*Arbitrage in Option Pricing*

All general laws are attended with inconveniences, when applied to particular cases.

— David Hume (1711–1776)

### Arbitrage

*• The no-arbitrage principle says there is no free lunch.*

*• It supplies the argument for option pricing.*

*• A riskless arbitrage opportunity is one that, without any*
initial investment, generates nonnegative returns under
all circumstances and positive returns under some.

*• In an eﬃcient market, such opportunities do not exist*
(for long).

### Portfolio Dominance Principle

*• Consider two portfolios A and B.*

*• A should be more valuable than B if A’s payoﬀ is at*
least as good as B’s under all circumstances and better
under some.

### Two Simple Corollaries

*• A portfolio yielding a zero return in every possible*
scenario must have a zero PV.

**– Short the portfolio if its PV is positive.**

**– Buy it if its PV is negative.**

**– In both cases, a free lunch is created.**

*• Two portfolios that yield the same return in every*
possible scenario must have the same price.^{a}

aAristotle, “those who are equal should have everything alike.”

### The PV Formula (p. 32) Justified

**Theorem 1 For a certain cash ﬂow C**_{1}*, C*_{2}*, . . . , C*_{n}*,*
*P =*

*n*
*i=1*

*C*_{i}*d(i).*

*• Suppose the price P*^{∗}*< P .*

*• Short the n zeros that match the security’s n cash ﬂows.*

*• The proceeds are P dollars.*

6 6 6 6 -

*C*1 *C*2 *C*3

*· · ·* *C**n*

? ? ? ?

*C*1 *C*2 *C*3

*· · ·*

*C**n*

6

*P*

?

*P*^{∗}

_{security}

_{zeros}

### The Proof (concluded)

*• Then use P** ^{∗}* of the proceeds to buy the security.

*• The cash inﬂows of the security will oﬀset exactly the*
obligations of the zeros.

*• A riskless proﬁt of P − P** ^{∗}* dollars has been realized now.

*• If P*^{∗}*> P , just reverse the trades.*

### Two More Examples

*• A put or a call must have a nonnegative value.*

**– Suppose otherwise and the option has a negative**
price.

**– Buy the option for a positive cash ﬂow now.**

**– It will end up with a nonnegative amount at**
expiration.

**– So an arbitrage proﬁt is realized now.**

### Two More Examples (continued)

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

**– This is true if the intrinsic value is zero (p. 204).**

**– Suppose the intrinsic value is positive but the claim**
is false.

**– So the American option is cheaper than its intrinsic**
value.

**– For the call: Short the stock and lend X dollars.**

**– For the put: Borrow X dollars and buy the stock.**

### Two More Examples (concluded)

*• (continued)*

**– In either case, the payoﬀ is the intrinsic value.**

**– At the same time, buy the option, promptly exercise**
it, and close the stock position.

*∗ For the call, call the lent money to exercise it.*

*∗ For the put, deliver the stock and use the received*
strike price to settle the debt.

**– The cost of buying the option is less than the**
intrinsic value.

**– So there is an immediate arbitrage proﬁt.**

### Relative Option Prices

*• These relations hold regardless of the model for stock*
prices.

*• Assume, among other things, that there are no*

transactions costs or margin requirements, borrowing and lending are available at the riskless interest rate, interest rates are nonnegative, and there are no

arbitrage opportunities.

*• Let the current time be time zero.*

*• PV(x) stands for the PV of x dollars at expiration.*

*• Hence PV(x) = xd(τ) where τ is the time to*

### Put-Call Parity

^{a}

*C = P + S − PV(X).* (26)

*• Consider the portfolio of:*

**– One short European call;**

**– One long European put;**

**– One share of stock;**

**– A loan of PV(X).**

*• All options are assumed to carry the same strike price X*
*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.

*• The loan is now X.*

*• 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.*

*• 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 (p. 200) 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.*

**– 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 2 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 (p. 205).

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

*• Recall C ≥ 0 (p. 204).*

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

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

*• Can explain the right ﬁgure on p. 177 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 4 (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 5 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 6 (Cox and Rubinstein (1985)) Any**

*piecewise linear payoﬀ function can be replicated using a*
*portfolio of calls and puts.*

**Corollary 7 Any suﬃciently well-behaved payoﬀ function***can be approximated by a portfolio of calls and puts.*

aSee Exercise 8.3.6 of the textbook.

### Convexity of Option Prices

^{a}

**Lemma 8 For three otherwise identical calls or puts with***strike prices X*1 *< X*2 *< X*3*,*

*C** _{X}*2

*≤ ωC*

*1 + (1*

_{X}*− ω) C*

*3*

_{X}*P** _{X}*2

*≤ ωP*

*1 + (1*

_{X}*− ω) P*

*3*

_{X}*Here*

*ω ≡ (X*3 *− X*2*)/(X*3 *− X*1*).*

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

aSee Lemma 8.5.1 of the textbook.

### The Intuition behind Lemma 8

^{a}

*• Set up the following portfolio:*

*ωC** _{X}*1

*− C*

*2 + (1*

_{X}*− ω) C*

*3*

_{X}*.*

*• This is a butterﬂy spread (p. 188).*

*• It has a nonnegative value as, for any S at maturity,*

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

*• Therefore,*

*ωC** _{X}*1

*− C*

*X*2 + (1

*− ω) C*

*X*3

*≥ 0.*

### Option on a Portfolio vs. Portfolio of Options

*• 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}*.

*• All options expire on the same date.*

### Option on a Portfolio vs. Portfolio of Options (concluded)

An option on a portfolio is cheaper than a portfolio of
options.^{a}

**Theorem 9 The call on the portfolio with a strike price**

*X ≡*

*i*

*ω*_{i}*X*_{i}*has a value at most*

*i*

*ω*_{i}*C*_{i}*.*
*The same 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.^{a}

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

aThe results were obtained as early as June 1969.

### 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** ^{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,* (27)

*B* = *uC*_{d}*− dC*_{u}

*(u − d) R* *.* (28)

*• 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 4 (p. 214).

*• So*

### 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.*

*• 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.*

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

### 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 ≡* *R − d*

*u − d* *.* (29)

*• 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*

### 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 or recombines.**

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

a

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* *,* (30)

*C** _{d}* =

*pC*

*+ (1*

_{ud}*− p) C*

_{dd}*R* *.*

*• Deltas can be derived from Eq. (27) on p. 232.*

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

*C* *− C*

### 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. (27)–(28) on p. 232.

### 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* *.*

### 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. (30) on p. 243.

*• 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}*.*

*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 (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}*≡*

_{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. 191. 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}*−*

*−1* *j−1* *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 ].*(31)

**– 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.**

a

### Hakansson’s Paradox

^{a}

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

aHakansson (1979).

*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*

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

*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*

*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}* (32)

= *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*

*b (j; n, pu/R) − Xe*^{−ˆrn}

*n*

*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 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 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*

### 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*

### 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.

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