### Consequences of Put-Call Parity

*• There is only one kind of European option.*

**– The other can be replicated from it in combination**
with stock and riskless lending or borrowing.

**– Combinations such as this create synthetic securities.**

*• S = C − P + PV(X): A stock is equivalent to a portfolio*
*containing a long call, a short put, and lending PV(X).*

*• C − P = S − PV(X): 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 3 An American call or a European call on a***non-dividend-paying stock is never worth less than its*
*intrinsic value.*

*• An American call cannot be worth less than its intrinsic*
value.^{a}

*• For European options, the put-call parity implies*
*C = (S − X) + (X − PV(X)) + P ≥ S − X.*

*• Recall C ≥ 0 (p. 218).*

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

### Intrinsic Value (concluded)

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

**Lemma 4 For European puts, P ≥ max(PV(X) − S, 0).**

*• Prove it with the put-call parity.*^{a}

*• Can explain the right ﬁgure on p. 190 why P < X − S*
*when S is small.*

aSee Lemma 8.3.2 of the textbook.

### Early Exercise of American Calls

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

**Theorem 5 (Merton, 1973) An American call on a***non-dividend-paying stock should not be exercised before*
*expiration.*

*• By Exercise 8.3.2 of the text, C ≥ max(S − PV(X), 0).*

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

*• But*

*max(S − PV(X), 0) ≥ 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, however.

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

**– And recall that we assume options are unprotected.**

### Early Exercise of American Calls: Dividend Case

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

**Theorem 6 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.

aSee Theorem 8.4.2 of the textbook.

### A General Result

^{a}

**Theorem 7 (Cox & Rubinstein, 1985) Any piecewise***linear payoﬀ function can be replicated using a portfolio of*
*calls and puts.*

**Corollary 8 Any suﬃciently well-behaved payoﬀ function***can be approximated by a portfolio of calls and puts.*

aSee Exercise 8.3.6 of the textbook.

*Option Pricing Models*

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.^{a}

**– Known as the Black-Scholes option pricing model.**

aThe results were obtained as early as June 1969. Merton and Scholes

### 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*^{Δ} * ^{ˆr}*: 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.**^{a}

*• Six pieces of information will suﬃce to determine the*
option value based on arbitrage considerations:

*S, u, d, X, ˆr, and the number of periods to*
expiration.

aSee Exercise 9.2.1 of the textbook.

### 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 1 if the stock price moves to

*Su.*

*• C** _{d}* is the call price at time 1 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*
*hSu + RB,* *up move,*

*hSd + RB,* *down move.*

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

*• 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,* (30)

*B* = *uC*_{d}*− dC*_{u}

*(u − d) R* *.* (31)

*• By the no-arbitrage principle, the European call should*
cost the same as the equivalent portfolio,^{a}

*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 5 (p. 226).

*• 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.*^{a}

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

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

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

a*More precisely, not directly dependent on q. Thanks to a lively class*
discussion on March 16, 2011.

### Pseudo Probability

*• After substitution and rearrangement,*

*hS + B =*

*R−d*
*u−d*

*C** _{u}* +

*u−R*
*u−d*

*C*_{d}

*R* *.*

*• Rewrite it as*

*hS + B =* *pC** _{u}* + (1

*− p) C*

_{d}*R* *,*

where

*p* =^{Δ} *R − d*

*u − d* *.* (32)

*• 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.* (33)

*• 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.*

### 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 or recombines.**

*• 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*

*+ (1*

_{uu}*− p) C*

*ud*

*R* *,* (34)

*C** _{d}* =

*pC*

*+ (1*

_{ud}*− p) C*

_{dd}*R* *.*

*• Deltas can be derived from Eq. (30) on p. 240.*

*• 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. (30)–(31) on p. 240.

### 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) 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* *.*

a

### Backward Induction

^{a}

*• 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. (34) on p. 250.

*• This recursive procedure is called backward induction.*

*• 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}*.*

aErnst Zermelo (1871–1953).

*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}
(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*^{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.

*• Similarly,*
*P =*

_{n}

*j=0*

_{n}

*j*

*p** ^{j}*(1

*− p)*

^{n−j}*× max*

*0, X − Su*^{j}*d*^{n−j}

*R*^{n}*.*

### Backward Induction (concluded)

*• Note that*

*p** _{j}* =

^{Δ}

_{n}

*j*

*p** ^{j}*(1

*− p)*

^{n−j}*R*

^{n}is the state price^{a} *for the state Su*^{j}*d*^{n−j}*, j = 0, 1, . . . , n.*

*• In general,*

option price =

*j*

*p*_{j}*× payoﬀ at state j.*

aRecall p. 205. One can obtain the undiscounted state price _{n}

*j*

*p** ^{j}*(1

*−*

*p)*

^{n−j}*—the risk-neutral probability—for the state Su*

^{j}*d*

^{n−j}*with (X*

_{M}*−*

*X*

*)*

_{L}

^{−1}*units of the butterfly spread where X*

_{L}*= Su*

^{j−1}*d*

^{n−j+1}*, X*

*=*

_{M}*Su*

^{j}*d*

^{n−j}*, and X*

*= Su*

^{j−1+1}*d*

*. See Bahra (1997).*

^{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*^{−ˆrn}*E** ^{π}*[

*D ].*(35)

**– E***means the expectation is taken under the*

^{π}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.*

### Binomial Distribution

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

*b(j; n, p)* =^{Δ}

*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.*

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

or, equivalently,

*a =*

*ln(X/Sd** ^{n}*)

*ln(u/d)*

*.*

### The Binomial Option Pricing Formula (concluded)

*• Hence,*

*C*

=

_{n}

*j=a*

_{n}

*j*

*p** ^{j}*(1

*− p)*

^{n−j}*Su*^{j}*d*^{n−j}*− X*

*R** ^{n}* (36)

= *S*

*n*
*j=a*

*n*
*j*

*(pu)** ^{j}*[ (1

*− p) d ]*

^{n−j}*R*

^{n}*−* *X*
*R*^{n}

*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).* (37)

### 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

^{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.

### 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}) because there are

*∼ n*^{2}*/2 nodes.*

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

**– 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 ,

## ?

0 Sud^{2}X

## D

max ,

## ?

0 Su d X^{2}

## D

max ,

## ?

0 Su^{3}X

## D

max ,

## ?

0 Sd^{3}X

## 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 (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(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(n*^{2}) time.

### 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.^{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-

### 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

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

*• First, ˆr = rτ/n.*

**– Each period is τ /n years long.**

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

*• Let*

*μ* =^{Δ} 1
*n* *E*

ln *S*_{τ}*S*

denote the expected value of the continuously

compounded rate of return per period of the BOPM.

*• Let*

*σ*^{2 Δ}= 1

*n* Var

ln *S*_{τ}*S*

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

*• Under the BOPM, it is not hard to show that*^{a}
*μ = q ln(u/d) + ln d,*

*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.*

aThe Bernoulli distribution.

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

*• The BOPM converges to the distribution only if*

*nμ = n[ q ln(u/d) + ln d ] → μτ,* (38)
*nσ*^{2} = *nq(1 − q) ln*^{2}*(u/d) → σ*^{2}*τ.* (39)

*• 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. 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).

### 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* *.* (40)

*• With Eqs. (40), it can be checked that*
*nμ = μτ,*

*nσ*^{2} =

1 *−* *μ*
*σ*

_{2} *τ*
*n*

*σ*^{2}*τ → σ*^{2}*τ.*

### 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).

b*Recall Eq. (33) on p. 245. So u and d are related to volatility.*

### 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*
*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.**

aMany papers and programs forget to check this condition!

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

*• 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 9 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*^{Δ} ^{rτ /n}*− d)/(u − d).*

*• Let n → ∞.*^{a}

*• Then μ = r − σ*^{2}*/2.*

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

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). In the latter*
*case, it would be r − σ*^{2}*/2 by Lemma 9.*

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

^{a}

**Theorem 10 (The Black-Scholes Formula)**

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

*τ ) − SN (−x),*
*where*

*x* =^{Δ} *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!

### 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 measure associated with
*N (x) and N (−x).*

### 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.*^{b}

aL. Chang & Palmer (2007).

bSee Exercise 9.3.8 of the textbook.

### 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.*

### TXO futures calls (September 25, 2015)

^{a}

**300**

**14** **200**
**9000**

**8500**
**16**

**8000** **100**

**7500**
**18**

**7000** **0**
**20**

**22**
**24**

ATM = $8132

aThe index futures closed at 8132. Plot supplied by Mr. Lok, U Hou

### 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.

*• 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*^{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.

### 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(S*_{τ}*/S) is*

_{τ}

0 *σ*^{2}*(t) dt*
*rather than σ*^{2}*τ .*

*• The annualized volatility to be used in the Black-Scholes*
formula should now be

_{τ}

0 *σ*^{2}*(t) dt*

*τ* *.*

aMerton (1973).

### Time-Dependent Instantaneous Volatility (concluded)

*• There is no guarantee that the implied volatility is*
constant.

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

*u = e*^{σ(t)}

*√**τ /n**,*
*d = e*^{−σ(t)}

*√**τ /n**.*

*• How to make the binomial tree combine?*^{a}

aAmin (1991); C. I. Chen (R98922127) (2011).

### 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