Toward the Black-Scholes Formula

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

Su²

Sud

Sdu

Sd²

2ⁿ Suⁿ

Su^{n − 1} Su³

Su²d Su²d

Sud² Su²d

Sud² Sud²

Sd³

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^rˆ.

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

σ² ≡ 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

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.

Toward the Black-Scholes Formula (continued)

• The BOPM converges to the distribution only if nbµ = n[ q ln(u/d) + ln d ] → µτ, nbσ² = nq(1 − q) ln²(u/d) → σ²τ.

• 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

rτ

n . (28)

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

nbσ² =

·

1 − ³ µ σ

´₂ τ n

¸

σ²τ → σ²τ.

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 > r²τ /σ² (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, σ²τ ).^a

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

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

aThe normal distribution with mean µτ and variance σ²τ .

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 − σ²/2) τ and variance σ²τ 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.

bIn 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)

Theorem 11 (The Black-Scholes Formula) C = SN (x) − Xe^−rτN (x − σ√

τ ), P = Xe^−rτN (−x + σ√

τ ) − SN (−x), where

x ≡ 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!

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?

aImplied 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^jd^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)u²

(S − D/R)ud

(S − D/R)d²

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)u²

(S − D/R)ud

(S − D/R)d²

The Ad-Hoc Approximation vs. P. 291

• 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)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,

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 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_τe^qτ 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, 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^−qτN (x) − Xe^−rτN (x − σ√

τ ), (29) P = Xe^−rτN (−x + σ√

τ ) − Se^−qτN (−x),

(29⁰) where

x ≡ ln(S/X) + ¡

r − q + σ²/2¢ τ σ√

τ .

• Formulas (29) and (29⁰) 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

• 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

S₀e^rt+ησ√t, Xe−r(T −t)+ησ√

T −t´ , S_min(t) = max

³

S₀e^{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

• 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

h

max

c

h

max

c

h

max

c

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

µi + j − 1 i

¶

pⁱ(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−1Xµ

i + j − 1 i

¶

pⁱ(1 − p)^jP_ie^{−(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 Suⁱd^j:

The Analysis (concluded)

(pP_u + (1 − p) P_d) e^−r∆t

= £ p¡

X − Suⁱ⁺¹d^j¢

+ (1 − p)¡

X − Suⁱd^j+1¢¤

e^−r∆t

= Xe^−r∆t − Suⁱd^j(pu + (1 − p) d) e^−r∆t

= Xe^−r∆t − Suⁱd^je^−q∆t

≤ Xe^−r∆t − Suⁱd^je^−r∆t

≤ X − Suⁱd^j.