Chapter 5: The binomial model

Question 1

Arbitrage in one period model with 2 commodities

Answer 1

• If we hold x units of cash at time 0, they become worth x(1 + r) at time 1.

At time t =0, a single unit of stock is worth s units of cash. At time 1, the value of a unit of stock is S_1 =
{sd with probability p_d
{su with probability p_u

Where p_d + p_u =1
And d less than u

If we hold y units of stock (worth ys) at time 0 they become worth yS_1 at time 1

• x and y can be negative =borrowed

Question 2

PORTFOLIO

Answer 2

Recall also that we use the term portfolio for the amount of cash/stock that we hold at some time.

A portfolio is a pair h = (x,y) ∈ R2, where x is the amount of cash and y is the number of (units of) of stock.

Question 3

DEF 5.1 value process /price process

Answer 3

Definition 5.1.1 The value process or price process of the portfolio h = (x, y) is the process V h given by

V_0^h = x + y s
V_1^h =x(1+r)+yS_1.

(S_1 is a RV)

Question 4

DEF 5.2.2 portfolio has ARBITRAGE POSSIBILITY

Answer 4

A portfolio h=(x,y) is an ARBITRAGE POSSIBILITY

V_0^h = 0

P[V_1^h ≥ 0] = 1.

P[V_1^h > 0] > 0.

We say that a market is arbitrage free if there do not exist any arbitrage possibilities.

• initially h is worth nothing e.g neg value x or y, cancel out
• h doesn’t lose money
• h might make money on the initial value

Question 5

PROP 5.1.3 arbitrage free one period market

Answer 5

Proposition 5.1.3 The one-period market is arbitrage free if and only if d < 1 + r < u.

Sometimes cash outperforms stock lower limit
d less than

Sometimes stock outperforms cash upper limit
bigger than u

Question 6

Proposition 5.1.3 The one-period market is arbitrage free if and only if d < 1 + r < u.

PROOF

Answer 6

Imply: since d is less than y if this condition fails then

Either: 1) if r less than or equal to d less than u
Choose h=(-S,1)
V_0 ^h = -S_T +1(S) =0

2) d less than u less than or equal to 1+r
V_0^h=S+(-1)S=0
choose n=(s,-1)
V^h_1= s(1+r)-S_1

(S_1 is either Su or Sd)

bigger thn or equal to S(1+r)-Su

hence P[V^h_1 ≥ 0] = 1. Further, with probability pu > 0 we have S1 = su, which means
V_h^1 > s(−(1 + r) + d) ≥ 0. Hence P[V^h_1 > 0] > 0. Thus, h is an arbitrage possibility.
If 0 < d < u ≤ 1 + r then we use the portfolio h’ = (s, −1), which has V^h’_0 = 0 and
V^h’_1 = s(1 + r) − S1 ≥ s(1 + r − u) ≥ 0,
hence P[V^h’_1 ≥ 0] = 1. Further, with probability p_d > 0 we have S_1 = sd, which means
V^h’_1 > s(−(1 + r) + u) ≥ 0. Hence P[V^h’_1 > 0] > 0. Thus, h’ is also an arbitrage possibility.

Remark 5.1.4 In both cases, at time 0 we borrow whichever commodity (cash or stock) will
grow slowest in value, immediately sell it and use the proceeds to buy the other, which we know will grow faster in value. Then we wait; at time 1 we own the commodity has grown fastest in value, so we sell it, repay our debt and have some profit left over.

(⇐) : Now, assume that d < 1 + r < u. We need to show that no arbitrage is possible. To
do so, we will show that if a portfolio has V^h_0 = 0 and V^h_1 ≥ 0 then it also has V^h_1 = 0.
So, let h = (x, y) be a portfolio such that V^h_0 = 0 and V^h_1 ≥ 0. We have
V^h_0 = x + ys = 0.
The value of h at time 1 is
V^h_1 = x(1 + r) + ysZ.
Using that x = −ys, we have

V^h_1 =
{ys(u − (1 + r)) if Z = u,
{ys(d − (1 + r)) if Z = d.

Since P[V^h_1 ≥ 0] = 1 this means that both (a) ys(u − (1 + r)) ≥ 0 and (b) ys(d − (1 + r)) ≥ 0.
If y < 0 then we contradict (a) because 1 + r < u. If y > 0 then we contradict (b) because
d < 1 + r. So the only option left is that y = 0, in which case V^h_0 = V^h_1 = 0.

Question 7

CONDITION FOR ONE PERIOD MODEL TO BE FREE OF ARBITRAGE

d less than 1+r less than u

introduce q_u and q_d

Answer 7

equiv cond

there exists q_u, q_d ∈ (0, 1) such that both
q_u + q_d = 1 and

1 + r = uq_u + dq_d

*1+r is weighted average of d and u

*solve q_u
=((1+r)-d)/(u-d)

q_d
=(u-(1+r))/(u-d)

Question 8

IN WORLD Q, RISK-NEUTRAL WORLD we define q_u and q_d

Answer 8

RISK_NEUTRAL PROBABILITIES

Q[S_1=sd]=q_d
Q[S_1=su]=q_u

S_1=
{su w.p q_u
{sd w.p q_d

Question 9

IN WORLD Q, RISK-NEUTRAL WORLD

EXPECTATIONS USING PROB MEASURE Q

Answer 9

PROB MEASURE P E^P
PROB MEASURE Q E^Q

Expectation does correctly calculate the value (i.e. price) of a single unit of
stock by taking an expectation. The point is that we (1) use E^Q rather than E^P and (2) then
discount according to the interest rate for DISCOUNTED STOCK PRICE

Question 10

IN WORLD Q, RISK-NEUTRAL WORLD

(1/(1+r))E^Q[S_1]

S_0 and S_1 relation

Answer 10

(1/(1+r))E^Q[S_1] = (1/(1+r))(suQ[S_1=su] +sdQ[S_1=sd)

=
(1/(1+r))(s)(uq_u + dq_d)
=s

The price of the stock at time 0 is S_0=s

The price of stock at time 0 is S_0=s

we have shown price S_1 of unit of stock at time 1 satisfies

S_0=
(1/(1+r))E^Q[S_1]

Question 11

DISCOUNTED STOCK PRICE

Answer 11

S_0=
(1/(1+r))E^Q[S_1]

This is a formula that is very well known to economists. It gives the stock price today (t = 0) as the expectation under Q of the stock price tomorrow (t = 1), discounted by the rate 1 + r at which it would earn interest.

Question 12

DEF 5.2.1 CONTINGENT CLAIM

Answer 12

A contingent claim is any random variable of the form X = Φ(S_1), where Φ:R to R is a deterministic function.

“PAYOFF OF THE CONTRACT” dep on price of stock at time 1

Question 13

FORWARD CONTRACT

Answer 13

example of a contingent claim
wn as the contract function. One example of a contingent
claim is a forward contract, in which the holder promises to buy a unit of stock at time 1 for
a fixed price K, known as the strike price.

In this case the contingent claim would be
Φ(S_1) = S1 − K,

the value of a unit of stock at time 1 minus the price paid for it

Question 14

EG 5.2.2 EUROPEAN CALL OPTION

Answer 14

its holder the right (but not the obligation) to
buy, at time 1, a single unit of stock for a fixed price K that is agreed at time 0. As for futures,
K is known as the strike price.

Value to the holder at time 1:
Suppose you hold a European call option at time 1

*if S_1 bigger than K
we could exercise our
right to buy a unit of stock at price K, immediately sell the stock for S1 and consequently earn S1 − K > 0 in cash.
*if S_1 K, Alternatively if S1 ≤ K then our option is worthless so don’t exercise

The contingent claim assoc to contract has 2 cases:

we consider when sd less than K less than su

In this case, the contingent claim for our European call option is
Φ(S1) =
{su − K if S_1 = su
{0 if S_1 = sd

In the first case our right to buy is worth exercising; in the second case it is not. A simpler way to write this contingent claim is
Φ(S1) = max(S1 − K, 0)

Question 15

DEF 5.2.3

REPLICATING PORTFOLIO

Answer 15

We say that a portfolio h is a replicating portfolio or hedging portfolio for the contingent claim Φ(S1) if V^h_1 = Φ(S1).

Question 16

Finding Φ(S_1)

Answer 16

Finding value Φ(S_1) at time 0 look for a gen way to construct trading strategies ie construct portfolio h=(x,y) st V_1^h= Φ(S_1) and by no arbitrage V_0^h=value of contract at time 0

If a contingent claim Φ(S1) has a replicating portfolio h, then the price of the Φ(S1) at
time 0 must be equal to the value of h at time 0.

Question 17

A market is COMPLETE

Answer 17

A market is COMPLETE if every CONTINGENT CLAIM can be REPLICATED

If the market is complete we can price

Question 18

EXAMPLE: 5.2.4

Suppose that s = 1, d =1/2, u = 2 and r =1/4, and that we are looking at the contingent claim
Φ(S_1) =
{1 if S1 = su,
{0 if S1 = sd

We wish to replicate Φ(S_1)

Answer 18

Representing as a tree

[price of contingent claim]
(stock price)

  (2) [1] (1) <
  (0. 5)   [0]

we want to replicate Φ(S_1)
ie want portfolio h=(x,y) st V_1^h= Φ(S_1)

*stock goes up
(1 + 1/4)x + 2y =1
*stock goes down
(1 + 1/4)x + 0.5y = 0

so x=-4/15 and y=2/3

Hence the price of our contingent claim Φ(S_1) at time 0 is V_0^h= (-4/15)+(1)(2/3)=2/5

ie we would pay 2/5 cash to hold this contract

Question 19

IN GENERAL

take an arbitrary contingent claim Φ(S1) and see if we can replicate it.

Answer 19

ie finding a portfolio h such that the value V^h_1 of the portfolio at time 1 is Φ(S1):
V^h_1 =
{Φ(su) if S1 = su,
{Φ(sd) if S1 = sd.

if we write h = (x, y) then we need
(1 + r)x + suy = Φ(su)
(1 + r)x + sdy = Φ(sd),

which is just a pair of linear equations to solve for (x, y). In matrix form,

[1 + r    su]
[1 + r    sd] *
[x]
[y]
=
[Φ(su)]
[Φ(sd)]

A unique solution exists when the determinant is non-zero, that is when (1+r)u−(1+r)d ≠ 0, or
equivalently when u ≠ d. So, in this case, we can find a replicating portfolio for any contingent claim.

It is an assumption of the model that d ≤ u, so we have that our one-period model is complete if d < u. Therefore prop 5.2.5

Question 20

PROPOSITION 5.2.5 complete one-period model

Answer 20

If the one-period model is arbitrage free then it is complete.

*when model was arbitrage free we had condition d less than 1+r less than u

Question 21

RISK_NEUTRAL VALUATION FORMULA’

DERIVATION

Answer 21

solving linear equations for replicating portfolio

to get the price of Φ(S1) at time 0
V^h_0= (1/(1+r))E^Q[Φ(S1) ]

*to find the price of Φ(S1) at 0 we take exp Q and discount one time step of interest by dividing by 1+r

• special case of previous wgere Φ(S1) =S_1 is 1 i.e. pricing the contingent claim corresponding to being given a single unit of stock

x =(1/(1 + r))[uΦ(sd) − dΦ(su)] /(u− d)

y = (1/s)(Φ(su) − Φ(sd))/(u − d)

which tells us that the price of Φ(S1) at time 0 should be
V^h_0 = x + sy
=(1/(1 + r)) [((1 + r)−d)/(u−d)Φ(su) + (u − (1 + r))/(u − d)Φ(sd)]

=(1/(1 + r))(q_uΦ(su) + q_dΦ(sd))
=(1/(1 + r)) E^Q[Φ(S_1)].

Question 22

PROPOSITION 5.2.6
For a given contingent claim

Then the (unique) replicating portfolio
h = (x, y) for Φ(S1)  is

Answer 22

Let Φ(S1) be a contingent claim. Then the (unique) replicating portfolio
h = (x, y) for Φ(S1) can be found by solving

V^h_1 = Φ(S1), which can be written as a pair of
linear equations:

(1 + r)x + suy = Φ(su)
(1 + r)x + sdy = Φ(sd).

The general solution is
x =(1/(1 + r))[uΦ(sd) − dΦ(su)] /(u− d)

y = (1/s)(Φ(su) − Φ(sd))/(u − d)

The value (and hence, the price) of Φ(S1) at time 0 is
V^h_0 = (1/(1 + r))E^Q[Φ(S1)]

Question 23

EXAMPLE 5.2.7

European call option value at time 0

Answer 23

For EU call option with strike price K ∈ (sd, su). We found contingent claim Φ(S_T)= max(S_T-K,0)

By prop 5.2.6 replicating portfolio h=(x,y) found by solving V_1^h= Φ(S_1)

(1+r)x + suy = su - K
(1+r)x + sdy = 0

SOL
x = [sd(K−su)]/
[(1+r)(su−sd)],
y = [su−K]/[su−sd] .

By the second part of prop 5.2.6

value of EU call option at time 0 is

1/(1+ r) E^Q[Φ(S1)] =
1/(1 + r)(qu(su − K) + qd(0))
=
(1/(1 + r))[(1 + r) − d]/(u − d)(su − K)

Question 24

financial derivative

Answer 24

A contract that specifies that buying/selling will occur, now or in the future, is known as a financial derivative, or simply derivative.

Question 25

options

Answer 25

Financial derivatives that give a choice between two options are often known simply as options.

Question 26

strike price

Answer 26

strike price to refer to a fixed price K that is agreed at time 0 (and paid at time 1).

before knowing stock value!

Question 27

FORWARD CONTRACT

Answer 27

• A forward or forward contract is the obligation to buy a single unit of stock at time 1 for a strike price K.

Question 28

EU CALL OPTION

Answer 28

A European call option is the right, but not the obligation, to buy a single unit of stock at time 1 for a strike price K.

Question 29

EU PUT OPTION

Answer 29

A European put option is the right, but not the obligation, to sell a single unit of stock at time 1 for a strike price K.

Question 30

Binomial model

Answer 30

Model has time points t=0,1,2,….,T

inside each time step we have a single step of the one-period model

*x units of cash at time t become worth x(1+r) cash at time t+1

*Value of single stock at time t is given by S_0=s
S_t=Z_tS_{t-1}

ie the value of stock is multiplies by a random var Z with dist P[Z=u]=p_u P[Z=d]=p_d

each time step Z is indep so (Z_t)^T_{t=1} iid sequence of RV with dist Z

TREE DIAGRAM FOR STOCK

Question 31

TREE DIAGRAM FOR STOCK

Answer 31

(su^2)
    (su)<    
(S)<        (sdu)
   (sd) <
           (sd^2)
t=0,1,2,..

The one-period model is simply the T = 1 case of the binomial model. Both
models are summarized on the formula sheet, see Appendix B.

Question 32

The filtration corresponding to the information available to a buyer/seller in the binomial
model is

Answer 32

Ft = σ(Z1, Z2, . . . , Zt).

the information in Ft contains changes in the stock price up to and including at time
t. This means that, S0, S1, . . . , St are all Ft measurable, but St+1 is not Ft measurable.

how much stock/cash to buy/sell at time t − 1, we do so without knowing
how the stock price will change during t − 1 7→ t. So we must do so using information only from F_{t−1}.
F_0={sample space, empty} = no info

Question 33

DEF 5.5.1

A portfolio strategy is a stochastic process

Answer 33

A portfolio strategy is a stochastic process
h_t = (x_t, y_t)
for t = 1, 2, . . . , T, such that h_t is Ft−1 measurable.

x_t amount of cash
y_t amount of stock
HELD DURING TIME STEP t-1 to t

Choice is made to hold amounts for t-1 to t based on knowing value of S0, S1, . . . , S_{t−1}, but without knowing St

Question 34

DEF 5.5.2
VALUE PROCESS
of the portfolio strategy h = (h_t)_ t=1 ^T

Answer 34

The value process of the portfolio strategy h = (h_t)_ t=1 ^T
is the stochastic process (Vt) given by
V^h_0 = x_1 + y_1S_0

V^h_t = x_t(1 + r) + y_tS_t

for t=1,2,…,T

At t=0, V_0^h is the value of the portfolio h_1

For r t ≥ 1, V^h_t is the value of the portfolio ht at time t, after the change in value of cash/stock that occurs during
t−1→ t.

The value process V^h_t is F_t measurable but it is not F_{t−1} measurable.

Question 35

NOTE SELF FINANCING PORTFOLIO

Answer 35

We will be especially interested in portfolio strategies that require an initial investment at
time 0 but, at later times t ≥ 1, 2, . . . , T − 1, any changes in the amount of stock/cash held will
pay for itself. We capture such portfolio strategies as self financing

The value of the portfolio at time t is equal to value at time t of cash/stock held between times t to t+1

in a self-financing portfolio
at the times t = 1, 2, . . . we can swap our stocks for shares (and vice versa) according to whatever the stock price turns out to be, but that is all we can do.

Question 36

DED 5.5.3 SELF-FINANCING

Answer 36

A portfolio strategy ht = (xt, yt) is said to be self-financing if
V^h_t = x_{t+1} + y_{t+1}S_t

for t = 0, 1, 2, . . . , T.

Question 37

DEF 5.5.4 ARBITRAGE POSSIBILITY

Answer 37

We say that a portfolio strategy (ht) is an arbitrage possibility if it is self-financing and satisfies
V^h_0 = 0
P[V^h_T ≥ 0] = 1.
P[V^h_T > 0] > 0.

arbitrage possibility requires that we invest nothing at times t = 0, 1, . . . , T − 1, but which gives us a positive probability of earning something at time T, with no risk at all of actually losing money

Question 38

PROPOSITION 5.5.5

arbitrage free binomial model

Answer 38

Proposition 5.5.5 The binomial model is arbitrage free if and only if d < 1 + r < u

the condition turns
out to be the same as for the one-period model.

Question 39

PROPOSITION 5.5.6

BINOMIAL
USING risk-neutral probabilities from In the one-period model, we use them to
define the risk-neutral world Q, in which on each time step the stock price moves up (by
u) with probability qu, or down (by d) with probability qd. This provides a connection to
martingales

Answer 39

If d < 1 + r < u, then under the probability measure Q, the process
Mt =
[ 1/[(1 + r)^t] ]S_t
is a martingale, with respect to the filtration (Ft).

Using Lemma 3.3.6 we have E^Q[M_0] = E^Q[M_1], which states that S_0 = (1/(1+r))E^Q[S_1]. This is precisely
DISCOUNTED STOCK PRICE
S_0=
(1/(1+r))E^Q[S_1]

Question 40

PROPOSITION 5.5.6

PROOF

If d < 1 + r < u, then under the probability measure Q, the process
Mt =
[ 1/[(1 + r)^t] ]S_t
is a martingale, with respect to the filtration (Ft).

Answer 40

We have commented above that S_t ∈ mF_t, and we also have d^t S0 ≤ St ≤ u^tS_0, so S_t is bounded and hence S_t ∈ L^1. Hence also M_t ∈ mF_t and M_t ∈ L^1. It remains to show that
E^Q[M_{t+1} | F_t] =
E^Q[ M_{t+1} 1{Z{t+1=u} + M_{t+1} 1{Zt+1=d}| Ft] =
E^Q[ ((uSt)/(1 + r)^{t+1}) 1{Z_{t+1}=u} + (dSt/(1 + r)^{t+1})1{Z{t+1}=d}| Ft]
=
(St/(1 + r)^{t+1}) (uE^Q[1{Z{t+1}=u}| Ft]
+ dE^Q[1{Z{t+1}=d}| Ft]
=
St/((1 + r)^{t+1})(uE^Q[1{Z{t+1}=u}
+ dE^Q[1{Z{t+1}=d}]
=
(S_t)/((1 + r)^t+1) (uQ[Z_{t+1} = u] + dQ [Z_{t+1} = d])
=
St/((1 + r)^{t+1}) (uq_u + dq_d)
=
[S_t/(1 + r)^{t+1}] (1 + r)
= Mt

Here, from the second to third line we take out what is known, using that St ∈ mFt
To deduce the third line we use linearity, and to deduce the fourth line we use that Z_{t+1} is independent of F_t

Lastly, we recall from (5.2) that uq_u +dq_d = 1 +r. Hence, (Mt) is a martingale with respect
to the filtration F_t, in the risk-neutral world Q.

Question 41

Hedging

Answer 41

We can adapt the derivatives from Section 5.3 to the binomial model, by simply replacing time
1 with time T. For example, in the binomial model a forward contract is the obligation to buy
a single unit of stock at time T for a strike price K that is agreed at time 0.

Question 42

DEF 5.6.1 contingent claim

Answer 42

A contingent claim is a random variable of the form Φ(ST ), where Φ : R →
R is a deterministic function.

For a forward contract, the contingent claim would be Φ(ST ) = ST − K.

Question 43

DEF 5.6.2
replicating portfolio
hedging stratehgy

Answer 43

We say that a portfolio strategy h = (ht)^T_t=1 is a replicating portfolio or hedging strategy for the contingent claim Φ(S_T ) if V^h_T = Φ(S_T ).

These match the definitions for the one-period model, except we now care about the value of the asset at time T (instead of time 1). We will shortly look at how to find replicating portfolios.

As in the one-period model, the binomial model is said to be complete if every contingent claim can be replicated. Further, as in the one-period model, the binomial model is complete if and only if it is free of arbitrage. With this in mind, for the rest of this section we assume that
d < 1 + r < u

Question 44

Lastly, as in the one-period model, our assumption that there is no arbitrage means that:

Answer 44

If a contingent claim Φ(ST ) has a replicating portfolio h = (ht)
T
t=1, then the price of the
Φ(ST ) at time 0 must be equal to the value of h0.

Question 45

COMPUTE PRICES AND REPLICATING PORTFOLIOS IN BINOMIAL MODEL

EXAMPLE
Let us take T = 3 and set
S_0 = 80, u = 1.5, d = 0.5, p_u = 0.6, p_d = 0.4.

To make the calculations easier, we’ll also take our interest rate to be r = 0. We’ll price a European call option with strike price K = 80. The contingent claim for this option, which is
Φ(ST ) = max(ST − K, 0).

Answer 45

STEP 1
work our the risk-neutral probabilities.

qu =(1+0−0.5)/(1.5−0.5) = 0.5
and qd = 1 − qu = 0.5

STEP 2 tree of possible values stock can tak at t=0,1,2,3
                    270
            180 <
           /           90
        120      /
80<       >60
       40         \30
           \       /
            20  \  
                    10

We then work out, at each of the nodes corresponding to time T = 3, what the value of our
contingent claim would be if this node were reached. We write these values in square
boxes:
                    270    [190]
            180 <
           /           90   [10]
        120      /
80<       >60
       40         \30     [0]
           \       /
            20  \  
                    10        [0]

STEP 3
Suppose we are sitting in one of the nodes at time t = 2, ‘current’ node. For example (labelled 180, the ‘current’ value of the stock). Looking forwards one step of time, if the stock price goes up option is worth 190, whereas if the stock price goes down option is worth 10. an instance of the one-period model. With
contingent claim
Φ~( su) = 190, Φ~(sd) = 10.

using one-period risk neutral valuation formula the value of call option at current node is
(1/(1+0))(190(0.5) + 10(0.5)) =100

         [100]     270    [190]
            180 <
           /           90   [10]
        120      /
80<       >60  [5]
       40         \30     [0]
           \       /
            20  \  
            [0]     10      [0]

again:
node 40 one step into the future Φ~( su) = 5, Φ~(sd) = 0.
call option value (1/(1+0))(5(0.5)+ 0(0.5)) =2.5

                   [100]     270    [190]
                  180 <
        [52.5]   /           90   [10]
[27.5]     120      /
    80<       >60  [5]
            40         \30     [0]
     [2.5]     \       /
                   20  \  
                  [0]     10      [0]

PRICE OF CALL OPTION AT t=0 is 27.5

Question 46

[100]     270    [190]
                  180 <
        [52.5]   /           90   [10]
[27.5]     120      /
    80<       >60  [5]
            40         \30     [0]
     [2.5]     \       /
                   20  \  
                  [0]     10      [0]

PRICE OF CALL OPTION AT t=0 is 27.5

Answer 46

STEP 4:
REPLICATING PORTFOLIO

at t=0
to replicate the contingent claim Φ~( su) = 52.5 and Φ~(sd) = 2.5 at time t = 1, equation (5.8) tells us that we want the portfolio
x_1 =
(1/(1 + 0)) [(1.5 · 2.5 − 0.5 · 52.5)/(1.5 − 0.5)]
= −22.5,
y_1 =(1/80) (52.5 − 2.5)/(1.5 − 0.5)
= 5/8

The value of this portfolio at time 0 is x_1 + 80y_1 =
−22.5 + 80 ·(5/8)
= 27.5

equal to initial value of call option.

We then work out which portfolio we would want to hold at each possible outcome. At all times we would be holding a portfolio with current value equal to the current value of the call option. Therefore, this gives a self-financing portfolio strategy that replicates Φ(S_T )

if stock went up t=0 to t=1: at t=1 node S_1=120
(x_1,y_1) portfolio now worth
x_1(1+0) +y_1(120) = -22.5+120(5/8) =52.5) equal to value of call current

We use (5.8) again to calculate the portfolio we
want to hold during time 1 → 2, this time with Φ~( su) = 100 and Φ~( sd) = 5, giving x_2 = −42.5
and y_2 =95/120 . You can check that the current value of the portfolio (x2, y2) is 52.5.

Next, suppose the stock price falls between t = 1 and t = 2, so our next node is S_2 = 60.
Our portfolio (x2, y2) now becomes worth
x2(1 + 0) + y2 · 60 = −42.5 +(95/120)· 60 = 5,

again equal to the value our of call option. For the final step, we must replicate the contingent
claim Φ~(su) = 10, Φ~(sd) = 0, which (5.8) tells us is done using x3 = −5 and y3 =1/6

Again, the current value of this portfolio is 5.

Lastly, the stock price rises again to S3 = 90. Our portfolio becomes worth
x3(1 + 0) + y3 · 90 = −5 +(1/6)· 90 = 10,
equal to the payoff from our call option.