Extensions of Options Theory
As I never learnt mathematics, so I have had to think.
— Joan Robinson (1903–1983)
Pricing Corporate Securities
^{a}• Interpret the underlying asset as the total value of the ﬁrm.
• The option pricing methodology can be applied to price corporate securities.
– The result is called the structural model.
• Assumptions:
– A ﬁrm can ﬁnance payouts by the sale of assets.
– If a promised payment to an obligation other than stock is missed, the claim holders take ownership of the ﬁrm and the stockholders get nothing.
aBlack & Scholes (1973); Merton (1974).
Risky ZeroCoupon Bonds and Stock
• Consider XYZ.com.
• Capital structure:
– n shares of its own common stock, S.
– Zerocoupon bonds with an aggregate par value of X.
• What is the value of the bonds, B?
• What is the value of the XYZ.com stock?
Risky ZeroCoupon Bonds and Stock (continued)
• On the bonds’ maturity date, suppose the total value of the ﬁrm V ^{∗} is less than the bondholders’ claim X.
• Then the ﬁrm declares bankruptcy, and the stock becomes worthless.
• If V ^{∗} > X, then the bondholders obtain X and the stockholders V ^{∗} − X.
V ^{∗} ≤ X V ^{∗} > X
Bonds V ^{∗} X
Stock 0 V ^{∗} − X
Risky ZeroCoupon Bonds and Stock (continued)
• The stock has the same payoﬀ as a call!
• It is a call on the total value of the ﬁrm with a strike price of X and an expiration date equal to the bonds’.
– This call provides the limited liability for the stockholders.
• The bonds are a covered call^{a} on the total value of the ﬁrm.
• Let V stand for the total value of the ﬁrm.
• Let C stand for a call on V .
aRecall p. 201.
Risky ZeroCoupon Bonds and Stock (continued)
• Thus
nS = C (market capitalization of XYZ.com), B = V − C.
• Knowing C amounts to knowing how the value of the ﬁrm is divided between stockholders and bondholders.
• Whatever the value of C, the total value of the stock and bonds at maturity remains V ^{∗}.
• The relative size of debt and equity is irrelevant to the ﬁrm’s current value V .
Risky ZeroCoupon Bonds and Stock (continued)
• From Theorem 10 (p. 296) and the putcall parity,^{a} nS = V N (x) − Xe^{−rτ}N (x − σ√
τ ), (51) B = V N (−x) + Xe^{−rτ}N (x − σ√
τ ). (52) – Above,
x =^{Δ} ln(V /X) + (r + σ^{2}/2)τ σ√
τ .
• The continuously compounded yield to maturity of the ﬁrm’s bond is
ln(X/B)
τ .
a
Risky ZeroCoupon Bonds and Stock (continued)
• Deﬁne the credit spread or default premium as the yield diﬀerence between risky and riskless bonds,
ln(X/B)
τ − r
= −1 τ ln
N (−z) + 1
ω N (z − σ√ τ )
. – ω = Xe^{Δ} ^{−rτ}/V .
– z = (ln ω)/(σ^{Δ} √
τ ) + (1/2) σ√
τ = −x + σ√ τ . – Note that ω is the debttototalvalue ratio.
Risky ZeroCoupon Bonds and Stock (concluded)
• In general, suppose the ﬁrm has a dividend yield at rate q and the bankruptcy costs are a constant proportion α of the remaining ﬁrm value.
• Then Eqs. (51)–(52) on p. 371 become, respectively, nS = V e^{−qτ}N (x) − Xe^{−rτ}N (x − σ√
τ ),
B = (1 − α)V e^{−qτ}N (−x) + Xe^{−rτ}N (x − σ√ τ ).
– Above,
x =^{Δ} ln(V /X) + (r − q + σ^{2}/2)τ σ√
τ .
A Numerical Example
• XYZ.com’s assets consist of 1,000 shares of Merck as of March 20, 1995.
– Merck’s market value per share is $44.5.
• XYZ.com’s securities consist of 1,000 shares of common stock and 30 zerocoupon bonds maturing on July 21, 1995.
• Each bond promises to pay $1,000 at maturity.
• n = 1, 000, V = 44.5 × n = 44, 500, and X = 30 × 1, 000 = 30, 000.
—Call— —Put—
Option Strike Exp. Vol. Last Vol. Last Merck 30 Jul 328 151/4 . . . . . .
44^{1/2} 35 Jul 150 9^{1/2} 10 ^{1/16}
441/2 40 Apr 887 43/4 136 1/16
441/2 40 Jul 220 51/2 297 1/4
44^{1/2} 40 Oct 58 6 10 ^{1/2}
44^{1/2} 45 Apr 3050 ^{7/8} 100 1^{1/8} 441/2 45 May 462 13/8 50 13/8
44^{1/2} 45 Jul 883 1^{15/16} 147 1^{3/4}
44^{1/2} 45 Oct 367 2^{3/4} 188 2^{1/16}
A Numerical Example (continued)
• The Merck option relevant for pricing is the July call with a strike price of X/n = 30 dollars.
• Such a call is selling for $15.25.
• So XYZ.com’s stock is worth 15.25 × n = 15, 250 dollars.
• The entire bond issue is worth
B = 44, 500 − 15, 250 = 29, 250 dollars.
– Or $975 per bond.
A Numerical Example (continued)
• The XYZ.com bonds are equivalent to a defaultfree zerocoupon bond with $X par value plus n written European puts on Merck at a strike price of $30.
– By the putcall parity.^{a}
• The diﬀerence between B and the price of the defaultfree bond is the value of these puts.
• The next table shows the total market values of the XYZ.com stock and bonds under various debt amounts X.
aSee p. 225.
Promised payment Current market Current market Current total to bondholders value of bonds value of stock value of firm
X B nS V
30,000 29,250.0 15,250.0 44,500
35,000 35,000.0 9,500.0 44,500
40,000 39,000.0 5,500.0 44,500
45,000 42,562.5 1,937.5 44,500
A Numerical Example (continued)
• Suppose the promised payment to bondholders is
$45,000.
• Then the relevant option is the July call with a strike price of 45, 000/n = 45 dollars.
• Since that option is selling for $1^{15/16}, the market value of the XYZ.com stock is (1 + 15/16) × n = 1, 937.5
dollars.
• The market value of the stock decreases as the debtequity ratio increases.
A Numerical Example (continued)
• There are conﬂicts between stockholders and bondholders.
• An option’s terms cannot be changed after issuance.
• But a ﬁrm can change its capital structure.
• There lies one key diﬀerence between options and corporate securities.
– Parameters such volatility,^{a} dividend, and strike price are under partial control of the stockholders or their boards.
aThis is called the asset substitution problem (Myers, 1977).
A Numerical Example (continued)
• Suppose XYZ.com issues 15 more bonds with the same terms to buy back stock.
• The total debt is now X = 45,000 dollars.
• The table on p. 378 says the total market value of the bonds should be $42,562.5.
• The new bondholders pay
42, 562.5 × (15/45) = 14, 187.5 dollars.
• The remaining stock is worth $1,937.5.
A Numerical Example (continued)
• The stockholders therefore gain
14, 187.5 + 1, 937.5 − 15, 250 = 875 dollars.
• The original bondholders lose an equal amount, 29, 250 − 30
45 × 42, 562.5 = 875.
– This is called claim dilution.^{a}
aFama & M. H. Miller (1972).
A Numerical Example (continued)
• Suppose the stockholders sell (1/3) × n Merck shares to fund a $14,833.3 cash dividend.
• The stockholders now have $14,833.3 in cash plus a call on (2/3) × n Merck shares.
• The strike price remains X = 30, 000.
• This is equivalent to owning 2/3 of a call on n Merck shares with a strike price of $45,000.
• n such calls are worth $1,937.5 (p. 378).
• So the total market value of the XYZ.com stock is (2/3) × 1, 937.5 = 1, 291.67 dollars.
A Numerical Example (concluded)
• The market value of the XYZ.com bonds is hence (2/3) × n × 44.5 − 1, 291.67 = 28, 375 dollars.
• Hence the stockholders gain
14, 833.3 + 1, 291.67 − 15, 250 ≈ 875 dollars.
• The bondholders watch their value drop from $29,250 to
$28,375, a loss of $875.
Further Topics
• Other Examples:^{a}
– Stock as compound call when company issues coupon bonds.
– Subordinated debts as bull call spreads.
– Warrants as calls.
– Callable bonds as American calls with 2 strike prices.
– Convertible bonds.
– Bonds with safety covenants as barrier options.
aCox & Rubinstein (1985); Geske (1977).
Further Topics (concluded)
• Securities issued by ﬁrms with a complex capital structure must be solved by trees.^{a}
aDai (B82506025, R86526008, D8852600), Lyuu, & C. Wang (F95922018) (2010).
Distance to Default (DTD)
^{a}• Let μ be the total value V ’s rate of expected return.
• From Eq. (51), on p. 371, the probability of default τ years from now equals
N (−DTD), where
DTD =^{Δ} ln(V /X) + (μ − σ^{2}/2)τ σ√
τ .
• V/X is called the leverage ratio.
aMerton (1974).
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.
H
Time Price
S Barrier hit
Barrier Options (continued)
• 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).
Barrier Options (concluded)
• 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).
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),
(53)
– 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.
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).
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.
• In conrast, the barrier in a barrier option is given by a contract.^{b}
aContributed by Mr. Yang, JuiChung (D97723002) on March 25, 2009.
bCox & Rubinstein (1985).
Interesting Observations
• Assume H < X.
• Replace S in the Merton pricing formula Eq. (43) on p.
326 for the call with H^{2}/S.
– Equation (53) on p. 392 for the downandin call becomes Eq. (43) when r − q = σ^{2}/2.
– Equation (53) becomes S/H times Eq. (43) when r − q = 0.
Interesting Observations (concluded)
• Replace S in the pricing formula for the downandin call, Eq. (53), with H^{2}/S.
– Equation (53) becomes Eq. (43) when r − q = σ^{2}/2.
– Equation (53) becomes H/S times Eq. (43) when r − q = 0.^{a}
• Why?^{b}
aContributed by Mr. Chou, MingHsin (R02723073) on April 24, 2014.
bApply the reﬂection principle (p. 700), Eq. (42) on p. 289, and Lemma 9 (p. 294).
Binomial Tree Algorithms
• Barrier options can be priced by binomial tree algorithms.
• Below is for the downandout option.
0 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 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.
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) 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).
Binomial Tree Algorithms (concluded)
^{a}100 150 200 250 300 350 400
#Periods 3
3.5 4 4.5 5 5.5
Downandin call value
aLyuu (1998).
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
A Heptanomial Tree (6 Periods Per Day)

_{1 day}
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).
Data! data! data!
— Arthur Conan Doyle (1892), The Adventures of Sherlock Holmes
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).
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.
Time Series of the Daily Euro–USD Exchange Rate
Distribution of the Daily Euro–USD Exchange Rate
Time Series of the Minutely Euro–USD Exchange Rate
Distribution of the Minutely Euro–USD Exchange Rate
Time Series of the Daily GBP–USD Exchange Rate
Distribution of the Daily GBP–USD Exchange Rate
Distribution of the Minutely GBP–USD Exchange Rate
Distribution of the Daily JPY–USD Exchange Rate
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 digital options (see p. 849).
– 0.7% were Asian options (see p. 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.
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
Foreign Exchange Options (concluded)
• Assume the exchange rate S is lognormally distributed.
• The formulas derived for stock index options in Eqs. (43) on p. 326 apply with the dividend yield equal to ˆr:
C = Se^{−ˆrτ}N (x) − Xe^{−rτ}N (x − σ√
τ ), (54) P = Xe^{−rτ}N (−x + σ√
τ ) − Se^{−ˆrτ}N (−x).
(54^{}) – Above,
x =^{Δ} ln(S/X) + (r − ˆr + σ^{2}/2) τ σ√
τ .
Distribution of the Logarithmic Euro–USD Exchange Rate
Distribution of the Logarithmic GBP–USD Exchange Rate
Distribution of the Logarithmic GBP–USD Exchange Rate (after the Collapse of Lehman Brothers and before Brexit)
Distribution of the Logarithmic JPY–USD Exchange Rate
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
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
.
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).
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}.
PathDependent Derivatives (concluded)
• 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 (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 )}.
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}
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).
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.
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
.
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), (55) P = Xe^{−rτ}N(−x + σa
√τ) − Se^{−q}^{a}^{τ}N(−x),
(55^{})
– 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 .
aSee Angus (1999), for example.
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)
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 .
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 +
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 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.
5 10 15 20
01000020000300004000050000
Stock Price Number of Averages
n=24
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.
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.^{b}
• Still other alternatives are possible (considering the distribution of averages on p. 440).
aHull & White (1993); Ritchken, Sankarasubramanian, & Vijh (1993).
bCalled loglinear interpolation.
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 .
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) ≤ Au < 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
.
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 A_{0}(j + 1, i) = · · · = A_{k}(j + 1, i).
– It will happen along extreme paths!
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).
• 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(kn^{2}).
– There are O(n^{2}) nodes.
– Each node has O(k) buckets.
Approximation Algorithm for Asian Options (continued)
• For the calculations at 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, ShihHang (R02723031) on April 9, 2014.
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.
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. 453 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 computed similarly.
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).