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# Trading Days and Calendar Days

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### Time-Dependent Instantaneous Volatility

a

• Suppose the (instantaneous) volatility can change over time but otherwise predictable: σ(t) instead of σ.

• In the limit, the variance of ln(Sτ/S) is

 τ

0 σ2(t) dt rather than σ2τ .

• The annualized volatility to be used in the Black-Scholes formula should now be

 τ

0 σ2(t) dt

τ .

(2)

### Time-Dependent Instantaneous Volatility (concluded)

• There is no guarantee that the implied volatility is constant.

• For the binomial model,u and d depend on time:

u = eσ(t)

τ /n,

d = e−σ(t)

τ /n.

• How to make the binomial tree combine?a

aAmin (1991); Chen (R98922127) (2011).

(3)

### Time-Dependent Short Rates

• Suppose the short rate (i.e., the one-period spot rate) changes over time but otherwise predictable.

• The riskless rate r in the Black-Scholes formula should be the spot rate with a time to maturity equal to τ .

• In other words,

r =

n−1

i=0 ri

τ ,

where ri is the continuously compounded, short rate measured in periods for period i.

(4)

### Trading Days and Calendar Days

• Interest accrues based on the calendar day.

• But σ is usually calculated based on trading days only.

– Stock price seems to have lower volatilities when the exchange is closed.a

• How to harmonize these two diﬀerent times into the Black-Scholes formula and binomial tree algorithms?b

aFama (1965); French (1980); French and Roll (1986).

bRecall p. 145 about dating issues.

(5)

### Trading Days and Calendar Days (continued)

• Think of σ as measuring the annualized volatility of stock price one year from now.

• Suppose a year has m (say 253) trading days.

• We can replace σ in the Black-Scholes formula witha

σ

 365

m × number of trading days to expiration number of calendar days to expiration .

aFrench (1984).

(6)

### Trading Days and Calendar Days (concluded)

• This works only for European options.

• How about binomial tree algorithms?a

aContributed by Mr. Lu, Zheng-Liang (D00922011) in 2015.

(7)

### Options on a Stock That Pays Dividends

• Early exercise must be considered.

• Proportional dividend payout model is tractable (see text).

– The dividend amount is a constant proportion of the prevailing stock price.

• In general, the corporate dividend policy is a complex issue.

(8)

### Known Dividends

• Constant dividends introduce complications.

• Use D to denote the amount of the dividend.

• Suppose an ex-dividend date falls in the ﬁrst period.

• At the end of that period, the possible stock prices are Su − D and Sd − D.

• Follow the stock price one more period.

• The number of possible stock prices is not three but four: (Su − D) u, (Su − D) d, (Sd − D) u, (Sd − D) d.

– The binomial tree no longer combines.

(9)

(Su − D) u

 Su − D

 

(Su − D) d S

(Sd − D) u

 

Sd − D



(Sd − D) d

(10)

• Use the Black-Scholes formula with the stock price reduced by the PV of the dividends.a

• This essentially decomposes the stock price into a riskless one paying known dividends and a risky one.

• The riskless component at any time is the PV of future dividends during the life of the option.

– Then, σ is the volatility of the process followed by the risky component.

• The stock price, between two adjacent ex-dividend dates, follows the same lognormal distribution.

aRoll (1977).

(11)

• Start with the current stock price minus the PV of future dividends before expiration.

• Develop the binomial tree for the new stock price as if there were no dividends.

• Then add to each stock price on the tree the PV of all future dividends before expiration.

• American option prices can be computed as before on this tree of stock prices.

(12)

S − D/R

*

j

(S − D/R)u

*

j

(S − D/R)d

*

j

(S − D/R)u2

(S − D/R)ud

(S − D/R)d2

(13)

### The Ad-Hoc Approximation vs. P. 298 (Step 2)

(S − D/R) + D/R = S

*

j

(S − D/R)u

*

j

(S − D/R)d

*

(S − D/R)u2

(S − D/R)ud

(S − D/R)d2

(14)

### The Ad-Hoc Approximation vs. P. 298

a

• The trees are diﬀerent.

• The stock prices at maturity are also diﬀerent.

– (Su − D) u, (Su − D) d, (Sd − D) u, (Sd − D) d (p. 298).

– (S − D/R)u2, (S − D/R)ud, (S − D/R)d2 (ad hoc).

• Note that, as d < R < u,

(Su − D) u > (S − D/R)u2, (Sd − D) d < (S − D/R)d2,

aContributed by Mr. Yang, Jui-Chung (D97723002) on March 18, 2009.

(15)

### The Ad-Hoc Approximation vs. P. 298 (concluded)

• So the ad hoc approximation has a smaller dynamic range.

• This explains why in practice the volatility is usually increased when using the ad hoc approximation.

(16)

### A General Approach

a

• A new tree structure.

• No approximation assumptions are made.

• A mathematical proof that the tree can always be constructed.

• The actual performance is quadratic except in pathological cases (see pp. 686ﬀ).

• Other approaches include adjusting σ and approximating the known dividend with a dividend yield.

aDai (R86526008, D8852600) and Lyuu (2004).

(17)

### Continuous Dividend Yields

• Dividends are paid continuously.

– Approximates a broad-based stock market portfolio.

• The payment of a continuous dividend yield at rate q reduces the growth rate of the stock price by q.

– A stock that grows from S to Sτ with a continuous dividend yield of q would grow from S to Sτe without the dividends.

• A European option has the same value as one on a stock with price Se−qτ that pays no dividends.a

(18)

### Continuous Dividend Yields (continued)

• The Black-Scholes formulas hold with S replaced by Se−qτ:a

C = Se−qτN (x) − Xe−rτN (x − σ√

τ ), (29) P = Xe−rτN (−x + σ√

τ ) − Se−qτN (−x),

(29) where

x ≡ ln(S/X) + 

r − q + σ2/2 τ σ√

τ .

• Formulas (29) and (29) remain valid as long as the dividend yield is predictable.

aMerton (1973).

(19)

### Continuous Dividend Yields (continued)

• To run binomial tree algorithms, replace u with ue−qΔt and d with de−qΔt, where Δt ≡ τ /n.

– The reason: The stock price grows at an expected rate of r − q in a risk-neutral economy.

• Other than the changes, binomial tree algorithms stay the same.

– In particular, p should use the original u and d!a

aContributed by Ms. Wang, Chuan-Ju (F95922018) on May 2, 2007.

(20)

### Continuous Dividend Yields (concluded)

• Alternatively, pick the risk-neutral probability as e(r−q) Δt − d

u − d , (30)

where Δt ≡ τ /n.

– The reason: The stock price grows at an expected rate of r − q in a risk-neutral economy.

• The u and d remain unchanged.

• Other than the change in Eq. (30), binomial tree

algorithms stay the same as if there were no dividends.

(21)

### Sensitivity Analysis of Options

(22)

Cleopatra’s nose, had it been shorter, the whole face of the world would have been changed.

— Blaise Pascal (1623–1662)

(23)

### Sensitivity Measures (“The Greeks”)

• How the value of a security changes relative to changes in a given parameter is key to hedging.

– Duration, for instance.

• Let x ≡ ln(S/X)+(r+σ2/2) τ σ

τ (recall p. 280).

• Recall that

N(y) = e−y2/2

√2π > 0,

the density function of standard normal distribution.

(24)

### Delta

• Deﬁned as

Δ ∂f

∂S. – f is the price of the derivative.

– S is the price of the underlying asset.

• The delta of a portfolio of derivatives on the same underlying asset is the sum of their individual deltas.

– Elementary calculus.

• The delta used in the BOPM (p. 226) is the discrete analog.

(25)

### Delta (concluded)

• The delta of a European call on a non-dividend-paying stock equals

∂C

∂S = N (x) > 0.

• The delta of a European put equals

∂P

∂S = N (x) − 1 < 0.

• The delta of a long stock is apparently 1.

(26)

0 50 100 150 200 250 300 350 Time to expiration (days) 0

0.2 0.4 0.6 0.8 1

Delta (call)

0 50 100 150 200 250 300 350 Time to expiration (days) -1

-0.8 -0.6 -0.4 -0.2 0

Delta (put)

0 20 40 60 80

Stock price 0

0.2 0.4 0.6 0.8 1

Delta (call)

0 20 40 60 80

Stock price -1

-0.8 -0.6 -0.4 -0.2 0

Delta (put)

Dotted curve: in-the-money call or out-of-the-money put.

Solid curves: at-the-money options.

Dashed curves: out-of-the-money calls or in-the-money puts.

(27)

### Delta Neutrality

• A position with a total delta equal to 0 is delta-neutral.

– A delta-neutral portfolio is immune to small price changes in the underlying asset.

• Creating one serves for hedging purposes.

– A portfolio consisting of a call and −Δ shares of stock is delta-neutral.

– Short Δ shares of stock to hedge a long call.

– Long Δ shares of stock to hedge a short call.

• In general, hedge a position in a security with delta Δ1

(28)

### Theta (Time Decay)

• Deﬁned as the rate of change of a security’s value with respect to time, or Θ ≡ −∂f/∂τ = ∂f/∂t.

• For a European call on a non-dividend-paying stock, Θ = −SN(x) σ

2

τ − rXe−rτN (x − σ√

τ ) < 0.

– The call loses value with the passage of time.

• For a European put, Θ = −SN(x) σ

2

τ + rXe−rτN (−x + σ√ τ ).

– Can be negative or positive.

(29)

0 50 100 150 200 250 300 350 Time to expiration (days) -60

-50 -40 -30 -20 -10 0

Theta (call)

0 50 100 150 200 250 300 350 Time to expiration (days) -50

-40 -30 -20 -10 0

Theta (put)

0 20 40 60 80

Stock price -6

-5 -4 -3 -2 -1 0

Theta (call)

0 20 40 60 80

Stock price -2

-1 0 1 2 3

Theta (put)

Dotted curve: in-the-money call or out-of-the-money put.

(30)

### Gamma

• Deﬁned as the rate of change of its delta with respect to the price of the underlying asset, or Γ ≡ ∂2Π/∂S2.

• Measures how sensitive delta is to changes in the price of the underlying asset.

• In practice, a portfolio with a high gamma needs be rebalanced more often to maintain delta neutrality.

• Roughly, delta ∼ duration, and gamma ∼ convexity.

• The gamma of a European call or put on a non-dividend-paying stock is

N(x)/(Sσ√

τ ) > 0.

(31)

0 20 40 60 80 Stock price

0 0.01 0.02 0.03 0.04

Gamma (call/put)

0 50 100 150 200 250 300 350 Time to expiration (days) 0

0.1 0.2 0.3 0.4 0.5

Gamma (call/put)

Dotted lines: in-the-money call or out-of-the-money put.

Solid lines: at-the-money option.

Dashed lines: out-of-the-money call or in-the-money put.

(32)

a

### (Lambda, Kappa, Sigma)

• Deﬁned as the rate of change of its value with respect to the volatility of the underlying asset

Λ ∂f

∂σ.

• Volatility often changes over time.

• A security with a high vega is very sensitive to small changes or estimation error in volatility.

• The vega of a European call or put on a non-dividend-paying stock is S√

τ N(x) > 0.

– So higher volatility always increases the option value.

aVega is not Greek.

(33)

### Vega (concluded)

• Note that if S = X, τ → 0 implies Λ → 0

(which answers the question on p. 284 for the Black-Scholes model).

• The Black-Scholes formula (p. 280) implies

C → S,

P → Xe−rτ, as σ → ∞.

(34)

0 20 40 60 80 Stock price

0 2 4 6 8 10 12 14

Vega (call/put)

50 100 150 200 250 300 350 Time to expiration (days) 0

2.5 5 7.5 10 12.5 15 17.5

Vega (call/put)

Dotted curve: in-the-money call or out-of-the-money put.

Solid curves: at-the-money option.

Dashed curve: out-of-the-money call or in-the-money put.

(35)

### Rho

• Deﬁned as the rate of change in its value with respect to interest rates

ρ ≡ ∂f

∂r .

• The rho of a European call on a non-dividend-paying stock is

Xτ e−rτN (x − σ√

τ ) > 0.

• The rho of a European put on a non-dividend-paying stock is

−Xτe−rτN (−x + σ√

τ ) < 0.

(36)

50 100 150 200 250 300 350 Time to expiration (days) 0

5 10 15 20 25 30 35

Rho (call)

50 100 150 200 250 300 350 Time to expiration (days) -30

-25 -20 -15 -10 -5 0

Rho (put)

0 20 40 60 80

Stock price 0

5 10 15 20 25

Rho (call)

0 20 40 60 80

Stock price -25

-20 -15 -10 -5 0

Rho (put)

Dotted curves: in-the-money call or out-of-the-money put.

Solid curves: at-the-money option.

Dashed curves: out-of-the-money call or in-the-money put.

(37)

### 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.

(38)

### 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.

aPelsser and Vorst (1994).

(39)

S/(ud)

S/d

S/u

Su/d

S

Sd/u

Su

Sd Suu/d

Sdd/u

Suuu/d

Suu

S

Sdd

(40)

### 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. 583).

(41)

### 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?

(42)

### 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. 582).

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

aBut see pp. 943ﬀ.

(43)

### Extensions of Options Theory

(44)

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

— Joan Robinson (1903–1983)

(45)

### 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.

(46)

### 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?

(47)

### 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

(48)

### 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 call 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 .

(49)

### 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

(50)

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

• From Theorem 11 (p. 280) and the put-call parity, nS = V N (x) − Xe−rτN (x − σ√

τ ), B = V N (−x) + Xe−rτN (x − σ√

τ ).

– 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 .

(51)

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

• 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.

(52)

### 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.

(53)

—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

(54)

### 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.

(55)

### 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.

• 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.

(56)

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

(57)

### 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.

(58)

### 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.

(59)

### 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. 345 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.

(60)

### 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.

(61)

### 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. 345).

• So the total market value of the XYZ.com stock is

(62)

### 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.

(63)

### 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 (R86526008, D8852600), Lyuu, and Wang (F95922018) (2010).

(64)

### 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 working in a bank. She was working for Lehman Brothers in HK as of April, 2006.

(65)

H

Time Price

S Barrier hit

(66)

### 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

aHaug (2006).

(67)

### 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 ),

(31)

– 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).

(68)

### 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).

(69)

### 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.

(70)

### Interesting Observations

• Assume H < X.

• Replace S in the pricing formula Eq. (29) on p. 307 for the call with H2/S.

• Equation (31) on p. 356 for the down-and-in call becomes Eq. (29) when r − q = σ2/2.

• Equation (31) becomes S/H times Eq. (29) when r − q = 0.

(71)

### Interesting Observations (concluded)

• Replace S in the pricing formula for the down-and-in call, Eq. (31), with H2/S.

• Equation (31) becomes Eq. (29) when r − q = σ2/2.

• Equation (31) becomes H/S times Eq. (29) when r − q = 0.a

• Why?

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

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### Binomial Tree Algorithms

• Barrier options can be priced by binomial tree algorithms.

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

0 H

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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

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### 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).

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### 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).

• A put gives its holder the right to sell a number of the underlying asset for the strike price.. • How to

• 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

• 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

• 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

zero-coupon bond prices, forward rates, or the short rate. • Bond price and forward rate models are usually non-Markovian

• P u is the price of the i-period zero-coupon bond one period from now if the short rate makes an up move. • P d is the price of the i-period zero-coupon bond one period from now

• A call gives its holder the right to buy a number of the underlying asset by paying a strike price.. • A put gives its holder the right to sell a number of the underlying asset

zero-coupon bond prices, forward rates, or the short rate.. • Bond price and forward rate models are usually non-Markovian

zero-coupon bond prices, forward rates, or the short rate.. • Bond price and forward rate models are usually non-Markovian

zero-coupon bond prices, forward rates, or the short rate. • Bond price and forward rate models are usually non-Markovian

• It works as if the call writer delivered a futures contract to the option holder and paid the holder the prevailing futures price minus the strike price.. • It works as if the

• 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

• The binomial interest rate tree can be used to calculate the yield volatility of zero-coupon bonds.. • Consider an n-period

• 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

• 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

• Now suppose the settlement date for a bond selling at par (i.e., the quoted price is equal to the par value) falls between two coupon payment dates. • Then its yield to maturity

– at a premium (above its par value) when its coupon rate c is above the market interest rate r;. – at par (at its par value) when its coupon rate is equal to the market

– at a premium (above its par value) when its coupon rate c is above the market interest rate r;. – at par (at its par value) when its coupon rate is equal to the market

• A yield curve plots the yields to maturity of coupon bonds against maturity.. • A par yield curve is constructed from bonds trading

• When a call is exercised, the holder pays the strike price in exchange for the stock.. • When a put is exercised, the holder receives from the writer the strike price in exchange

• A call gives its holder the right to buy a number of the underlying asset by paying a strike price.. • A put gives its holder the right to sell a number of the underlying asset

• When a call is exercised, the holder pays the strike price in exchange for the stock.. • When a put is exercised, the holder receives from the writer the strike price in exchange

zero-coupon bond prices, forward rates, or the short rate.. • Bond price and forward rate models are usually non-Markovian