying a continuous dividend yield of r.

– The futures price at time 0 is (p. 442) F = Se^rT.

– From Lemma 10 (p. 282), the expected value of S at time Δt in a risk-neutral economy is

Se^rΔt.

– So the expected futures price at time Δt is

Binomial Model for Forward and Futures Options (continued)

• The above observation continues to hold even if S pays a dividend yield!^a

– By Eq. (53) on p. 450, the futures price at time 0 is F = Se^{(r−q) T}.

– From Lemma 10 (p. 282), the expected value of S at time Δt in a risk-neutral economy is

Se^{(r−q) Δt}.

– So the expected futures price at time Δt is Se^{(r−q) Δt}e(r−q)(T −Δt) = Se^{(r−q) T} = F.

Binomial Model for Forward and Futures Options (concluded)

• Now, under the BOPM, the risk-neutral probability for the futures price is

p_f ≡ (1 − d)/(u − d) by Eq. (38) on p. 313.

– The futures price moves from F to F u with

probability p_f and to F d with probability 1 − p_f. – Note that the original u and d are used!

• The binomial tree algorithm for forward options is

Spot and Futures Prices under BOPM

• The futures price is related to the spot price via F = Se^rT

if the underlying asset pays no dividends.

• Recall the futures price F moves to F u with probability p_f per period.

• So the stock price moves from S = F e^−rT to F ue^{−r(T −Δt)} = Sue^rΔt

with probability p_f per period.

Spot and Futures Prices under BOPM (concluded)

• Similarly, the stock price moves from S = F e^−rT to Sde^rΔt

with probability 1 − p_f per period.

• Note that

S(ue^rΔt)(de^rΔt) = Se^2rΔt = S.

• So this binomial model is not the CRR tree.

• This model may not be suitable for pricing barrier options (why?).

Negative Probabilities Revisited

• As 0 < pf < 1, we have 0 < 1 − p_f < 1 as well.

• The problem of negative risk-neutral probabilities is solved:

– Build the tree for the futures price F of the futures contract expiring at the same time as the option.

– Let the stock pay a continuous dividend yield of q.

– By Eq. (53) on p. 450, calculate S from F at each node via

S = F e−(r−q)(T −t).