### Problems; the Smile

*• Options written on the same underlying asset usually do*
not produce the same implied volatility.

*• A typical pattern is a “smile” in relation to the strike*
price.

– The implied volatility is lowest for at-the-money options.

– It becomes higher the further the option is in- or out-of-the-money.

### Problems; the Smile (concluded)

*• To address this issue, volatilities are often combined to*
produce a composite implied volatility.

*• This practice is not sound theoretically.*

*• The existence of different implied volatilities for options*
on the same underlying asset shows the Black-Scholes
model cannot be literally true.

### 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}

*– σ measures the volatility of stock price one year from*
now (regardless of what happens in between).

*• How to incorporate these two different ways of day*

count into the Black-Scholes formula and binomial tree algorithms?

### Trading Days and Calendar Days (concluded)

*• Suppose a year has 260 trading days.*

*• A quick and dirty way is to replace σ with*^{a}
*σ*

s 365 260

number of trading days to expiration
number of calendar days to expiration *.*

*• How about binomial tree algorithms?*

aFrench (1984).

### Binomial Tree Algorithms for American Puts

*• Early exercise has to be considered.*

*• The binomial tree algorithm starts with the terminal*
payoffs

*max(0, X − Su*^{j}*d** ^{n−j}*)
and applies backward induction.

*• At each intermediate node, it checks for early exercise*
by comparing the payoff if exercised with the

continuation value.

### Bermudan Options

*• Some American options can be exercised only at discrete*
time points instead of continuously.

*• They are called Bermudan options.*

*• Their pricing algorithm is identical to that for American*
options.

*• The only exception is early exercise is considered for*
only those nodes when early exercise is permitted.

### 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.

### 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.

*(Su − D) u*

*%*
*Su − D*

*%* *&*

*(Su − D) d*
*S*

*(Sd − D) u*

*&* *%*

*Sd − D*

*&*

*(Sd − D) d*

### An Ad-Hoc Approximation

*• Use the Black-Scholes formula with the stock price*
reduced by the PV of the dividends (Roll, 1977).

*• 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.

*– σ equal to the volatility of the process followed by*
the risky component.

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

### An Ad-Hoc Approximation (concluded)

*• 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.

### 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.

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

aDai and Lyuu (2004).

### 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 S** _{τ}* with a continuous

*dividend yield of q would grow from S to S*

_{τ}*e*

*without the dividends.*

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

### Continuous Dividend Yields (continued)

*• The Black-Scholes formulas hold with S replaced by*
*Se** ^{−qτ}* (Merton, 1973):

*C = Se*^{−qτ}*N (x) − Xe*^{−rτ}*N (x − σ√*

*τ ),* (24)
*P = Xe*^{−rτ}*N (−x + σ√*

*τ ) − Se*^{−qτ}*N (−x),*

(24* ^{0}*)
where

*x ≡* *ln(S/X) +* ¡

*r − q + σ*^{2}*/2*¢
*τ*
*σ√*

*τ* *.*

*• Formulas (24) and (24’) remain valid as long as the*
dividend yield is predictable.

### Continuous Dividend Yields (continued)

*• To run binomial tree algorithms, replace u with ue*^{−q∆t}*and d with de*^{−q∆t}*, where ∆t ≡ 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.

### Continuous Dividend Yields (concluded)

*• Alternatively, pick the risk-neutral probability as*
*e*^{(r−q) ∆t}*− d*

*u − d* *,* (25)

*where ∆t ≡ 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. (25), binomial tree*
algorithms stay the same.

*Sensitivity Analysis of Options*

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

— Blaise Pascal (1623–1662)

### Sensitivity Measures (“The Greeks”)

*• Understanding 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+σ*^{2}*/2) τ*
*σ**√*

*τ* (recall p. 244).

*• Note that*

*N*^{0}*(y) = (1/√*

*2π ) e*^{−y}^{2}^{/2}*> 0,*

the density function of standard normal distribution.

### Delta

*• Defined as ∆ ≡ ∂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.

*• The delta used in the BOPM is the discrete analog.*

### 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 1.*

### 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 −∆ shares of*
stock is delta-neutral.

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

*• In general, hedge a position in a security with a delta of*

∆_{1} by shorting ∆_{1}*/∆*_{2} units of a security with a delta

### Theta (Time Decay)

*• Defined as the rate of change of a security’s value with*
*respect to time, or Θ ≡ −∂f /∂τ .*

*• For a European call on a non-dividend-paying stock,*
*Θ = −SN*^{0}*(x) σ*

2*√*

*τ* *− rXe*^{−rτ}*N (x − σ√*

*τ ) < 0.*

– The call loses value with the passage of time.

*• For a European put,*
*Θ = −SN*^{0}*(x) σ*

2*√*

*τ* *+ rXe*^{−rτ}*N (−x + σ√*
*τ ).*

### Gamma

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

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

*• A portfolio with a high gamma needs in practice be*
rebalanced more often to maintain delta neutrality.

*• Delta ∼ duration; gamma ∼ convexity.*

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

*√*

### Vega

^{a}

### (Lambda, Kappa, Sigma)

*• Defined as the rate of change of its value with respect to*
*the volatility of the underlying asset Λ ≡ ∂Π/∂σ.*

*• Volatility often changes over time.*

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

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

*τ N*^{0}*(x) > 0.*

– Higher volatility increases option value.

a

### Rho

*• Defined as the rate of change in its value with respect to*
*interest rates ρ ≡ ∂Π/∂r.*

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

*Xτ e*^{−rτ}*N (x − σ√*

*τ ) > 0.*

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

*−Xτ e*^{−rτ}*N (−x + σ√*

*τ ) < 0.*

### Numerical Greeks

*• Needed when closed-form formulas do not exist.*

*• Take delta as an example.*

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

*2∆S* *.*

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

### An Alternative Numerical Delta

^{a}

*• Use intermediate results of the binomial tree algorithm.*

*• When the algorithm reaches the end of the first period,*
*f*_{u}*and f** _{d}* are computed.

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

*• Delta is approximated by*

*f*_{u}*− f*_{d}*Su − Sd.*

*• Almost zero extra computational effort.*

*S/(ud)*

*S/d*

*S/u*

*Su/d*

*S*

*Sd/u*

*Su*

*Sd*
*Suu/d*

*Sdd/u*

*Suuu/d*

*Suu*

*S*

*Sdd*

*Sddd/u*

### Numerical Gamma

*• At the stock price (Suu + Sud)/2, delta is*
*approximately (f*_{uu}*− f*_{ud}*)/(Suu − Sud).*

*• At the stock price (Sud + Sdd)/2, delta is*
*approximately (f*_{ud}*− f*_{dd}*)/(Sud − Sdd).*

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

*f**uu**−f**ud*

*Suu−Sud* *−* _{Sud−Sdd}^{f}^{ud}^{−f}^{dd}*(Suu − Sdd)/2* *.*

*• Alternative formulas exist.*

### Finite Difference Fails for Numerical Gamma

*• Numerical differentiation gives*

*f (S + ∆S) − 2f (S) + f (S − ∆S)*

*(∆S)*^{2} *.*

*• It does not work (see text).*

*• Why did the binomial tree version work?*

### Other Numerical Greeks

*• The theta can be computed as*
*f*_{ud}*− f*

*2(τ /n)* *.*

– In fact, the theta of a European option will be shown to be computable from delta and gamma (see p. 490).

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

*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*
pricing corporate securities.

*• Assume:*

– 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.

### 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?*

### 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*

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

*• The stock 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 the call.*

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

*• Thus nS = C and 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 Zero-Coupon Bonds and Stock (continued)

*• From Theorem 8 (p. 244) and the put-call parity,*
*nS* *= V N (x) − Xe*^{−rτ}*N (x − σ√*

*τ ),*
*B* *= V N (−x) + Xe*^{−rτ}*N (x − σ√*

*τ ).*

– Above,

*x ≡* *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* *.*

### 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

µ

*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.*

### 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 = 1000, V = 44.5 × n = 44500, and*
*X = 30 × n = 30000.*

—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

### 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 = 15250 dollars.*

*• The entire bond issue is worth*

*B = 44500 − 15250 = 29250 dollars.*

– Or $975 per bond.

### 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.*

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 45000/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 = 1937.5*

dollars.

*• The market value of the stock decreases as the*
debt-equity 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.

*• So parameters such volatility, dividend, and strike price*
are under partial control of the stockholders.

### 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. 293 says the total market value of the*
bonds should be $42,562.5.

*• The new bondholders pay 42562.5 × (15/45) = 14187.5*
dollars.

*• The remaining stock is worth $1,937.5.*

### A Numerical Example (continued)

*• The stockholders therefore gain*

*14187.5 + 1937.5 − 15250 = 875*
dollars.

*• The original bondholders lose an equal amount,*
*29250 −* 30

45 *× 42562.5 = 875.* (26)

### A Numerical Example (continued)

*• Suppose the stockholders distribute $14,833.3 cash*
*dividends by selling (1/3) × n Merck shares.*

*• They now have $14,833.3 in cash plus a call on*
*(2/3) × n Merck shares.*

*• The strike price remains X = 30000.*

*• This is equivalent to owning two-thirds of a call on n*
Merck shares with a total strike price of $45,000.

*• n such calls are worth $1,937.5 (p. 293).*

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

### A Numerical Example (concluded)

*• The market value of the XYZ.com bonds is hence*
*(2/3) × n × 44.5 − 1291.67 = 28375 dollars.*

*• Hence the stockholders gain*

*14833.3 + 1291.67 − 15250 ≈ 875*
dollars.

*• The bondholders watch their value drop from $29,250 to*

$28,375, a loss of $875.

### Other Examples

*• Subordinated debts as bull call spreads.*

*• Warrants as calls.*

*• Callable bonds as American calls with 2 strike prices.*

*• Convertible bonds.*

### 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 student told me on March 26, 2004, that she did not un- derstand why I covered barrier options until she started working in a

### 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 kinds of barrier options.*

### 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}

µ*H*
*S*

¶_{2λ}

*N (x) − Xe*^{−rτ}

µ*H*
*S*

¶_{2λ−2}

*N (x − σ**√*
*τ ),*

(27)

*– x ≡* ^{ln(H}^{2}*/(SX))+(r−q+σ*^{2}*/2) τ*
*σ**√*

*τ* .

*– λ ≡ (r − q + σ*^{2}*/2)/σ*^{2}.

*• A European down-and-out call can be priced via the*
in-out parity.

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

^{a}

*• Assume X ≤ H.*

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

*Xe*^{−rτ}

µ*H*

*S*

¶_{2λ−2}

*N (−x + σ**√*

*τ ) − Se*^{−qτ}

µ*H*

*S*

¶_{2λ}

*N (−x).*

*• A European up-and-out put can be priced via the in-out*
parity.

aMerton (1973).

### Interesting Observations

*• Assume H < X.*

*• Replace S in the pricing formula for the down-and-in*
*call, Eq. (27) on p. 303, with H*^{2}*/S.*

*• Equation (27) becomes Eq. (24) on p. 261 when*
*r − q = σ*^{2}*/2.*

*• Equation (27) becomes S/H times Eq. (24) on p. 261*
*when r − q = 0.*

### Binomial Tree Algorithms

*• Barrier options can be priced by binomial tree*
algorithms.

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

0 *H*

*H*
8

16

4

32

8

2

64

16

4

1

4.992

12.48

1.6

27.2

4.0

0

58

10

0

0
*X*

0.0

*S = 8, X = 6, H = 4, R = 1.25, u = 2, and d = 0.5.*

### Binomial Tree Algorithms (concluded)

*• But convergence is erratic because H is not at a price*
level on the tree (see plot on next page).

– Typically, the barrier has to be adjusted to be at a price level.

*• Solutions will be presented later.*

100 150 200 250 300 350 400

#Periods 3

3.5 4 4.5 5 5.5

Down-and-in call value