### Put-Call Parity (Castelli, 1877)

*C = P + S − PV(X).* (19)

*• Consider the portfolio of one short European call, one*
long European put, one share of stock, and a loan of
*PV(X).*

*• All options are assumed to carry the same strike price*
*and time to expiration, τ .*

*• The initial cash flow is therefore*

*C − P − S + PV(X).*

### The Proof (continued)

*• At expiration, if the stock price S*_{τ}*≤ X, the put will be*
*worth X − S** _{τ}* and the call will expire worthless.

*• After the loan, now X, is repaid, the net future cash*
flow is zero:

*0 + (X − S*_{τ}*) + S*_{τ}*− X = 0.*

*• On the other hand, if S*_{τ}*> X, the call will be worth*
*S*_{τ}*− X and the put will expire worthless.*

*• After the loan, now X, is repaid, the net future cash*
flow is again zero:

### The Proof (concluded)

*• The net future cash flow is zero in either case.*

*• The no-arbitrage principle implies that the initial*

investment to set up the portfolio must be nil as well.

### Consequences of Put-Call Parity

*• There is only one kind of European option because the*
other can be replicated from it in combination with the
underlying stock and riskless lending or borrowing.

– Combinations such as this create synthetic securities.

*• S = C − P + PV(X) says a stock is equivalent to a*

portfolio containing a long call, a short put, and lending
*PV(X).*

*• C − P = S − PV(X) implies a long call and a short put*
amount to a long position in stock and borrowing the
PV of the strike price (buying stock on margin).

### Intrinsic Value

*Lemma 1 An American call or a European call on a*
*non-dividend-paying stock is never worth less than its*
*intrinsic value.*

*• The put-call parity implies*

*C = (S − X) + (X − PV(X)) + P ≥ S − X.*

*• Recall C ≥ 0.*

*• It follows that C ≥ max(S − X, 0), the intrinsic value.*

*• An American call also cannot be worth less than its*
intrinsic value.

### Intrinsic Value (concluded)

A European put on a non-dividend-paying stock may be worth less than its intrinsic value (p. 161).

*Lemma 2 For European puts, P ≥ max(PV(X) − S, 0).*

*• Prove it with the put-call parity.*

*• Can explain the right figure on p. 161 why P < X − S*
*when S is small.*

### Early Exercise of American Calls

European calls and American calls are identical when the underlying stock pays no dividends.

*Theorem 3 (Merton (1973)) An American call on a*
*non-dividend-paying stock should not be exercised before*
*expiration.*

*• By an exercise in text, C ≥ max(S − PV(X), 0).*

*• If the call is exercised, the value is the smaller S − X.*

### Remarks

*• The above theorem does not mean American calls*
should be kept until maturity.

*• What it does imply is that when early exercise is being*
*considered, a better alternative is to sell it.*

*• Early exercise may become optimal for American calls*
on a dividend-paying stock.

– Stock price declines as the stock goes ex-dividend.

### Early Exercise of American Calls: Dividend Case

Surprisingly, an American call should be exercised only at a few dates.

*Theorem 4 An American call will only be exercised at*
*expiration or just before an ex-dividend date.*

In contrast, it might be optimal to exercise an American put even if the underlying stock does not pay dividends.

### Convexity of Option Prices

*Lemma 5 For three otherwise identical calls or puts with*
*strike prices X*_{1} *< X*_{2} *< X*_{3}*,*

*C*_{X}_{2} *≤ ωC*_{X}_{1} *+ (1 − ω) C*_{X}_{3}
*P*_{X}_{2} *≤ ωP*_{X}_{1} *+ (1 − ω) P*_{X}_{3}
*Here*

*ω ≡ (X*_{3} *− X*_{2}*)/(X*_{3} *− X*_{1}*).*

*(Equivalently, X*_{2} *= ωX*_{1} *+ (1 − ω) X*_{3}*.)*

### The Intuition behind Lemma 5

^{a}

*• Consider ωC*_{X}_{1} *+ (1 − ω) C*_{X}_{3} *− C*_{X}_{2}.

*• This is a butterfly spread (p. 171).*

*• It has a nonnegative value as*

*ω max(S−X*_{1}*, 0)+(1−ω) max(S−X*_{3}*, 0)−max(S−X*_{2}*, 0) ≥ 0.*

*• Therefore, ωC*_{X}_{1} *+ (1 − ω) C*_{X}_{3} *− C*_{X}_{2} *≥ 0.*

*• In the limit, ∂*^{2}*C/∂X*^{2} *≥ 0.*

aContributed by Mr. Cheng, Jen-Chieh (B96703032) on March 17, 2010.

### Option on Portfolio vs. Portfolio of Options

An option on a portfolio of stocks is cheaper than a portfolio of options.

*Theorem 6 Consider a portfolio of non-dividend-paying*
*assets with weights ω*_{i}*. Let C*_{i}*denote the price of a*

*European call on asset i with strike price X*_{i}*. Then the call*
*on the portfolio with a strike price X ≡* P

*i* *ω*_{i}*X*_{i}*has a value*
*at most* P

*i* *ω*_{i}*C*_{i}*. All options expire on the same date.*

The same result holds for European puts.

*Option Pricing Models*

If the world of sense does not fit mathematics, so much the worse for the world of sense.

— Bertrand Russell (1872–1970) Black insisted that anything one could do with a mouse could be done better with macro redefinitions of particular keys on the keyboard.

— Emanuel Derman,
*My Life as a Quant (2004)*

### The Setting

*• The no-arbitrage principle is insufficient to pin down the*
exact option value.

*• Need a model of probabilistic behavior of stock prices.*

*• One major obstacle is that it seems a risk-adjusted*
interest rate is needed to discount the option’s payoff.

*• Breakthrough came in 1973 when Black (1938–1995)*
and Scholes with help from Merton published their
celebrated option pricing model.

– Known as the Black-Scholes option pricing model.

### Terms and Approach

*• C: call value.*

*• P : put value.*

*• X: strike price*

*• S: stock price*

*• ˆr > 0: the continuously compounded riskless rate per*
period.

*• R ≡ e*^{r}^{ˆ}: gross return.

*• Start from the discrete-time binomial model.*

### Binomial Option Pricing Model (BOPM)

*• Time is discrete and measured in periods.*

*• If the current stock price is S, it can go to Su with*
*probability q and Sd with probability 1 − q, where*
*0 < q < 1 and d < u.*

*– In fact, d < R < u must hold to rule out arbitrage.*

*• Six pieces of information suffice to determine the option*
*value based on arbitrage considerations: S, u, d, X, ˆr,*
and the number of periods to expiration.

### S

### Su q

### 1 q

### Sd

### Call on a Non-Dividend-Paying Stock: Single Period

*• The expiration date is only one period from now.*

*• C** _{u}* is the call price at time one if the stock price moves

*to Su.*

*• C** _{d}* is the call price at time one if the stock price moves

*to Sd.*

*• Clearly,*

*C*_{u}*= max(0, Su − X),*
*C*_{d}*= max(0, Sd − X).*

C

Cu= max( 0, Su X ) q

1 q

Cd = max( 0, Sd X )

### Call on a Non-Dividend-Paying Stock: Single Period (continued)

*• Set up a portfolio of h shares of stock and B dollars in*
riskless bonds.

*– This costs hS + B.*

*– We call h the hedge ratio or delta.*

*• The value of this portfolio at time one is either*
*hSu + RB or hSd + RB.*

*• Choose h and B such that the portfolio replicates the*
payoff of the call,

*hSu + RB* *= C*_{u}*,*
*hSd + RB* *= C*_{d}*.*

### Call on a Non-Dividend-Paying Stock: Single Period (concluded)

*• Solve the above equations to obtain*
*h =* *C*_{u}*− C*_{d}

*Su − Sd* *≥ 0,* (20)

*B =* *uC*_{d}*− dC*_{u}

*(u − d) R* *.* (21)

*• By the no-arbitrage principle, the European call should*
*cost the same as the equivalent portfolio, C = hS + B.*

*• As uC*_{d}*− dC*_{u}*< 0, the equivalent portfolio is a levered*
long position in stocks.

### American Call Pricing in One Period

*• Have to consider immediate exercise.*

*• C = max(hS + B, S − X).*

*– When hS + B ≥ S − X, the call should not be*
exercised immediately.

*– When hS + B < S − X, the option should be*
exercised immediately.

*• For non-dividend-paying stocks, early exercise is not*
optimal by Theorem 3 (p. 191).

*• So C = hS + B.*

### Put Pricing in One Period

*• Puts can be similarly priced.*

*• The delta for the put is (P*_{u}*− P*_{d}*)/(Su − Sd) ≤ 0, where*
*P*_{u}*= max(0, X − Su),*

*P*_{d}*= max(0, X − Sd).*

*• Let B =* ^{uP}_{(u−d) R}^{d}^{−dP}* ^{u}*.

*• The European put is worth hS + B.*

*• The American put is worth max(hS + B, X − S).*

– Early exercise is always possible with American puts.

### Risk

*• Surprisingly, the option value is independent of q.*

*• Hence it is independent of the expected gross return of*
*the stock, qSu + (1 − q) Sd.*

*• It therefore does not directly depend on investors’ risk*
preferences.

*• The option value depends on the sizes of price changes,*
*u and d, which the investors must agree upon.*

*• Note that the set of possible stock prices is the same*
*whatever q is.*

*Can You Figure Out u, d without Knowing q?*

^{a}

*• Yes, you can under BOPM.*

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

*u is available*

z }| {

*S, Su,* *Su*^{2}*, Su*^{3}*, Su*^{3}*d*

| {z }

*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.

### Pseudo Probability

*• After substitution and rearrangement,*

*hS + B =*

³*R−d*
*u−d*

´

*C** _{u}* +

³*u−R*
*u−d*

´
*C*_{d}

*R* *.*

*• Rewrite it as*

*hS + B =* *pC*_{u}*+ (1 − p) C*_{d}

*R* *,*

where

*p ≡* *R − d*
*u − d* *.*

*• As 0 < p < 1, it may be interpreted as a probability.*

### Risk-Neutral Probability

*• The expected rate of return for the stock is equal to the*
riskless rate ˆ*r under p as pSu + (1 − p) Sd = RS.*

*• The expected rates of return of all securities must be the*
riskless rate when investors are risk-neutral.

*• For this reason, p is called the risk-neutral probability.*

*• The value of an option is the expectation of its*

discounted future payoff in a risk-neutral economy.

*• So the rate used for discounting the FV is the riskless*
rate in a risk-neutral economy.

### Binomial Distribution

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

*b(j; n, p) ≡*

µ*n*
*j*

¶

*p*^{j}*(1 − p)** ^{n−j}* =

*n!*

*j! (n − j)!* *p*^{j}*(1 − p)*^{n−j}*.*
*– n! = n × (n − 1) · · · 2 × 1 with the convention 0! = 1.*

*• Suppose you toss 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.

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 one can be obtained by applying*
the same logic:

*C** _{u}* =

*pC*

_{uu}*+ (1 − p) C*

_{ud}*R* *,* (22)

*C** _{d}* =

*pC*

_{ud}*+ (1 − p) C*

_{dd}*R* *.*

*• Deltas can be derived from Eq. (20) on p. 206.*

*• For example, the delta at C** _{u}* is

*C*

_{uu}*− C*

_{ud}*.*

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

*• We now reach the current period.*

*• An equivalent portfolio of h shares of stock and $B*
*riskless bonds can be set up for the call that costs C*_{u}*(C*_{d}*, resp.) if the stock price goes to Su (Sd, resp.).*

*• The values of h and B can be derived from*
Eqs. (20)–(21) on p. 206.

*• That is, compute*

*pC*_{u}*+ (1 − p) C*_{d}*R*

as the price.

### Early Exercise

*• Since the call will not be exercised at time one even if it*
*is American, C*_{u}*≥ Su − X and C*_{d}*≥ Sd − X.*

*• Therefore,*

*hS + B* = *pC*_{u}*+ (1 − p) C*_{d}

*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* *.*

### Backward Induction of Zermelo (1871–1953)

*• 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. (22) on p. 218.

*• This recursive procedure is called backward induction.*

*• Now, C equals*

*[ p*^{2}*C*_{uu}*+ 2p(1 − p) C*_{ud}*+ (1 − p)*^{2}*C*_{dd}*](1/R*^{2})

= *[ p*^{2} max¡

*0, Su*^{2} *− X*¢

*+ 2p(1 − p) max (0, Sud − X)*
*+(1 − p)*^{2} max¡

*0, Sd*^{2} *− X*¢

*]/R*^{2}*.*

*S*_{0}
1

*

j

*S*_{0}*u*
*p*

*

j

*S*_{0}*d*
*1 − p*

*

j

*S*_{0}*u*^{2}
*p*^{2}

*S*_{0}*ud*

*2p(1 − p)*

*S*_{0}*d*^{2}

### Backward Induction (concluded)

*• In the n-period case,*
*C =*

P_{n}

*j=0*

¡_{n}

*j*

¢*p*^{j}*(1 − p)*^{n−j}*× max*¡

*0, Su*^{j}*d*^{n−j}*− X*¢

*R*^{n}*.*

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

risk-neutral economy.

*• The value of a European put is*
*P =*

P_{n}

*j=0*

¡_{n}

*j*

¢*p*^{j}*(1 − p)*^{n−j}*× max*¡

*0, X − Su*^{j}*d** ^{n−j}*¢

*R*^{n}*.*

### 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*^{−ˆ}^{rn}*E*^{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

### Self-Financing

*• Delta changes over time.*

*• The maintenance of an equivalent portfolio is dynamic.*

*• The maintaining of an equivalent portfolio does not*
depend on our correctly 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.

– Changes in value are due entirely to capital gains.

### 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 such that*
*Su*^{a}*d*^{n−a}*≥ X,*

or, equivalently,

*a =*

»*ln(X/Sd** ^{n}*)¼

*.*

### The Binomial Option Pricing Formula (concluded)

*• Hence,*

*C*

=

P_{n}

*j=a*

¡_{n}

*j*

¢*p*^{j}*(1 − p)** ^{n−j}* ¡

*Su*^{j}*d*^{n−j}*− X*¢

*R** ^{n}* (23)

*= S*

X*n*
*j=a*

µ*n*
*j*

¶*(pu)*^{j}*[ (1 − p) d ]*^{n−j}*R*^{n}

*−* *X*
*R*^{n}

X*n*
*j=a*

µ*n*
*j*

¶

*p*^{j}*(1 − p)*^{n−j}

*= S*

X*n*
*j=a*

*b (j; n, pu/R) − Xe*^{−ˆ}^{rn}

X*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)

60 (0.189)

20 (0.027) Binomial process for the stock price

(probabilities in parentheses)

360 (0.49)

120 (0.42)

40 (0.09) 240

(0.7)

80 (0.3)

85.069 (0.82031)

390

30

0

0 Binomial process for the call price

(hedge ratios in parentheses)

235 (1.0)

17.5 (0.25)

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)

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.*

*• 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.*

### 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}).

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

*– Can be further reduced to O(n) by reusing space*

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

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

## c

0,Sud^{2}X

## h

max

## c

0,Su d^{2}X

## h

max

## c

0,Su^{3}X

## h

max

## c

0,Sd^{3}X

## h

1 p

1 p

1 p

1 p

1 p

1 p

### Further 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 ]:*

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

*• It runs in O(n) steps.*

### Optimal Algorithm (concluded)

*• With the b(j; n, p) available, the risk-neutral valuation*
formula (23) on p. 227 is trivial to compute.

*• 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

### On 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.

*• 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)

*• Use*
b

*µ ≡* 1
*n* *E*

·

ln *S*_{τ}*S*

¸

and b*σ*^{2} *≡* 1

*n* Var

·

ln *S*_{τ}*S*

¸

to denote, resp., the expected value and variance of the continuously compounded rate of return per period.

*• 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).*

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

*• 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.*

*• 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.*

*• Impose ud = 1 to make nodes at the same horizontal*
level of the tree have identical price (review p. 237).

– Other choices are possible (see text).

### 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* *. (24)*

*• With Eqs. (24),*

*nbµ = µτ,*
*nbσ*^{2} =

·

*1 −* *³ µ*
*σ*

´_{2} *τ*
*n*

¸

*σ*^{2}*τ → σ*^{2}*τ.*

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

*• The no-arbitrage inequalities d < R < u may not hold*
under Eqs. (24) on p. 246.

– 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}*,*

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

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

– Other solutions will be presented later.

aMany papers forget to check this!

### 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*
*the normal distribution with mean µτ and variance*
*σ*^{2}*τ .*

*• So ln S** _{τ}* approaches the normal distribution with mean

*µτ + ln S and variance σ*

^{2}

*τ .*

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

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

*Lemma 7 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 → ∞.*

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

*• By Lemma 7 (p. 249) and Eq. (18) on p. 151, the*
expected stock price at expiration in a risk-neutral
*economy is Se** ^{rτ}*.

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

a*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 8 (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!