# The Binomial Option Pricing Formula

## Full text

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### Binomial Distribution

• Denote the binomial distribution with parameters n and p by

b(j; n, p) =Δ

n j



pj(1 − p)n−j = n!

j! (n − j)! pj(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.

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### The Binomial Option Pricing Formula

• The stock prices at time n are

Sun, Sun−1d, . . . , Sdn.

• 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 Sujdn−j ≥ X,

or, equivalently,

a =

ln(X/Sdn) ln(u/d)

 .

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### The Binomial Option Pricing Formula (concluded)

• Hence,

C

=

n

j=a

n

j

pj(1 − p)n−j 

Sujdn−j − X

Rn (36)

= S

n j=a

n j

(pu)j[ (1 − p) d ]n−j Rn

X Rn

n j=a

n j



pj(1 − p)n−j

= S

n j=a

b (j; n, pu/R) − Xe−ˆrn

n j=a

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

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### Numerical Examples

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

• u = 1.5 and d = 0.5.

• r = 18.232% per period (R = e0.18232 = 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.

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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)

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### 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.

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### Numerical Examples (continued)

Time 1:

• Suppose the stock price moves to \$240.

• The new delta is 0.90625.

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.

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### 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!

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### 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

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### 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.

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### 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.

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### 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.

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### Binomial Tree Algorithms for European Options

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

• The total running time is O(n2) because there are

∼ n2/2 nodes.

• The memory requirement is O(n2).

– 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.

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+[2][0]

+[2][1]

+[2][2]

+[1][0]

+[1][1]

+[0][0]

F

F

F F

F F

max ,

0 5K@2 :

max ,

0 5K @ :2

max ,

0 5K3 :

max ,

0 5@3 :

1 F

1 F

1 F

1 F

1 F

1 F

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0

0 0

All zeros

:

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### 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).

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### 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

 pa(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

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### 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(Sn − 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(n2) time.

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S

Su

Sd

Su2

Sud

Sdu

Sd2

2n

n

Sun Sun − 1 Su3

Su2d Su2d

Sud2 Su2d

Sud2 Sud2

Sd3

Sun − 1d

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### 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)!

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### 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

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

• First, ˆr = rτ/n.

– Each period is τ /n years long.

– The period gross return R = eˆr.

• Let

μ =Δ 1 n E

ln Sτ S

denote the expected value of the continuously

compounded rate of return per period of the BOPM.

• Let

σ2 Δ= 1

n Var

ln Sτ S

denote the variance of that return.

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

• Under the BOPM, it is not hard to show thata μ = q ln(u/d) + ln d,

σ2 = q(1 − q) ln2(u/d).

• Assume the stock’s true continuously compounded rate of return over τ years has mean μτ and variance σ2τ .

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

aRecall the Bernoulli distribution.

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

• The BOPM converges to the distribution only if

nμ = n[ q ln(u/d) + ln d ] → μτ, (38) 2 = nq(1 − q) ln2(u/d) → σ2τ. (39)

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

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### 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).

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

• The above requirements can be satisﬁed by

u = eσ

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,

2 =

1 μ σ

2 τ n

σ2τ → σ2τ.

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### 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).

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

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### 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 eσ

τ/n > erτ/n,

i.e., when n > r2τ /σ2 (check it).

– So it goes away if n is large enough.

– Other solutions can be found in the textbookb or will be presented later.

aMany papers and programs forget to check this condition!

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

• The central limit theorem says ln(Sτ/S) converges to N (μτ, σ2τ ).a

• So ln Sτ approaches N (μτ + ln S, σ2τ ).

• Conclusion: Sτ has a lognormal distribution in the limit.

aThe normal distribution with mean μτ and variance σ2τ .

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

Lemma 9 The continuously compounded rate of return ln(Sτ/S) approaches the normal distribution with mean (r − σ2/2) τ and variance σ2τ in a risk-neutral economy.

• Let q equal the risk-neutral probability p = (eΔ rτ/n − d)/(u − d).

• Let n → ∞.a

• Then μ = r − σ2/2.

aSee Lemma 9.3.3 of the textbook.

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

• The expected stock price at expiration in a risk-neutral economy isa

Se.

• The stock’s expected annual rate of returnb is thus the riskless rate r.

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

bIn the sense of (1/τ ) ln E[ Sτ/S ] (arithmetic average rate of return) not (1/τ )E[ ln(Sτ/S) ] (geometric average rate of return). In the latter case, it would be r − σ2/2 by Lemma 9.

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

a

Theorem 10 (The Black-Scholes Formula) C = SN (x) − Xe−rτN (x − σ√

τ ), P = Xe−rτN (−x + σ√

τ ) − SN (−x), where

x =Δ ln(S/X) + 

r + σ2/2 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!

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### 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).

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### 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 = eσ

τ/n,

d = e−σ

τ/n, r = rτ /n.ˆ

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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).

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### 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.

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### 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.

aImplied volatility is hard to compute when τ is small (why?).

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### 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).

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### 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.

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### 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

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### 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.

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### Binomial Tree Algorithms for American Puts

• Early exercise has to be considered.

• The binomial tree algorithm starts with the terminal payoﬀs

max(0, X − Sujdn−j) and applies backward induction.

• At each intermediate node, it compares the payoﬀ if exercised and the continuation value.

• It keeps the larger one.

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### 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.

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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 .

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### Time-Dependent Instantaneous Volatility (concluded)

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

u = eσ(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).

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### 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

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### 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.a

• Will the binomial tree fail to combine?

aThat is, one-period forward rate.

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### 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.

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### 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 .

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### 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.

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### 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.

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### 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.

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(Su − D) u

 Su − D

 

(Su − D) d S

(Sd − D) u

 

Sd − D



(Sd − D) d

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• 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.

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• 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.

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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

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### 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)u2

(S − D/R)ud

(S − D/R)d2

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### 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)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,

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### 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.

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### 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

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### 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

aIn pricing European options, only the distribution of Sτ matters.

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### Continuous Dividend Yields (continued)

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

C = Se−qτN (x) − Xe−rτN (x − σ√

τ ), (41) P = Xe−rτN (−x + σ√

τ ) − Se−qτN (−x),

(41) where

x =Δ ln(S/X) + 

r − q + σ2/2 τ σ√

τ .

• Formulas (41) and (41) remain valid as long as the dividend yield is predictable.

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### 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.

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### 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.

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### 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.

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