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# Finite Diﬀerence Fails for Numerical Gamma

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### Numerical Greeks

• Needed when closed-form formulas do not exist.

• Take delta as an example.

• A standard method computes the ﬁnite diﬀerence, f (S + ΔS) − f (S − ΔS)

2ΔS .

• The computation time roughly doubles that for evaluating the derivative security itself.

(2)

### An Alternative Numerical Delta

a

• Use intermediate results of the binomial tree algorithm.

• When the algorithm reaches the end of the ﬁrst period, fu and fd are computed.

• These values correspond to derivative values at stock prices Su and Sd, respectively.

• Delta is approximated by

fu − fd Su − Sd.

• Almost zero extra computational eﬀort.

a

(3)

S/(ud)

S/d

S/u

Su/d

S

Sd/u

Su

Sd Suu/d

Sdd/u

Suuu/d

Suu

S

Sdd

Sddd/u

(4)

### Numerical Gamma

• At the stock price (Suu + Sud)/2, delta is approximately (fuu − fud)/(Suu − Sud).

• At the stock price (Sud + Sdd)/2, delta is approximately (fud − fdd)/(Sud − Sdd).

• Gamma is the rate of change in deltas between (Suu + Sud)/2 and (Sud + Sdd)/2, that is,

fuu−fud

Suu−Sud Sud−Sddfud−fdd (Suu − Sdd)/2 .

• Alternative formulas exist (p. 601).

(5)

### Finite Diﬀerence Fails for Numerical Gamma

• Numerical diﬀerentiation gives

f (S + ΔS) − 2f (S) + f (S − ΔS)

(ΔS)2 .

• It does not work (see text for the reason).

• In general, calculating gamma is a hard problem numerically.

• But why did the binomial tree version work?

(6)

### Other Numerical Greeks

• The theta can be computed as fud − f

2(τ /n) .

– In fact, the theta of a European option can be derived from delta and gamma (p. 600).

• For vega and rho, there seems no alternative but to run the binomial tree algorithm twice.a

aBut see pp. 959ﬀ.

(7)

### Extensions of Options Theory

(8)

As I never learnt mathematics, so I have had to think.

— Joan Robinson (1903–1983)

(9)

### Pricing Corporate Securities

a

• Interpret the underlying asset as the total value of the ﬁrm.

• The option pricing methodology can be applied to pricing 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.

(10)

### Risky Zero-Coupon Bonds and Stock

• Consider XYZ.com.

• Capital structure:

– n shares of its own common stock, S.

– Zero-coupon 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?

(11)

### Risky Zero-Coupon 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

(12)

### Risky Zero-Coupon 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 calla 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 .

aSee p. 183.

(13)

### Risky Zero-Coupon Bonds and Stock (continued)

• Thus

nS = C,

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

(14)

### Risky Zero-Coupon Bonds and Stock (continued)

• From Theorem 11 (p. 284) and the put-call parity,a nS = V N (x) − Xe−rτN (x − σ√

τ ), (43) B = V N (−x) + Xe−rτN (x − σ√

τ ). (44) – Above,

x ≡ ln(V /X) + (r + σ2/2)τ σ√

τ .

• The continuously compounded yield to maturity of the ﬁrm’s bond is

ln(X/B)

τ .

a

(15)

### Risky Zero-Coupon 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−rτ/V .

– z ≡ (ln ω)/(σ

τ ) + (1/2) σ√

τ = −x + σ√ τ . – Note that ω is the debt-to-total-value ratio.

(16)

### Risky Zero-Coupon 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 then remaining ﬁrm value.

• Then Eqs. (43)–(44) on p. 351 become, respectively, nS = V e−qτN (x) − Xe−rτN (x − σ√

τ ),

B = (1 − α)V e−qτN (−x) + Xe−rτN (x − σ√ τ ).

– Above,

x ≡ ln(V /X) + (r − q + σ2/2)τ σ√

τ .

(17)

### 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 zero-coupon 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.

(18)

—Call— —Put—

Option Strike Exp. Vol. Last Vol. Last Merck 30 Jul 328 151/4 . . . . . .

441/2 35 Jul 150 91/2 10 1/16 441/2 40 Apr 887 43/4 136 1/16 441/2 40 Jul 220 51/2 297 1/4

441/2 40 Oct 58 6 10 1/2

441/2 45 Apr 3050 7/8 100 11/8 441/2 45 May 462 13/8 50 13/8

441/2 45 Jul 883 115/16 147 13/4

441/2 45 Oct 367 23/4 188 21/16

(19)

### 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.

(20)

### A Numerical Example (continued)

• The XYZ.com bonds are equivalent to a default-free zero-coupon bond with \$X par value plus n written European puts on Merck at a strike price of \$30.

– By the put-call parity.a

• The diﬀerence between B and the price of the default-free 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. 208.

(21)

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

(22)

### 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 \$115/16, 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 debt-equity ratio increases.

(23)

### 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, dividend, and strike price are under partial control of the stockholders.

(24)

### 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. 358 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.

(25)

### 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 and Miller (1972).

(26)

### A Numerical Example (continued)

• Suppose the stockholders sell (1/3) × n Merck shares to fund a \$14,833.3 cash dividend.

• They 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 total strike price of \$45,000.

• n such calls are worth \$1,937.5 (p. 358).

• So the total market value of the XYZ.com stock is (2/3) × 1, 937.5 = 1, 291.67 dollars.

(27)

### 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.

(28)

### Further Topics

• Other Examples:

– Subordinated debts as bull call spreads.

– Warrants as calls.

– Callable bonds as American calls with 2 strike prices.

– Convertible bonds.

• Securities with a complex liability structure must be solved by trees.a

aDai (B82506025, R86526008, D8852600), Lyuu, and Wang (F95922018) (2010).

(29)

### Barrier Options

a

• 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 MBA student in ﬁnance told me on March 26, 2004, that she did not understand why I covered barrier options until she started

(30)

H

Time Price

S Barrier hit

(31)

### 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

a

(32)

### A Formula for Down-and-In Calls

a

• Assume X ≥ H.

• The value of a European down-and-in call on a stock paying a dividend yield of q is

Se−qτ

H S



N(x) − Xe−rτ

H S

2λ−2

N(x − σ τ),

(45)

– 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).

aMerton (1973).

(33)

### A Formula for Up-and-In Puts

a

• Assume X ≤ H.

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

Xe−rτ

H S

2λ−2

N(−x + σ

τ) − Se−qτ

H S



N(−x).

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

aMerton (1973).

(34)

### 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 derived during backward induction.

• But the barrier in a barrier option is given a priori.

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

(35)

### Interesting Observations

• Assume H < X.

• Replace S in the pricing formula Eq. (37) on p. 311 for the call with H2/S.

• Equation (45) on p. 369 for the down-and-in call becomes Eq. (37) when r − q = σ2/2.

• Equation (45) becomes S/H times Eq. (37) when r − q = 0.

(36)

### Interesting Observations (concluded)

• Replace S in the pricing formula for the down-and-in call, Eq. (45), with H2/S.

• Equation (45) becomes Eq. (37) when r − q = σ2/2.

• Equation (45) becomes H/S times Eq. (37) when r − q = 0.a

• Why?

aContributed by Mr. Chou, Ming-Hsin (R02723073) on April 24, 2014.

(37)

### Binomial Tree Algorithms

• Barrier options can be priced by binomial tree algorithms.

• Below is for the down-and-out option.

0 H

(38)

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.

(39)

### 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 to a node price.

– This “eﬀective barrier” changes as n increases.

• In fact, the binomial tree is O(1/√

n) convergent.b

• Solutions will be presented later.

aBoyle and Lau (1994).

bLin (R95221010) (2008).

(40)

### Binomial Tree Algorithms (concluded)

a

100 150 200 250 300 350 400

#Periods 3

3.5 4 4.5 5 5.5

Down-and-in call value

aLyuu (1998).

(41)

### 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 save time or space?a

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

(42)

-

 1 day

(43)

### 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) is equal to x units of currency B

(44)

### 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.

(45)

### 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. 778).

– 0.7% were Asian options (see p. 389).

(46)

### 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.

(47)

### 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

(48)

### Foreign Exchange Options (concluded)

• Assume the exchange rate S is lognormally distributed.

• The formulas derived for stock index options in Eqs. (37) on p. 311 apply with the dividend yield equal to ˆr:

C = Se−ˆrτN (x) − Xe−rτN (x − σ√

τ ), (46) P = Xe−rτN (−x + σ√

τ ) − Se−ˆrτN (−x).

(46) – Above,

x ≡ ln(S/X) + (r − ˆr + σ2/2) τ σ√

τ .

(49)

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

(50)

### Path-Dependent Derivatives

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

• S0 is the known price at time zero.

• Sn is the price at expiration.

• The standard European call has a terminal value depending only on the last price, max(Sn − X, 0).

• Its value thus depends only on the underlying asset’s terminal price regardless of how it gets there.

(51)

### 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

 1 n + 1

n i=0

Si − X, 0

 .

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

max



X − 1

n + 1

n i=0

Si, 0

 .

(52)

### 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.

• 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 and Sandmann, 2003).

(53)

### Path-Dependent Derivatives (continued)

• A lookback call option on the minimum has a terminal payoﬀ of

Sn − min

0≤i≤nSi.

• A lookback put on the maximum has a terminal payoﬀ of

0≤i≤nmax Si − Sn.

(54)

### Path-Dependent Derivatives (concluded)

• The ﬁxed-strike lookback option provides a payoﬀ of – max(max0≤i≤n Si − X, 0) for the call.

– max(X − min0≤i≤n Si, 0) for the put.

• Lookback calls and puts on the average (instead of a constant X) are called average-strike options.

(55)

### Average-Rate Options

• Average-rate options are notoriously hard to price.

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

• A naive algorithm enumerates the 2n paths for an n-period binomial tree and then averages the payoﬀs.a

• But the complexity is exponential.

• The Monte Carlo methodb and approximation algorithms are some of the alternatives left.

aDai (B82506025, R86526008, D8852600) and Lyuu (2007) reduce it to 2O(n ).

(56)

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

(57)

### States and Their Transitions

• The tuple

(i, S, P )

captures the statea 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 (Kao (R89723057) and Lyuu, 2003).

(58)

### 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 exponential-time algorithm.

(59)

### 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



(S0S1 · · · Sn)1/(n+1) − X, 0 , max



X − (S0S1 · · · Sn)1/(n+1), 0

 .

(60)

### Pricing Some Path-Dependent Options (concluded)

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

C = Se−qaτN (x) − Xe−rτN (x − σa

τ ), (47) P = Xe−rτN (−x + σa

τ ) − Se−qaτN (−x),

(47) – With the volatility set to σa ≡ σ/√

3 .

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

– x ≡ ln(S/X)+(r−qaa2/2)τ σa

τ .

(61)

### An Approximate Formula for Asian Calls

a

C = e−rτ

S τ

τ

0 eμt+σ2t/2N

−γ + (σt/τ)(τ − t/2) τ /3

 dt

−XN

 −γ τ /3

 , where

• μ ≡ r − σ2/2.

• γ is the unique value that satisﬁes S

τ

τ

0 e3γσt(τ−t/2)/τ2+μt+σ2[ t−(3t23)(τ−t/2)2]/2 dt = X.

a

(62)

### Approximation Algorithm for Asian Options

• Based on the BOPM.

• Consider a node at time j with the underlying asset price equal to S0uj−idi.

• Name such a node N(j, i).

• The running sum j

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

S0(1 +

 j 

u + u2 + · · · + uj−i + uj−id + · · · + uj−idi)

= S0 1 − uj−i+1

1 − u + S0uj−id 1 − di 1 − d .

(63)

Path with maximum running average

Path with minimum running average

N

(64)

### Approximation Algorithm for Asian Options (continued)

• Divide this value by j + 1 and call it Amax(j, i).

• Similarly, the running sum has a minimum value of

S0(1 +

 j 

d + d2 + · · · + di + diu + · · · + diuj−i)

= S0 1 − di+1

1 − d + S0diu 1 − uj−i 1 − u .

• Divide this value by j + 1 and call it Amin(j, i).

• Amin and Amax are running averages.

(65)

### 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



∼ 2j

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.

a

(66)

5 10 15 20

01000020000300004000050000

Number of Averages

n=24

(67)

### Approximation Algorithm for Asian Options (continued)

• But all averages must lie between Amin(j, i) and Amax(j, i).

• Pick k + 1 equally spaced values in this range and treat them as the true and only running averages:

Am(j, i) ≡

k − m k



Amin(j, i) +

m k



Amax(j, i) for m = 0, 1, . . . , k.

(68)

m Amin(j,i)

Amax(j,i) Am(j,i)

(69)

### 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. 403).

• Generally, k must scale with at least n to show convergence behavior.c

aHull and White (1993).

bCalled log-linear interpolation.

c B82506025, R86526008, D8852600), Huang (F83506075), and

• 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

• 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

• What is delivered is now 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

Lemma 2 An American call or a European call on a non-dividend-paying stock is never worth less than its intrinsic value.. • An American call cannot be worth less than its

• 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

• 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

• 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

• 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