Option on a Non-Dividend-Paying Stock: Multi-Period

Binomial Distribution

• Denote the binomial distribution with parameters n and p by

b(j; n, p) ≡

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.

Option on a Non-Dividend-Paying Stock: Multi-Period

• Consider a call with two periods remaining before expiration.

• Under the binomial model, the stock can take on three possible prices at time two: Suu, Sud, and Sdd.

– There are 4 paths.

– But the tree combines.

• At any node, the next two stock prices only depend on the current price, not the prices of earlier times.^a

aIt is Markovian.

S

Su

Sd

Suu

Sud

Sdd

Option on a Non-Dividend-Paying Stock: Multi-Period (continued)

• Let C_uu be the call’s value at time two if the stock price is Suu.

• Thus,

C_uu = max(0, Suu − X).

• C_ud and C_dd can be calculated analogously, C_ud = max(0, Sud − X),

C_dd = max(0, Sdd − X).

C

Cu

Cd

Cuu= max( 0, Suu X )

Cud = max( 0, Sud X )

Cdd = max( 0, Sdd X )

Option on a Non-Dividend-Paying Stock: Multi-Period (continued)

• The call values at time 1 can be obtained by applying the same logic:

C_u = pC_uu + (1 − p) Cud

R , (26)

C_d = pC_ud + (1 − p) C_dd

R .

• Deltas can be derived from Eq. (23) on p. 228.

• For example, the delta at C_u is C_uu − C_ud Suu − Sud.

Option on a Non-Dividend-Paying Stock: Multi-Period (concluded)

• We now reach the current period.

• Compute

pC_u + (1 − p) C_d R

as the option price.

• The values of delta h and B can be derived from Eqs. (23)–(24) on p. 228.

Early Exercise

• Since the call will not be exercised at time 1 even if it is American, C_u ≥ Su − X and C_d ≥ Sd − X.

• Therefore,

hS + B = pC^u + (1 − p) Cd

R ≥ [ pu + (1 − p) d ] S − X R

= S − X

R > S − X.

– The call again will not be exercised at present.^a

• So

C = hS + B = pC_u + (1 − p) C_d

R .

aConsistent with Theorem 4 (p. 210).

Backward Induction

• The above expression calculates C from the two successor nodes C_u and C_d and none beyond.

• The same computation happened at C_u and C_d, too, as demonstrated in Eq. (26) on p. 239.

• This recursive procedure is called backward induction.

• C equals

[ p²Cuu + 2p(1 − p) Cud + (1 − p)²Cdd](1/R²)

= [ p² max

0, Su² − X

+ 2p(1 − p) max (0, Sud − X) +(1 − p)² max

0, Sd² − X

]/R².

S0

1

*

j

S0u p

*

j

S0d 1 − p

*

j

S0u² p²

S0ud

2p(1 − p)

S0d² (1 − p)²

Backward Induction (continued)

• In the n-period case, C =

_n

j=0

_n

j

p^j(1 − p)^n−j × max

0, Su^jd^n−j − X

Rⁿ .

– The value of a call on a non-dividend-paying stock is the expected discounted payoff at expiration in a

risk-neutral economy.

• Similarly, P =

_n

j=0

_n

j

p^j(1 − p)^n−j × max

0, X − Su^jd^n−j

Rⁿ .

Backward Induction (concluded)

• Note that

p_j ≡

_n

j

 p^j(1 − p)^n−j Rⁿ

is the state price^a for the state Su^jd^n−j, j = 0, 1, . . . , n.

• In general,

option price = 

j

p_j × payoff at state j.

aRecall p. 187. One can obtain the undiscounted state price _n

j

p^j(1− p)^n−j—the risk-neutral probability—for the state Su^jd^n−j with (X_M − X_L)⁻¹ units of the butterfly spread where X_L = Su^j−1d^n−j+1, X_M = Su^jd^n−j, and X_H = Su^j−1+1d^n−j−1.

Risk-Neutral Pricing Methodology

• Every derivative can be priced as if the economy were risk-neutral.

• For a European-style derivative with the terminal payoff function D, its value is

e^−ˆrnE^π[D ].

– E^π means the expectation is taken under the risk-neutral probability.

• The “equivalence” between arbitrage freedom in a model and the existence of a risk-neutral probability is called the (first) fundamental theorem of asset pricing.

Self-Financing

• Delta changes over time.

• The maintenance of an equivalent portfolio is dynamic.

• But it does not depend on predicting future stock prices.

• The portfolio’s value at the end of the current period is precisely the amount needed to set up the next portfolio.

• The trading strategy is self-financing because there is neither injection nor withdrawal of funds throughout.^a

– Changes in value are due entirely to capital gains.

aExcept at the beginning, of course, when you have to put up the option value C or P before the replication starts.

Hakansson’s Paradox

• If options can be replicated, why are they needed at all?

aHakansson (1979).

Can You Figure Out u, d without Knowing q?

• Yes, you can, under BOPM.

• Let us observe the time series of past stock prices, e.g.,

u is available

 

S, Su, Su², Su  ³, Su³d

d is available

, . . .

• So with sufficiently long history, you will figure out u and d without knowing q.

aContributed by Mr. Hsu, Jia-Shuo (D97945003) on March 11, 2009.

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 Su^jd^n−j ≥ X,

or, equivalently,

a =

ln(X/Sdⁿ) ln(u/d)

.

The Binomial Option Pricing Formula (concluded)

• Hence,

C

=

_n

j=a

_n

j

p^j(1 − p)^n−j 

Su^jd^n−j − X

Rⁿ (27)

= S

n j=a

n j

(pu)^j[ (1 − p) d ]^n−j Rⁿ

− X Rⁿ

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).

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 and invest $85.069 in the replicating portfolio with 0.82031 shares of stock required by 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 dollars.

Numerical Examples (continued)

Time 3 (the case of rising price):

• The stock price moves to $180.

• The call we wrote finishes in the money.

• For a loss of 180 − 150 = 30 dollars, close out the

position by either buying back the call or buying a share of stock for delivery.

• 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

• Replicate an option using stocks and bonds.

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

• Hedge the options we issued.

– Set up a portfolio to replicate the call with $85.069 to counterbalance its values exactly.^b

• · · ·

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

bHedge and replication are mirror images.

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.

C[2][0]

C[2][1]

C[2][2]

C[1][0]

C[1][1]

C[0][0]

p

p

p p

p p

max ,

?

D

max ,

?

D

max ,

?

D

max ,

?

D

1 p

1 p

1 p

1 p

1 p

1 p

Further Time Improvement for Calls

0

0 0

All zeros

X

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. 251 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(n²) time.

The Bushy Tree

S

Su

Sd

Su²

Sud

Sdu

Sd²

2ⁿ

n

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 r to match the empirical results as n → ∞.ˆ

• First, ˆr = rτ/n.

– The period gross return R = e^ˆr.

Toward the Black-Scholes Formula (continued)

• Let

μ ≡ 1 n E



ln S_τ S



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

• Let

σ² ≡ 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

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.

Toward the Black-Scholes Formula (continued)

• The BOPM converges to the distribution only if nμ = n[ q ln(u/d) + ln d ] → μτ, nσ² = 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. 263).

– 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

τ

n . (28)

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

nσ² =



1 − 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. 232.

• 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. 274.

– 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 > r²τ /σ² (check it).

– So it goes away if n is large enough.

– Other solutions will be presented later.

aMany papers and programs forget to check this condition!

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. 278) and Eq. (21) on p. 160.

bIn the sense of (1/τ ) ln E[ Sτ/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, 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 (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?

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.