5 Binomial Trees

Lecture 6:Binomial Option Pricing Model

This lecture shows that if stock prices move in aparticular way we can replicate the payoff of an optionby dynamically trading the underlying stock and a bond.We will construct a replicating portfolio of the stock anda bond that has exactly the same payoff as the option.By no-arbitrage, the option must then have the sameprice as this replicating portfolio.

I. Example

II. Binomial Option Pricing Model

A. Binomial Stock PricesB. One-Period ModelC. Risk-Neutral Pricing and ProbabilitiesD. Two-Period Model

III. Dynamic Replication

IV. Multi-Period Models

A. Three-Period ModelB. n-Period Model

Binomial Option Pricing Model

Binomial Option Pricing

� The option pricing methodology used most on thestreet

� Developed by Cox, Ross and Rubinstein.

– “Option Pricing: A Simplified Approach”

� 1979, Journal of Financial Economics

� More flexible than Black-Scholes pricing,

– But based on the same key ideas.

– In particular, the use of “replicating portfolios”.

B35100 Robert Novy-Marx


We’ve used replicating portfolios in class already.

� How did we price a forward contract?

– We constructed a “synthetic forward.”

– I.e., a portfolio of traded assets that has the samepayoffs as the forward.

– Then priced the forward by invoking NA.

� We’d like to do the same thing with options.

– But there’s a problem: an option’s payoff is non-

linear in the underlying.

� The forward’s payoff was linear in the underlying;that’s what made replication so easy.

– No (static) portfolio of traded securities (e.g., theunderlying and bonds) is going to have the sameterminal payoff as the option.

� So what do we do?



We make a seemingly ridiculous simplification:

� We’re going to assume than in the future the stockwill take on one of only two possible prices.

� It’s actually not that bad an assumption.

– We’ll justify it in a little while, but first...

I. Example: a stock

Suppose the spot $60. We’re going to assume thatover the next “period” the stock price will

� fall to $30,

� or rise to $90.

We’ll also assume the risk-free simple interest rateover the period is 20%.

S D 60

Su D 90

Sd D 30



At the end of this period, a ATM call option (K D $60)is worth either $0 or $30:

c

cu D 30

cd D 0

Now let’s look at a portfolio

� long a half share of the underlying stock,

� and short $12:50 in bonds.

S=2 � 12:50

90=2 � 15 D 30

30=2 � 15 D 0

� It replicates the call’s payoffs.

� What is the portfolio worth initially?



The price of the portfolio is initially

1

2� $60 � $12:50 D $17:50:

The portfolio replicates the payoff of the call option,so the call must, by NA, have the same price, $17:50.

� What if the price of the call is $18:50?

Transaction Payoff at t Payoff at T

� What if the price of the call is $16:50?

Transaction Payoff at t Payoff at T



How do we “guess” the replicating portfolio?

We need to

� buy � shares of the underlying,

� and borrow D dollars

– i.e., short D worth of bonds,

such that

90� � 1:2D D 30

30� � 1:2D D 0:

In the previous example we “guessed” � D 0:5 andD D $12:5 because they solve these two equations,

90� � 1:2D D 30

�.30� � 1:2D D 0 /

60� D 30 ) � D 1=2

) 15 D 1:2D

) D D 12:5:



Option �:

Again, the delta of an option is the sensitivity of theoption price to the stock price.

In our example

� D@C

@S

D30 � 15

90 � 60D

1

2

D0 � 15

30 � 60D

1

2:

We know that � will always be between 0 and 1.

� Remember: in our numeric example,

C0 D 0:5 � S0 � 12:5 D 17:50:

� Why is it always be between 0 and 1?

Can we see this graphically?



S(T)

Payoff

Underlying

CK(T)

Cu

Cd

Sd

Su

Cu - Cd

Su - Sd

The sensitivity of the call to the underlying:

� DCu � Cd

Su � Sd

There should also be a short bond position.

� The y-intercept isn’t zero.



Why this works

Why can we find a replicating portfolio for the call?

� It works because the option payoff is linear in theunderlying if we assume only two possible futurestock prices.

– In technical jargon, the stock and the bond thenspan the option’s possible payoffs.

Because there’s a linear relation between the priceof the option and the underlying, we can think aboutthe sensitivity of the option’s price to the price of theunderlying.

� This sensitivity is called the hedge ratio, or“delta”, of the call.

– That’s why we choose to use the � symbol.

� We chose � such that the replicating portfolio hasthe same sensitivity to the stock price as the calloption.



Where do we use the probabilities of the up and downmoves?

� DCu � Cd

Su � Sd

� We don’t!

Important: The option price we got did not dependon any probabilities!

� At least not directly.

We don’t need the probabilities of the price moves tocompute the option price

� Our pricing only depended on NA

– I.e., constructing the replicating portfolio.

– It doesn’t depend on the likelihood of anyoutcomes.

– It’s model-independent.

� The methodology, not the tree.

Question: why doesn’t the option price depend onany probabilities, or the expected return of the stock?

� What’s the intuition?



� If we priced options by discounting future payoffs,

– i.e., like we do with stocks and bonds,

then we would need to use the probabilities ofdifferent outcomes.

� But we don’t. We price options as a function of thestock price, using NA.

– The current stock price already reflects whatcan happen to the stock price in the future.

– We don’t need to consider this again in derivingthe relation between the option price and thestock price.

Does this mean that news which leads to an increasein the probability of a stock price up-move doesn’taffect the call price?

� Of course not!

– But it affects the call price through its impact onthe current stock price.

� Expectations about future prices are embeddedin the current price.



II. Binomial Option Pricing Model

(A) Binomial Stock Prices

Obviously, stock prices can take on more than twovalues. However, we can increase the number ofoutcomes by shortening each period and takingmore steps. This results in a binomial tree:

S0

Su

Suu

Suuu

Suud

Sd

Sud

SuddS

ddS

ddd

We’ll argue (in lecture 7) that it’s reasonable toassume that stock returns are normally distributed.� With enough steps a multi-period tree can

approximate log-normal returns arbitrarily well.

To price an option with a multi-period tree, wejust solve the one-period tree repeatedly. Sowe’ll derive the general solution to the one-periodbinomial model first.



(B) One-Period Model

The payoffs of the stock are:

S

Su D uS

Sd D dS

where

u D 1 C rgood

d D 1 C rbad

:

and rgood

and rbad

are the returns to the stock whenit goes up and down, respectively.

To prevent arbitrage we’ll also require that

d < 1 C r < u

where r is the risk-free borrowing/lending rate.



The payoffs of a call option (at maturity) are:

c

cu D .uS � K/C

cd D .dS � K/C

where XC � maxŒX; 0�:

The payoffs of a portfolio with � shares of the stockand D dollars of borrowing are:

�S � D

�uS � .1 C r/D

�dS � .1 C r/D



To replicate the call option we require that:

�uS � .1 C r/D D cu

�dS � .1 C r/D D cd

Computing the hedge ratio, or delta, yields:

�.uS � dS/ D cu � cd

) � Dcu � cd

S.u � d/:

If you think of it as

� Dcu � cd

Su � Sd

it makes it clear that � D @c@S

.



The amount of borrowing can be computed from:

�uS � .1 C r/D D cu

) D D�uS � cu

1 C r:

Then using � Dcu�cd

S.u�d/we get

.1 C r/D D�

cu�cd

S.u�d/

�

uS � cu

D.cu � cd/u � .u � d/cu

.u � d/

Ddcu � ucd

u � d:



The call is a levered claim on the underlying.

� How do we know this?

Because � and D are always positive:

� Dcu � cd

S.u � d/

� 0;

and

D Ddcu � ucd

.1 C r/.u � d/

D.duS � dK/C � .udS � uK/C

.1 C r/.u � d/

� 0:

Therefore, the replicating portfolio for a call on astock consists of

� a long position in the stock, which is

� partially financed through borrowing.



Question:

What is the replicating portfolio of a put option withpayoffs pu and pd?

� Replicating the put requires

�uS � .1 C r/D D pu

�dS � .1 C r/D D pd

) � Dpu � pd

S.u � d/:

� Then �uS � .1 C r/D D pu )

.1 C r/D Ddpu � upd

u � d:

Just like for the call, only now � � 0 and D � 0.

� The replicating portfolio for the put is a leveredshort position in the underlying.



Numeric Example: Intel is priced at 20

� Next period will either be priced at 18 or 23

� The simple, one-period risk-free rate is 8%

What’s the price of the one-period ATM call?

Well, what is the price of the replicating portfolio?

� How much Intel is it long?

� How big is the debt position?

So what’s it worth?



(C) Risk-Neutral Pricing and Probabilities

“Risk-neutral pricing” is a methodology thatsimplifies everything we’ve been doing.

� We’re going to develop it three different ways

1. As an algebraic convenience.

– This is the “easiest” way to think about it.

– But you don’t really learn anything.

� It ignores all the economic content.

� It doesn’t help us when we want to do thesame things in a more realistic setting.

2. As pricing in a “risk-neutral” world.

– This is how it developed historically.

� It is quite powerful, and provides intuition.

3. As a better way to replicate.

– This methodology hints at the most powerfultools in modern finance.

� It really involves calculating “state prices.”



1. R-N Pricing as an algebraic convenience.

Remember: we priced the call by constructing,and then pricing, the replicating portfolio,

c D �S � D

where

� Dcu � cd

.u � d/S

D Ddcu � ucd

.1 C r/.u � d/:

� Substituting for � and D gives the call price asa function of

1. The tree “primatives”: u, d , and r

2. The next-period payoffs: cu and cu

c Dcu � cd

u � d�

dcu � ucd

.1 C r/.u � d/

D.1 C r/.cu � cd/ � .dcu � ucd /

.1 C r/.u � d/:



� There was nothing special about the call.

– The same methodology works for any asset.

� So for any asset V

V D.1 C r/.Vu � Vd/ � .dVu � uVd/

.1 C r/.u � d/

where Vu and Vd are the asset value nextperiod in the up and down states for theunderlying.

� Now do some algebra

V D

�

1

1 C r

�

.1 C r � d/Vu C .u � .1 C r//Vd

.u � d/

D

�

1Cr�du�d

�

Vu C�

u�.1Cr/

u�d

�

Vd

1 C r:



� Finally, if we define, as an algebraic convenience,

q �.1 C r/ � d

u � d;

then

V D

�

1Cr�du�d

�

Vu C�

u�.1Cr/

u�d

�

Vd

1 C r

DqVu C .1 � q/Vd

1 C r:

� What does this “say”?

� It says

Vt DE

Qt

�

VtC1

�

1 C r:

where EQt Œ�� means expectation assuming the

stock goes up with probability q. (Note: q ¤ p!)



� That is, the call’s price is just the discounted“expected” payoff of the option.

– Here “expected” payoff “assumes” that thestock goes up with probability q.

� Of course, that isn’t “really” the probability ofan up move.

� We have not actually said (or assumed)anything about the “real” probability yet.

� We call q the risk-neutral probability of thestock price up-move,

– or sometimes the “pseudo probability”.

� We call the true probability, p, the objectiveprobability,

– or sometimes the “physical probability”.

� Why do we call q the risk-neutral probability?

– We’ll get back to this in a little while.



Important for us: it makes pricing easier.

c Dqcu C .1 � q/cd

1 C r

where

q D.1 C r/ � d

u � d

� It’s easier than calculating, and then pricing,the replicating portfolio

– Even when we have formulae for the stock,bond positions in the replicating portfolio

c D �S � D

where

� Dcu � cd

.u � d/S

D Ddcu � ucd

.1 C r/.u � d/:



Intel example redux

Whats the price of the ATM call?

� Remember:

– S D 20

– Price next period is 18 or 23

– Simple, one-period rate is 8%



2. R-N pricing in a “risk-neutral world”.

� For a minute, let’s forget entirely about the“real world”.

– The “real world” in which:

� We see S , u, d , and r .

� People are risk-averse.

� Let’s hypothesize a fictional world.

– In this fictional world:

� Everyone is risk-neutral.

� We see the exact same S , u, d , and r .

– How can that be?

� p must be different in this fictional world.

� Question: what is the price of a call in thisfictional, risk-neutral world?

– It must be exactly the same as in the “realworld”!

� Why?



� No-arbitrage considerations imply the relationbetween the option’s price and S , u, d and r .

– For example, we saw

c D �S � D

where

� Dcu � cd

S.u � d/

D Ddcu � ucd

.1 C r/.u � d/:

� By assumption, S , u, d and r are exactly thesame in the risk-neutral world.

� No-arbitrage relationships are independent ofinvestors’ risk preferences

) the same pricing relationship holds in afictitious world in which:

– Prices are the same.

– Investors are risk-neutral.



Question: How do we compute the price of anoption (or any asset) in a risk-neutral world?

� We could replicate the payoffs by constructingthe replicating portfolio.

– But we don’t have to!

� Pricing options in this fictitious, risk-neutralworld is easier.

– Pricing any asset is easier.

– We can “forget” that we’re replicating payoffs.

� Asset price are discounted, expected payoffs:

c Dq cu C .1 � q/ cd

1 C r

p Dq pu C .1 � q/ pd

1 C r;

where q is the probability of the up-move in therisk-neutral world.



� But what is q, the probability the stock pricegoes up?

– Everyone’s risk-neutral

– Expected return to the stock = risk-free rate:

quS C .1 � q/dS D Et ŒStC1� D .1 C r/S:

So

qu C .1 � q/d D 1 C r

) q D.1 C r/ � d

u � d:

� Once we have the risk-neutral probabilities, it’seasy to price any asset.

– Not just options, though they are what we’reinterested in, primarily.



� That is, in the risk-neutral world, given any

asset V ,

V Dq Vu C .1 � q/ Vd

1 C r;

where Vu and Vd are the payoffs to theasset when the stock goes up and down,respectively.

� So, in the risk-neutral world


1 C r

where

q D.1 C r/ � d

u � d:

� But what does this tells us about c in the “realworld”?



� It tells us exactly the price of the call in the “realworld”!


1 C r

where

q D.1 C r/ � d

u � d:

� Remember, we assumed prices in this risk-neutral economy were exactly the same asthose we observe in the real economy.

� Ultimately, the option price in the risk-neutraleconomy just depends on NA.

– We could construct replicating portfolio.

� They only depend on prices.

� They don’t depend on risk preferences at all.

� So options priced the same in the real world.

– Same replicating portfolios in both worlds.

– Same prices for the assets in the replicatingportfolios in both worlds.



3. R-N Pricing as a better way to replicate.

Again: we priced the call by constructing, andthen pricing, the replicating portfolio,

c D �S � D:

� Here � is you holdings in the stock, and�D=.1 C r/ is your holdings in the bond.

– But � and D aren’t immediately obvious.

� It would have been a lot easier to replicate thecall with:

Au

1

0

Ad

0

1

� You don’t even have to stop to think,

c D cuAu C cdAd :



We don’t have Au and Ad though, you say?

� But we can construct them!

– How?

� We replicate them.

– Using S and B.

Au

1

0

The replicating portfolio for Au has the portfolioweights

�u DAu

u � Aud

Su � Sd

D1

S.u � d/

Du DdAu

u � uAud

.1 C r/.u � d/D

d

.1 C r/.u � d/:



Ad

0

1

Similarly, the replicating portfolio for Ad has theportfolio weights

�u D�1

S.u � d/

Du D�u

.1 C r/.u � d/:

� But is it really worth it?

– We constructed Au and Ad ,

� using S and B,

– so that we can replicate other security’spayoffs using Au and Ad ,

� instead of S and B.



� It’s absolutely worth it!

� To price assets using Au and Ad we don’tneed to know their replicating portfolios.

– We only need their prices!

� What are their prices?

– Price their replicating portfolios:

Au D�

1S.u�d/

�

S � d.1Cr/.u�d/

D1 C r � d

.1 C r/.u � d/

Dq

1 C r

Ad D�

�1S.u�d/

�

S � �u.1Cr/.u�d/

Du � .1 C r/

.1 C r/.u � d/

D1 � q

1 C r:



� That is, the prices of Au and Ad are thediscounted risk-neutral probabilities of the upand down moves, respectively.

� Now it should be obvious what the price of anasset V is.

– Price the replicating portfolio.

� If V pays Vu when the underlying goes up,

� and pays Vd when it goes down, then

V D VuAu C VdAd

D Vu

�

q

1Cr

�

C Vd

�

1�q

1Cr

�

DqVu C .1 � q/Vd

1 C r:

� In practice we never worry about the replicatingportfolios for Au and Ad .

– Just their prices (undiscounted for time), andthese are easy to calculate.

� They’re q and 1 � q.



� That is, on one level, risk neutral probabilitiesare just a back-door way of constructingreplicating portfolios.

– Using particularly convienient portfolios of theunderlying and bonds.

� But they have a concrete economic interpretation.

– The risk-neutral probability of the up-move isclosely related to the state price of the upmove.

� Remember,

Au Dq

1 C r:

– That is, q=.1 C r/ is today’s price for a dollartomorrow that you only receive in the “state ofthe world” that the stock price is high.

– And .1 � q/=.1 C r/ is today’s price for a dollartomorrow that you only receive in the “state ofthe world” that the stock price is low.

� So what is q?



We can say a little more if we know p

� The objective probability of the up move

Remember, the R-N probabilities price the stock:

S

Su

Sd

q

1 � q

S DqSu C .1 � q/Sd

1 C r

� “Altering” the probabilities was actually arbitrary

– It’s just a matter of interpretation

� We could have “altered” the prices

– I.e., adjusted them for risk



The objective probabilities also price the stock

� If we use risk-adjusted payoffs

S

�

q

p

�

Su

�

1�q

1�p

�

Sd

p

1 � p

S Dp �

�

q

p

�

Su C .1 � p/ ��

1�q

1�p

�

Sd

1 C r

� Remember, one interpretation of q:

– q=.1 C r/ D today’s price of the up dollar

� Another interpretation of q:

– q=p D tomorrow’s value of a dollar in the upstate, relative to today’s value of a dollar

� Typically think q < p, so q=p < 1

� Stock goes up when the market goes up, soyou’re richer, value a marginal dollar less



(D) Two-Period Model

The tree for the stock:

S

uS

uuS

udS

dS

ddS

The tree for the call:

c

cu

cuu D .uuS � K/C

cud D .udS � K/C

cd

cdd D .ddS � K/C



Dynamic Programming

� I.e., start at the end, work backwards

From the one-period model:

cu Dq cuu C .1 � q/ cud

1 C r

cd Dq cud C .1 � q/ cdd

1 C r:

Once we know cu and cd , we have a one-periodmodel.


1 C r

Dq

q cuuC.1�q/ cud

1CrC .1 � q/

q cudC.1�q/ cdd

1Cr

1 C r

Dq2cuu C 2q .1 � q/cud C .1 � q/2cdd

.1 C r/2:



That is, the call price is the discounted risk-neutralexpected payoff at maturity,

ct DE

QŒ ctC2 �

.1 C r/2

DP

QŒcuu� cuu C P

QŒcud � cud C P

QŒcdu� cdu C P

QŒcdd � cdd

.1 C r/2

where the risk-neutral probabilities for the finalpayoffs are

PQŒ cuu � D q2

PQŒ cud � D q .1 � q/

PQŒ cdu � D q .1 � q/

PQ

Œ cdd � D .1 � q/2:

Note: the superscript-Q on the expectation orprobability operator denotes “under the risk-neutralmeasure”

� Just a fancy way of saying “use q instead of p”



Example: ABS’s stock is trading at $10 per share.

� In each of the next two years the stock price willeither

– go up by 20%,

– or go down by 10%.

� The simple annual risk-free rate is 10%.

What’s the price of a two-year ATM call?

� First, draw the stock and option price-trees:



� What are the option prices in one year?

� So, what is today’s option price?



III. Dynamic Replication

� Risk-neutral pricing is a powerful tool for pricingoptions.

– Easy to implement, and fairly flexible.

– That’s why it’s used on the street.

� However, the methodology obscures what’s reallyunderlying binomial option pricing.

– Binomial pricing works through NA replication ofan option’s payoff.

– Risk-neutral pricing works because it gives thecost of replicating the option’s final payoff.

� Also, if the payoff is path-dependent we need tothink about the whole tree

– I.e., if the option’s payoff depends on not just theprice of the underlying at maturity, but also onhow it got there, we can’t dispense with the tree.

We showed how to replicate a call on a one-periodtree. Now, we’ll show how to dynamically replicate acall option in a multi-period binomial model.



Example continued ...

Consider our two-period example again. In theexample

u D 1:2

d D 0:9

r D 0:1

) q D.1 C 0:1/ � 0:9

1:2 � 0:9D

2

3

The stock price tree was

10

12

14:40

10:80

9

8:10



The call price tree is

c

cu

4:40

0:80

cd

0

Risk-neutral calculations give

cu D

23

� 4:40 C 13

� 0:80

1:1D 2:91

cd D

23

� 0:80 C 13

� 0

1:1D 0:48

) c D

23

� 2:91 C 13

� 0:48

1:1D 1:91:



So the call price on the tree evolves like

1:91

2:91

4:40

0:80

0:48

0

Question: how does the replicating portfolio evolveover time?



� At the uuu-node (time 111)

The replicating portfolio for a call has �u shares ofstock and Du dollars of borrowing:

�u Dcuu � cud

uuS � udS

D4:4 � 0:8

14:4 � 10:8D 1

Du Ddcuu � ucud

.1 C r/.u � d/

D0:9 � 4:4 � 1:2 � 0:80

1:1 � .1:2 � 0:9/D 9:09:

Check that one share of stock and $9:09 ofborrowing replicates the option payoff at time 2:

1:2 � 12 � 1:1 � 9:09 D 4:4

0:9 � 12 � 1:1 � 9:09 D 0:8:

The cost of the u-node replicating portfolio is

1 � 12 � 9:09 D 2:91:



� At the ddd -node (time 111)

The replicating portfolio for a call has �d shares ofstock and Dd dollars of borrowing:

�d Dcud � cdd

udS � ddS

D0:8 � 0

10:8 � 8:10D 0:296

Dd Ddcud � ucdd

.1 C r/.u � d/

D0:9 � 0:80 � 1:2 � 0

1:1 � .1:2 � 0:9/D 2:18:

Check that 0:296 shares of stock and $2:18 ofborrowing replicates the option payoff at time 2:

1:2 � .0:296 � 9/ � 1:1 � 2:18 D 0:8

0:9 � .0:296 � 9/ � 1:1 � 2:18 D 0:

The cost of the d -node replicating portfolio is

0:296 � 9 � 2:18 D 0:48:



� At time 000

The replicating portfolio should satisfy

�0 Dcu � cd

uS � dS

D2:91 � 0:48

12 � 9D 0:808

D0 Ddcu � ucd

.1 C r/.u � d/

D0:9 � 2:91 � 1:2 � 0:48

1:1 � .1:2 � 0:9/D 6:17:

Check that 0:81 shares of stock and $6:17 ofborrowing replicates the option payoff at time-1:

1:2 � .0:808 � 10/ � 1:1 � 6:17 D 2:91

0:9 � .0:808 � 10/ � 1:1 � 6:17 D 0:48:

The cost of the time-0 replicating portfolio is

0:808 � 10 � 6:17 D 1:91:



� The replicating portfolio develops dynamically overtime

� D 0:808

D D $6:17

� D 1

D D $9:09

� D 0:296

D D $2:18

That’s we call it dynamic replication.

� We cannot replicate the payoff of the option atmaturity using a buy-and-hold strategy.

– That is, we can’t use static replication, like wedid for forwards.

� To replicate the payoff of an option we have torebalance the replicating portfolio every period.

– “every period” really = every price change.

– Note: the strategy is also self-financing.

� Note that calls are a “strategy” that “doubles up”

– when the stock goes up you “buy” more

– when the stock goes down you “sell” some



How do we arbitrage an option mispricing?

Suppose that the ATM call was priced at $2.

� At time-0

– Write the call

� It’s over priced.

– Construct the replicating portfolio. Cost: $1:91.

– Put the $0:09 in the bank (i.e., buy bonds).

� At time-1, rebalance the replicating portfolio.

– If the stock went up, buy more stock

– If the stock went down, sell some stock

� At time-2, you have to payoff on the call you’reshort.

– But liquidating the replicating portfolio exactlycovers this liability.

You’re up the $0:09 mispricing, plus interest.

What would you do if the call were selling for $1:85?



Another example: the ATM put

p

pu

0

0

pd

1:90

Clearly pu D 0. Risk-neutral calculations give

pd D

23

� 0 C 13

� 1:90

1:1D 0:58

) p D

23

� 0 C 13

� 0:58

1:1D 0:17:

0:17

0

0

0

0:58

1:90



The replicating portfolio at the d -node:

�d Dpdu � pdd

Sdu � Sdd

D0 � 1:90

10:80 � 8:10D �0:704

Dd Ddpdu � updd

.1 C r/.u � d/

D0 � 1:2 � 1:90

1:1 � 0:3D �6:91

Note: �0:704 � 9 C 6:91 D 0:58.

At the time-0:

�0 Dpu � pd

Su � Sd

D0 � 0:58

12 � 9D �0:192

D0 Ddpu � upd

.1 C r/.u � d/D

0 � 1:2 � 0:58

1:1 � 0:3D �2:09

And �0:192 � 10 C 2:09 D 0:17.



� Again, the replicating portfolio develops dynamicallyover time

� D �0:192

D D �$2:09

� D 0D D 0

� D �0:704

D D �$6:91

� Note that puts are also a “strategy” that “doublesup”

– when the stock goes down you short “more”

� that is, if things go your way you increase yourposition

– when the stock goes up you buy some back

� if things go against you, you decrease yourposition



Also, put-call parity holds in our examples.

� Put-call parity says

cK � pK D S � K � B:

The replicating portfolio for the portfolio of options is

� D 0:808�.�0:192/ D 1

DD6:17�.�2:09/D8:26

� D 1�0 D 1DD9:09�0D9:09

�D0:296�.�0:704/D1

DD2:18�.�6:91/D9:09

Payoff tree for synthetic forward, made either way:

1:91�0:17D1:7410�8:26

2:91�0D2:9112�9:09

4:40�0D4:4014:40�10

0:80�0D0:8010:80�10

0:48�0:58D�0:099�9:09

0�1:90D�1:908:10�10



IV. Multi-Period Models

A. Three-Period Model

SuS

uuSuuuS

uudS

dSudS

uddSddS

dddS

c D1

.1 C r/3

�

q3 cuuu C 3q2 .1 � q/ cuud

C 3q .1 � q/2 cudd C .1 � q/3 cddd

�

� We could keep going; gets tedious very quickly.

� We need a general solution

– For counting the number of paths to each payoffat maturity.

– The method needs to account for each path’s“probability.”



B. The General n-Period Model

let

n D number of steps,

j D number of up-moves to n;

) n � j D number of down-moves to n:

Then the number of paths that lead to cuj dn�j is

nŠ

j Š.n � j /Šwhere nŠ D n � .n � 1/ � .n � 2/ � � � 2 � 1

So, the option price in an n-period model is:

c D1

.1 C r/n

nX

j D0

�

nŠ

j Š.n � j /Š

�

qj .1 � q/n�j cuj dn�j

wherecuj dn�j D

�

uj dn�j S � K�C

and 1 C r is the simple one period discount rate.



Let a denote the fewest number of up-moves thestock needs to make for the call to finish in the money.

� I.e., pick a such that

S � u a�1 d .n�aC1/ � K

S � u a d .n�a/ � K:

Then, the price of the call can be written

c D1

.1 C r/n

nX

j Da

�

nŠj Š.n�j /Š

�

qj .1 � q/n�j Œuj dn�j S � K�

D1

.1 C r/n

"

nX

j Da

�

nŠj Š.n�j /Š

�

qj .1 � q/n�j uj dn�j S

�

nX

j Da

�

nŠj Š.n�j /Š

�

qj .1 � q/n�j K

#

:



Can write this as two terms:

c D

0

@

nX

j Da

�

nŠj Š.n�j /Š

�

qj .1 � q/n�j�

uj dn�j

.1Cr/n

�

1

A S

�

0

@

nX

j Da

�

nŠj Š.n�j /Š

�

qj .1 � q/n�j

1

A

K

.1 C r/n:

Some interpretation: 1=.1 C r/n D Bt;T

� So the equation describes a replicating portfolio

– Long the stock: # of shares

nX

j Da

�

nŠj Š.n�j /Š

�

qj .1 � q/n�j�

uj dn�j

.1Cr/n

�

– Short bonds with a face = K: # of bonds

nX

j Da

�

nŠj Š.n�j /Š

�

qj .1 � q/n�j



We can also interpret these positions:

Bond position,Pn

j Da

�

nŠj Š.n�j /Š

�

q j .1 � q/n�j is the

“probability” that the call finishes in-the-money.

� The probability “under the risk-neutral measure”,

nX

j Da

�

nŠj Š.n�j /Š

�

qj .1 � q/n�j D PQ

t ŒST > K�:

Stock position,

nX

j Da

�

nŠj Š.n�j /Š

�

qj .1 � q/n�j�

uj dn�j

.1Cr/n

�

D PQ

t ŒST > K� EQ

t

h

ST =St

.1Cr/n

ˇ

ˇ

ˇST > K

i

;

� The probability of exercise, times the discountedgrowth in the stock price in the “good” states of theworld



So the price of the call is

ct D

0

@

nX

j Da

�

nŠj Š.n�j /Š

�

qj .1 � q/n�j�

uj dn�j

.1Cr/n

�

1

A St

�

0

@

nX

j Da

�

nŠj Š.n�j /Š

�

qj .1 � q/n�j

1

A KBt;T

D EQ

t

h�

ST =St

.1Cr/n

�

1ST >K

i

St � PQ

t ŒST > K� KBt;T

This looks a lot like Black-Scholes!

� The Black-Scholes value of a call:

c D N.d1/ S � N.d2/ Ke�rT

� We can get to Black-Scholes by letting n ! 1

– and doing a lot of algebra

� We’ll get there a different way



How many steps do we need?

� Binomial option prices.

– As a function of number of periods used in thecalculation.

– Holding all other variables, including time-to-maturity, constant (i.e., as n ", �t #).


5 Binomial Trees

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