### 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 (27) on p. 248 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*

2

### The Bushy Tree

*S*

*Su*

*Sd*

*Su*^{2}

*Sud*

*Sdu*

*Sd*^{2}

2^{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.

*• Any proper calibration of the model parameters makes*
the BOPM converge to the continuous-time model.

*• We now skim through the proof.*

### 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*
ˆ

*r to match the empirical results as n goes to infinity.*

*• First, ˆr = rτ /n.*

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

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

*• Let*

b

*µ ≡* 1
*n* *E*

·

ln *S*_{τ}*S*

¸

denote the expected value of the continuously compounded rate of return per period.

*• Let*

b

*σ*^{2} *≡* 1

*n* Var

·

ln *S*_{τ}*S*

¸

denote the variance of the that return.

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

*• Under the BOPM, it is not hard to show that*
b

*µ = q ln(u/d) + ln d,*
b

*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.*

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

*• The BOPM converges to the distribution only if*
*nbµ = n[ q ln(u/d) + ln d ] → µτ,*
*nbσ*^{2} *= nq(1 − q) ln*^{2}*(u/d) → σ*^{2}*τ.*

*• 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. 260).

– Other choices are possible (see text).

*– Exact solutions for u, d, q are also feasible: 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*

r*τ*

*n* *. (28)*

*• With Eqs. (28), it can be checked that*
*nbµ = µτ,*

*nbσ*^{2} =

·

*1 −* *³ µ*
*σ*

´_{2} *τ*
*n*

¸

*σ*^{2}*τ → σ*^{2}*τ.*

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

*• The choices (28) result in the CRR binomial model.*^{a}

*• A more common choice for the probability is actually*
*q =* *R − d*

*u − d* *.*
by Eq. (25) on p. 230.

*• Their numerical properties are essentially identical.*

aCox, Ross, and Rubinstein (1979).

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

*• The no-arbitrage inequalities d < R < u may not hold*
under Eqs. (28) on p. 271.

– If this happens, the risk-neutral probability may 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 will be presented later.

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

*• What is the limiting probabilistic distribution of the*
*continuously compounded rate of return ln(S*_{τ}*/S)?*

*• 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 10 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*^{rτ /n}*− d)/(u − d).*

*• Let n → ∞.*^{a}

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 10 (p. 275) and Eq. (21) on p. 161.

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).*

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

^{a}

Theorem 11 (The Black-Scholes Formula)
*C* *= SN (x) − Xe*^{−rτ}*N (x − σ√*

*τ ),*
*P* *= Xe*^{−rτ}*N (−x + σ√*

*τ ) − SN (−x),*
*where*

*x ≡* *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!

### 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**,*
ˆ

*r = rτ /n.*

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 (see text).*

aChang and Palmer (2007).

### 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}

*• 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.*

### 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.

*• So?*

### 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 incorporate these two different ways of day*

count into the Black-Scholes formula and binomial tree
algorithms?^{b}

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

bRecall p. 146 about dating issues.

### Trading Days and Calendar Days (concluded)

*• Think of σ as measuring the volatility of stock price one*
year from now (regardless of what happens in between).

*• Suppose a year has 260 trading days.*

*• So a heuristic is to replace σ in the Black-Scholes*
formula with^{a}

*σ*
s

365

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

*• How about binomial tree algorithms?*

### 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.

### 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.*

*(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. 291 (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. 291 (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. 291

^{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. 291).

*– (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}*,*

a

### The Ad-Hoc Approximation vs. P. 291 (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. 686ff).

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

### 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}

aIn pricing European options, we care only about the distribution of

### Continuous Dividend Yields (continued)

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

^{a}

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

*τ ),* (29)
*P = Xe*^{−rτ}*N (−x + σ√*

*τ ) − Se*^{−qτ}*N (−x),*

(29* ^{0}*)
where

*x ≡* *ln(S/X) +* ¡

*r − q + σ*^{2}*/2*¢
*τ*
*σ√*

*τ* *.*

*• Formulas (29) and (29** ^{0}*) 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.

*– 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) ∆t}*− d*

*u − d* *,* (30)

*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. (30), binomial tree*

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

### Curtailing the Range of Tree Nodes

^{a}

*• Those nodes can be skipped if they are extremely*
unlikely to be reached.

*• The probability that the price will move more than a*
certain number of standard deviations from its initial
value is negligible.

*• Similarly, suppose the stock price at maturity is known.*

*• The probability that the price will move outside that*
number of standard deviations in working backward is
also negligible.

a

### Curtailing the Range of Tree Nodes (continued)

*• In summary, for certain stock prices, the strike price is so*
low or so high that it could not realistically be reached.

*• For these prices the option value is basically*
deterministic.

*• By working only within the said range of stock prices,*
we can save time without significant loss of accuracy.

### Curtailing the Range of Tree Nodes (concluded)

*• For time t, where 0 < t < T , the maximum and*
*minimum stock prices S*_{max}*(t) and S*_{min}*(t) are:*

*S*_{max}*(t) = min*

³

*S*_{0}*e*^{rt+ησ}^{√}^{t}*, Xe**−r(T −t)+ησ**√*

*T −t*´
*,*
*S*_{min}*(t) = max*

³

*S*_{0}*e*^{rt−ησ}^{√}^{t}*, Xe**−r(T −t)−ησ**√*

*T −t*´
*.*

### Traversal Sequence

*• Can the standard quadratic-time binomial tree*
algorithm for American options be improved?

– By an order?

– By a constant factor?

*• In any case, it helps to skip nodes.*

*• Note the traversal sequence of backward induction on*
the tree.

– It is by time (recall p. 259).

### Diagonal Traversal of the Tree

^{a}

*• Suppose we traverse the tree diagonally.*

*• Convince yourself that this procedure is well-defined.*

*• An early-exercise node is trivial to evaluate.*

– The difference of the strike price and the stock price.

*• A non-early-exercise node must be evaluated by*
backward induction.

aCurran (1995).

1 p

1 p 1 p

P[2][0]

P[2][1]

P[2][2]

P[1][0]

P[1][1]

P[0][0]

p

p

p p

p p

max

## c

0,X Sud^{2}

## h

max

## c

0,X Sd^{3}

## h

max

## c

0,X Su d^{2}

## h

max

## c

0,X Su^{3}

## h

START

1 p

1 p

1 p

### Diagonal Traversal of the Tree (continued)

Two properties of the propagation of early exercise nodes (E) and non-early-exercise nodes (C) during backward induction are:

1. A node is an early-exercise node if both its successor nodes are exercised early.

*• A terminal node that is in-the-money is considered*
an early exercise node.

2. If a node is a non-early-exercise node, then all the earlier nodes at the same horizontal level are also

non-early-exercise nodes.

*.*

E *

*.* *.* *-*

E C C

*-* *-* *.*

E *

Rule 1 Rule 2 *-*

### Diagonal Traversal of the Tree (continued)

*• Nothing is achieved if the whole tree needs to be*
explored.

*• We need a stopping rule.*

*• The traversal stops when a diagonal D consisting*
*entirely of non-early-exercise nodes is encountered.*

– By Rule 2, all early-exercise nodes have been found.

### Diagonal Traversal of the Tree (continued)

*• When the algorithm finds an early exercise node N in*
traversing a diagonal, it can stop immediately and move
on to the next diagonal.

*– By Rule 2, the node to the right of N must also be*
an early exercise node.

– By Rule 1 and induction, the rest of the nodes on the current diagonal must all be early-exercise nodes.

– They are hence computable on the fly when needed.

### Diagonal Traversal of the Tree (continued)

*• Also by Rule 1, the traversal can start from the*

zero-valued terminal node just above the strike price.

*• The upper triangle above the strike price can be skipped*
since its nodes are all zero valued.

Visited nodes **0**

0 0 0 0

Traverse from here
**Stop at diagonal D**

Strike price Strike price

Early exercise nodes by Rule 1 Early exercise nodes by Rule 1

Exercise boundary Exercise boundary

Early exercise nodes Early exercise nodes

20 40 60 80 100Volatility 5

10 15 20 25

Percent of nodes visited by the diagonal method

### Diagonal Traversal of the Tree (continued)

*• It remains to calculate the option value.*

*• It is the weighted sum of the discounted option values of*
*the nodes on D.*

– How does the payoff influence the root?

– We cannot go from the root to a node at which the
*option will be exercised without passing through D.*

*• The weight is the probability that the stock price hits*
*the diagonal for the first time at that node.*

### Diagonal Traversal of the Tree (concluded)

*• For a node on D which is the result of i up moves and*
*j down moves, the said probability is*

µ*i + j − 1*
*i*

¶

*p*^{i}*(1 − p)*^{j}*.*

– A valid path must pass through the node which is
*the result of i up moves and j − 1 down moves.*

*• Call the option value on this node P** _{i}*.

*• The desired option value then equals*

*a−1*Xµ

*i + j − 1*
*i*

¶

*p*^{i}*(1 − p)*^{j}*P*_{i}*e*^{−(i+j) r∆t}*.*

### The Analysis

*• Since each node on D has been evaluated by that time,*
*this part of the computation consumes O(n) time.*

*• The space requirement is also linear in n since only the*
diagonal has to be allocated space.

*• This idea can save computation time when D does not*
take long to find.

### The Analysis (continued)

*• Rule 2 is true with or without dividends.*

*• Suppose now that the stock pays a continuous dividend*
*yield q ≤ r (or r ≤ q for calls by parity).*

*• Recall p =* ^{e}^{(r−q)∆t}_{u−d}* ^{−d}*.

*• Rule 1 continues to hold since, for a current stock price*
*of Su*^{i}*d** ^{j}*:

### The Analysis (concluded)

*(pP*_{u}*+ (1 − p) P*_{d}*) e*^{−r∆t}

= £
*p*¡

*X − Su*^{i+1}*d** ^{j}*¢

*+ (1 − p)*¡

*X − Su*^{i}*d** ^{j+1}*¢¤

*e*^{−r∆t}

*= Xe*^{−r∆t}*− Su*^{i}*d*^{j}*(pu + (1 − p) d) e*^{−r∆t}

*= Xe*^{−r∆t}*− Su*^{i}*d*^{j}*e*^{−q∆t}

*≤ Xe*^{−r∆t}*− Su*^{i}*d*^{j}*e*^{−r∆t}

*≤ X − Su*^{i}*d*^{j}*.*