### Binomial Distribution

*• Denote the binomial distribution with parameters n*
*and p by*

*b(j; n, p)* =^{Δ}

*n*
*j*

*p** ^{j}*(1

*− p)*

*=*

^{n−j}*n!*

*j! (n − j)!* *p** ^{j}*(1

*− p)*

^{n−j}*.*

**– n! = 1 × 2 × · · · × n.****– Convention: 0! = 1.**

*• Suppose you ﬂip a coin n times with p being the*
probability of getting heads.

*• Then b(j; n, p) is the probability of getting j heads.*

### The Binomial Option Pricing Formula

*• The stock prices at time n are*

*Su*^{n}*, Su*^{n−1}*d, . . . , Sd*^{n}*.*

*• Let a be the minimum number of upward price moves*
for the call to ﬁnish in the money.

*• So a is the smallest nonnegative integer j such that*
*Su*^{j}*d*^{n−j}*≥ X,*

or, equivalently,

*a =*

*ln(X/Sd** ^{n}*)

*ln(u/d)*

*.*

### The Binomial Option Pricing Formula (concluded)

*• Hence,*

*C*

=

_{n}

*j=a*

_{n}

*j*

*p** ^{j}*(1

*− p)*

^{n−j}*Su*^{j}*d*^{n−j}*− X*

*R** ^{n}* (36)

= *S*

*n*
*j=a*

*n*
*j*

*(pu)** ^{j}*[ (1

*− p) d ]*

^{n−j}*R*

^{n}*−* *X*
*R*^{n}

*n*
*j=a*

*n*
*j*

*p** ^{j}*(1

*− p)*

^{n−j}= *S*

*n*
*j=a*

*b (j; n, pu/R) − Xe*^{−ˆrn}

*n*
*j=a*

*b(j; n, p).* (37)

### Numerical Examples

*• A non-dividend-paying stock is selling for $160.*

*• u = 1.5 and d = 0.5.*

*• r = 18.232% per period (R = e*^{0.18232}*= 1.2).*

**– Hence p = (R − d)/(u − d) = 0.7.**

*• Consider a European call on this stock with X = 150*
*and n = 3.*

*• The call value is $85.069 by backward induction.*

*• Or, the PV of the expected payoﬀ at expiration:*

*390 × 0.343 + 30 × 0.441 + 0 × 0.189 + 0 × 0.027*

*(1.2)*^{3} *= 85.069.*

160

540 (0.343)

180 (0.441)

(0.189)60

(0.027)20 Binomial process for the stock price

(probabilities in parentheses)

(0.49)360

(0.42)120

40 (0.09) (0.7)240

80 (0.3)

85.069 (0.82031)

390

30

0

0 Binomial process for the call price

(hedge ratios in parentheses)

(1.0)235

(0.25)17.5

0 (0.0) 141.458

(0.90625)

10.208 (0.21875)

### Numerical Examples (continued)

*• Mispricing leads to arbitrage proﬁts.*

*• Suppose the option is selling for $90 instead.*

*• Sell the call for $90.*

*• Invest $85.069 in the replicating portfolio with 0.82031*
shares of stock as required by the delta.

*• Borrow 0.82031 × 160 − 85.069 = 46.1806 dollars.*

*• The fund that remains,*

90 *− 85.069 = 4.931 dollars,*
is the arbitrage proﬁt, as we will see.

### Numerical Examples (continued)

Time 1:

*• Suppose the stock price moves to $240.*

*• The new delta is 0.90625.*

*• Buy*

*0.90625 − 0.82031 = 0.08594*

*more shares at the cost of 0.08594 × 240 = 20.6256*
dollars ﬁnanced by borrowing.

*• Debt now totals 20.6256 + 46.1806 × 1.2 = 76.04232*
dollars.

### Numerical Examples (continued)

*• The trading strategy is self-ﬁnancing because the*
portfolio has a value of

*0.90625 × 240 − 76.04232 = 141.45768.*

*• It matches the corresponding call value!*

### Numerical Examples (continued)

Time 2:

*• Suppose the stock price plunges to $120.*

*• The new delta is 0.25.*

*• Sell 0.90625 − 0.25 = 0.65625 shares.*

*• This generates an income of 0.65625 × 120 = 78.75*
dollars.

*• Use this income to reduce the debt to*

*76.04232 × 1.2 − 78.75 = 12.5*

### Numerical Examples (continued)

Time 3 (the case of rising price):

*• The stock price moves to $180.*

*• The call we wrote ﬁnishes in the money.*

*• Close out the call’s short position by buying back the*
call or buying a share of stock for delivery.

*• This results in a loss of 180 − 150 = 30 dollars.*

*• Financing this loss with borrowing brings the total debt*
*to 12.5 × 1.2 + 30 = 45 dollars.*

*• It is repaid by selling the 0.25 shares of stock for*
*0.25 × 180 = 45 dollars.*

### Numerical Examples (concluded)

Time 3 (the case of declining price):

*• The stock price moves to $60.*

*• The call we wrote is worthless.*

*• Sell the 0.25 shares of stock for a total of*
*0.25 × 60 = 15*

dollars.

*• Use it to repay the debt of 12.5 × 1.2 = 15 dollars.*

### Applications besides Exploiting Arbitrage Opportunities

^{a}

*• Replicate an option using stocks and bonds.*

**– Set up a portfolio to replicate the call with $85.069.**

*• Hedge the options we issued.*

**– Use $85.069 to set up a portfolio to replicate the call**
to counterbalance its values exactly.^{b}

*• · · ·*

*• Without hedge, one may end up forking out $390 in the*
worst case!^{c}

aThanks to a lively class discussion on March 16, 2011.

bHedging and replication are mirror images.

cThanks to a lively class discussion on March 16, 2016.

### Binomial Tree Algorithms for European Options

*• The BOPM implies the binomial tree algorithm that*
applies backward induction.

*• The total running time is O(n*^{2}) because there are

*∼ n*^{2}*/2 nodes.*

*• The memory requirement is O(n*^{2}).

**– Can be easily reduced to O(n) by reusing space.**^{a}

*• To price European puts, simply replace the payoﬀ.*

aBut watch out for the proper updating of array entries.

+[2][0]

+[2][1]

+[2][2]

+[1][0]

+[1][1]

+[0][0]

F

F

F F

F F

max ,

### ?

0 5K@^{2}:

### D

max ,

### ?

0 5K @ :^{2}

### D

max ,

### ?

0 5K^{3}:

### D

max ,

### ?

0 5@^{3}:

### D

1 F

1 F

1 F

1 F

1 F

1 F

### Further Time Improvement for Calls

0

0 0

All zeros

:

### Optimal Algorithm

*• We can reduce the running time to O(n) and the*
*memory requirement to O(1).*

*• Note that*

*b(j; n, p) =* *p(n − j + 1)*

(1 *− p) j* *b(j − 1; n, p).*

### Optimal Algorithm (continued)

*• The following program computes b(j; n, p) in b[ j ]:*

*• It runs in O(n) steps.*

1: *b[ a ] :=* _{n}

*a*

*p** ^{a}*(1

*− p)*

*;*

^{n−a}2: **for j = a + 1, a + 2, . . . , n do**

3: *b[ j ] := b[ j − 1 ] × p × (n − j + 1)/((1 − p) × j);*

4: **end for**

### Optimal Algorithm (concluded)

*• With the b(j; n, p) available, the risk-neutral valuation*
formula (36) on p. 261 is trivial to compute.

*• But we only need a single variable to store the b(j; n, p)s*
as they are being sequentially computed.

*• This linear-time algorithm computes the discounted*
*expected value of max(S*_{n}*− X, 0).*

*• The above technique cannot be applied to American*
options because of early exercise.

*• So binomial tree algorithms for American options*
*usually run in O(n*^{2}) time.

### The Bushy Tree

*S*

*Su*

*Sd*

*Su*^{2}

*Sud*

*Sdu*

*Sd*^{2}

2^{n}

*n*

*Su*^{n}*Su*^{n }^{− 1}
*Su*^{3}

*Su*^{2}*d*
*Su*^{2}*d*

*Sud*^{2}
*Su*^{2}*d*

*Sud*^{2}
*Sud*^{2}

*Sd*^{3}

*Su*^{n }^{− 1}*d*

### Toward the Black-Scholes Formula

*• The binomial model seems to suﬀer from two unrealistic*
assumptions.

**– The stock price takes on only two values in a period.**

**– Trading occurs at discrete points in time.**

*• As n increases, the stock price ranges over ever larger*
numbers of possible values, and trading takes place
nearly continuously.^{a}

*• Need to calibrate the BOPM’s parameters u, d, and R*
to make it converge to the continuous-time model.

*• We now skim through the proof.*

aContinuous-time trading may create arbitrage opportunities in prac- tice (Budish, Cramton, & Shim, 2015)!

### Toward the Black-Scholes Formula (continued)

*• Let τ denote the time to expiration of the option*
measured in years.

*• Let r be the continuously compounded annual rate.*

*• With n periods during the option’s life, each period*
*represents a time interval of τ /n.*

*• Need to adjust the period-based u, d, and interest rate*
*r to match the empirical results as n → ∞.*ˆ

### Toward the Black-Scholes Formula (continued)

*• First, ˆr = rτ/n.*

**– Each period is τ /n years long.**

**– The period gross return R = e*** ^{ˆr}*.

*• Let*

*μ* =^{Δ} 1
*n* *E*

ln *S*_{τ}*S*

denote the expected value of the continuously

compounded rate of return per period of the BOPM.

*• Let*

*σ*^{2 Δ}= 1

*n* Var

ln *S*_{τ}*S*

denote the variance of that return.

### Toward the Black-Scholes Formula (continued)

*• Under the BOPM, it is not hard to show that*^{a}
*μ = q ln(u/d) + ln d,*

*σ*^{2} = *q(1 − q) ln*^{2}*(u/d).*

*• Assume the stock’s true continuously compounded rate*
*of return over τ years has mean μτ and variance σ*^{2}*τ .*

*• Call σ the stock’s (annualized) volatility.*

aRecall the Bernoulli distribution.

### Toward the Black-Scholes Formula (continued)

*• The BOPM converges to the distribution only if*

*nμ = n[ q ln(u/d) + ln d ] → μτ,* (38)
*nσ*^{2} = *nq(1 − q) ln*^{2}*(u/d) → σ*^{2}*τ.* (39)

*• We need one more condition to have a solution for u, d, q.*

### Toward the Black-Scholes Formula (continued)

*• Impose*

*ud = 1.*

**– It makes nodes at the same horizontal level of the**
tree have identical price (review p. 273).

**– Other choices are possible (see text).**

*• Exact solutions for u, d, q are feasible if Eqs. (38)–(39)*
are replaced by equations: 3 equations for 3 variables.^{a}

aChance (2008).

### Toward the Black-Scholes Formula (continued)

*• The above requirements can be satisﬁed by*

*u = e*^{σ}

*√**τ/n**, d = e*^{−σ}

*√**τ/n**, q =* 1

2 + 1 2

*μ*
*σ*

*τ*

*n* *.* (40)

*• With Eqs. (40), it can be checked that*
*nμ = μτ,*

*nσ*^{2} =

1 *−* *μ*
*σ*

_{2} *τ*
*n*

*σ*^{2}*τ → σ*^{2}*τ.*

### Toward the Black-Scholes Formula (continued)

*• The choices (40) result in the CRR binomial model.*^{a}

*• With the above choice, even if u and d are not*
calibrated, the mean is still matched!^{b}

aCox, Ross, & Rubinstein (1979).

b*Recall Eq. (33) on p. 245. So u and d are related to volatility exclu-*
*sively. They do not depend on r.*

### Toward the Black-Scholes Formula (continued)

*• The no-arbitrage inequalities d < R < u may not hold*
under Eqs. (40) on p. 284 or Eq. (32) on p. 244.

**– If this happens, the probabilities lie outside [ 0, 1 ].**^{a}

*• The problem disappears when n satisﬁes*
*e*^{σ}

*√**τ/n* *> e*^{rτ/n}*,*

*i.e., when n > r*^{2}*τ /σ*^{2} (check it).

**– So it goes away if n is large enough.**

**– Other solutions can be found in the textbook**^{b} or will
be presented later.

aMany papers and programs forget to check this condition!

### Toward the Black-Scholes Formula (continued)

*• The central limit theorem says ln(S*_{τ}*/S) converges to*
*N (μτ, σ*^{2}*τ ).*^{a}

*• So ln S*_{τ}*approaches N (μτ + ln S, σ*^{2}*τ ).*

*• Conclusion: S** _{τ}* has a lognormal distribution in the limit.

a*The normal distribution with mean μτ and variance σ*^{2}*τ .*

### Toward the Black-Scholes Formula (continued)

**Lemma 9 The continuously compounded rate of return***ln(S*_{τ}*/S) approaches the normal distribution with mean*
*(r − σ*^{2}*/2) τ and variance σ*^{2}*τ in a risk-neutral economy.*

*• Let q equal the risk-neutral probability*
*p* *= (e*^{Δ} ^{rτ/n}*− d)/(u − d).*

*• Let n → ∞.*^{a}

*• Then μ = r − σ*^{2}*/2.*

aSee Lemma 9.3.3 of the textbook.

### Toward the Black-Scholes Formula (continued)

*• The expected stock price at expiration in a risk-neutral*
economy is^{a}

*Se*^{rτ}*.*

*• The stock’s expected annual rate of return*^{b} is thus the
*riskless rate r.*

aBy Lemma 9 (p. 289) and Eq. (28) on p. 175.

b*In the sense of (1/τ ) ln E[ S**τ**/S ] (arithmetic average rate of return)*
*not (1/τ )E[ ln(S**τ**/S) ] (geometric average rate of return). In the latter*
*case, it would be r − σ*^{2}*/2 by Lemma 9.*

### Toward the Black-Scholes Formula (continued)

^{a}

**Theorem 10 (The Black-Scholes Formula)**
*C* = *SN (x) − Xe*^{−rτ}*N (x − σ√*

*τ ),*
*P* = *Xe*^{−rτ}*N (−x + σ√*

*τ ) − SN (−x),*
*where*

*x* =^{Δ} *ln(S/X) +*

*r + σ*^{2}*/2*
*τ*
*σ√*

*τ* *.*

aOn a United flight from San Francisco to Tokyo on March 7, 2010, a real-estate manager mentioned this formula to me!

### Toward the Black-Scholes Formula (concluded)

*• See Eq. (37) on p. 261 for the meaning of x.*

*• See Exercise 13.2.12 of the textbook for an interpretation*
*of the probability associated with N (x) and N (−x).*

### BOPM and Black-Scholes Model

*• The Black-Scholes formula needs 5 parameters: S, X, σ,*
*τ , and r.*

*• Binomial tree algorithms take 6 inputs: S, X, u, d, ˆr,*
*and n.*

*• The connections are*

*u = e*^{σ}

*√**τ/n**,*

*d = e*^{−σ}

*√**τ/n**,*
*r = rτ /n.*ˆ

5 10 15 20 25 30 35 n

11.5 12 12.5 13

Call value

0 10 20 30 40 50 60 n

15.1 15.2 15.3 15.4 15.5

Call value

*• S = 100, X = 100 (left), and X = 95 (right).*

### BOPM and Black-Scholes Model (concluded)

*• The binomial tree algorithms converge reasonably fast.*

*• The error is O(1/n).*^{a}

*• Oscillations are inherent, however.*

*• Oscillations can be dealt with by the judicious choices of*
*u and d.*^{b}

aL. Chang & Palmer (2007).

bSee Exercise 9.3.8 of the textbook.

### Implied Volatility

*• Volatility is the sole parameter not directly observable.*

*• The Black-Scholes formula can be used to compute the*
market’s opinion of the volatility.^{a}

* – Solve for σ given the option price, S, X, τ , and r*
with numerical methods.

**– How about American options?**

a*Implied volatility is hard to compute when τ is small (why?).*

### Implied Volatility (concluded)

*• Implied volatility is*

the wrong number to put in the wrong formula to
get the right price of plain-vanilla options.^{a}

*• Just think of it as an alternative to quoting option*
prices.

*• Implied volatility is often preferred to historical*
volatility in practice.

**– Using the historical volatility is like driving a car**
with your eyes on the rearview mirror?

aRebonato (2004).

### 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.

*• Other patterns have also been observed.*

### TXO Calls (September 25, 2015)

^{a}

**300**

**14** **200**
**9000**

**8500**
**16**

**8000** **100**

**7500**
**18**

**7000** **0**
**20**

**22**
**24**

ATM = $8132

aThe underlying Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) closed at 8132. Plot supplied by Mr. Lok, U Hou

### Solutions to the Smile

*• 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.

### 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 compares the payoﬀ if*
*exercised and the continuation value.*

*• It keeps the larger one.*

### 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.

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

### 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(S*_{τ}*/S) is*

_{τ}

0 *σ*^{2}*(t) dt*
*rather than σ*^{2}*τ .*

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

_{τ}

0 *σ*^{2}*(t) dt*

*τ* *.*

### Time-Dependent Instantaneous Volatility (concluded)

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

*u = e*^{σ(t)}

*√**τ/n**,*

*d = e*^{−σ(t)}

*√**τ/n**.*

*• But how to make the binomial tree combine?*^{a}

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

### Volatility (1990–2016)

^{a}

2-Jan-90 2-Jan-91 2-Jan-92 4-Jan-93 3-Jan-94 3-Jan-95 2-Jan-96 2-Jan-97 2-Jan-98 4-Jan-99 3-Jan-00 2-Jan-01 2-Jan-02 2-Jan-03 2-Jan-04 3-Jan-05 3-Jan-06 3-Jan-07 2-Jan-08 2-Jan-09 4-Jan-10 3-Jan-11 3-Jan-12 2-Jan-13 2-Jan-14 2-Jan-15 4-Jan-16 0

10 20 30 40 50 60 70 80 90

VIX

CBOE S&P 500 Volatility Index

### 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* *r**i*

*τ* *,*

*where r** _{i}* is the continuously compounded short rate

*measured in periods for period i.*

^{a}

*• Will the binomial tree fail to combine?*

aThat is, one-period forward rate.

### 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); K. French (1980); K. French & Roll (1986).

bRecall p. 158 about dating issues.

### 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 with*^{a}

*σ*

365

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

aD. French (1984).

### 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.

### 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.^{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.

### 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.

### The Ad-Hoc Approximation vs. P. 312 (Step 1)

*S − D/R*

*

j

*(S − D/R)u*

*

j

*(S − D/R)d*

*

j

*(S − D/R)u*^{2}

*(S − D/R)ud*

*(S − D/R)d*^{2}

### The Ad-Hoc Approximation vs. P. 312 (Step 2)

*(S − D/R) + D/R = S*

*

j

*(S − D/R)u*

*

j

*(S − D/R)d*

*

j

*(S − D/R)u*^{2}

*(S − D/R)ud*

*(S − D/R)d*^{2}

### The Ad-Hoc Approximation vs. P. 312

^{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. 312).

**– (S − D/R)u**^{2}*, (S − D/R)ud, (S − D/R)d*^{2} (ad hoc).

*• Note that, as d < R < u,*

*(Su − D) u > (S − D/R)u*^{2}*,*
*(Sd − D) d < (S − D/R)d*^{2}*,*

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

### The Ad-Hoc Approximation vs. P. 312 (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.

### 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. 765ﬀ).

*• Other approaches include adjusting σ and approximating*
the known dividend with a dividend yield.^{b}

aDai (B82506025, R86526008, D8852600) & Lyuu (2004). Also Arealy

& Rodrigues (2013).

bGeske & Shastri (1985). It works well for American options but not

### 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.*^{a}

a*In pricing European options, only the distribution of S**τ* matters.

### Continuous Dividend Yields (continued)

*• So the Black-Scholes formulas hold with S replaced by*
*Se** ^{−qτ}*:

^{a}

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

*τ ),* (41)
*P = Xe*^{−rτ}*N (−x + σ√*

*τ ) − Se*^{−qτ}*N (−x),*

(41* ^{}*)
where

*x* =^{Δ} *ln(S/X) +*

*r − q + σ*^{2}*/2*
*τ*
*σ√*

*τ* *.*

*• Formulas (41) and (41** ^{}*) remain valid as long as the
dividend yield is predictable.

### Continuous Dividend Yields (continued)

*• To run binomial tree algorithms, replace u with ue*^{−qΔt}*and d with de*^{−qΔt}*, where Δt* *= τ /n.*^{Δ}

**– 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.

### Continuous Dividend Yields (concluded)

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

*u − d* *,* (42)

*where Δt* *= τ /n.*^{Δ}

**– 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. (42), binomial tree*

*algorithms stay the same as if there were no dividends.*

### Distribution of Logarithmic Returns of TAIEX

### Exercise Boundaries of American Options (in the Continuous-Time Model)

^{a}

*• The exercise boundary is a nondecreasing function of t*
for American puts (see the plot next page).

*• The exercise boundary is a nonincreasing function of t*
for American calls.

aSee Section 9.7 of the textbook for the tree analog.