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5 Multi-period BOPM, N>1: A Path Integral Approach

# 5 Multi-period BOPM, N>1: A Path Integral Approach

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494

OPTIONS

FIGURE 14.9 The BOPM for N=3

Option Price
Cuuu
Cuu
Cu

Path #1
Cuud

Cud Path #2

Path #3

C0

Cudd

Cd
Cdd

Cddd
Time

0

1

2

3

Then, we move back another step to N=1 and determine Cu and Cd by
reapplying the single-period BOPM. Finally, we determine C0 from a single
application of the single-period BOPM, applied to the now known values of
Cu and Cd .
There is another approach which is based on the paths of the process and
does not calculate the full option value at each node. Rather, it looks at the
tree in terms of individual paths and just values the contribution of cash ﬂows
at the end of path segments to the full nodal value, which is at the beginning
of the given path.
This approach is consistent with the way the formula is usually written,

j =0,1,…N

C (N ,j ) (p′) j (1 − p′)N − jC u jd N − j

(BOPM, all N)

rN
and helps to elucidate it. We want to be able to fully explain each term in
this formula.
In order to tackle the general case, we will start with the BOPM for N=3
to get the basic idea of valuation in terms of individual paths. So, re-consider
Figure 14.9.

OPTION PRICING IN DISCRETE TIME, PART 2

495

As an example, we focus on a terminal option value at N=3, say Cuud
indicated by the color orange. We ﬁrst ask how many paths terminate at Cuud .
The answer is clearly the three paths indicated or C(3, 2) paths where N=3
and the number of up moves over the life of the option is j=2. We list all the
paths terminating at Cuud .
1. Path 1 (all orange) is C0CuCuuCuud.
2. Path 2 (orange, blue, blue) is C0CuCudCuud .
3. Path 3 (purple, purple, blue) is C0CdCudCuud .
We want to move backwards along each path to the previous node on each
path. This time, instead of calculating the full option value at each previous
node using the single-period BOPM, we will calculate the contribution to the
full nodal value made by the the value at the end of each path segment. This
is easier to show than to explain in words, but it will become clear as we
work through our example.
Using our example again, re-consider Path 1 and start at Cuud . Moving
backward to the previous node on that path along the path segment CuuCuud ,
we easily arrive at Cuu . Then, we partially apply the single-period BOPM to
calculate the contribution to the full option value of Cuu made by what is at the
end of the path segment, Cuud .
This means that we are at Cuu and that we apply the appropriate risk-neutral
probability to Cuud and discount by the risk-free rater (as opposed to including
Cuuu as well and calculating the full option value Cuu ). That is, we are simply
calculating that part of the full option value Cuu contributed by the end point
of Path 1, namely by Cuud .
The ﬁrst step is, of course, to determine the correct risk-neutral probability.
Since the path segment CuuCuud corresponds to a down move from Cuu, the
risk-neutral probability is (1–p′) where p′ is deﬁned as before.
Then the value of the contribution of Cuud to Cuu is its correctly discounted
present value. This is clearly (1–p′)*Cuud/r where p′ is the risk-neutral
probability. Keep in mind that this path contribution is part, but not all, of the
full value of Cuu.
Next, we examine the previous path segment of Path 1, CuCuu , imagining
that we are now standing at Cu. Here, we want to calculate the contribution
to the full option value Cu made by the path segment CuCuu . The answer is
p′*Cuu /r since CuCuu constitutes an up move from Cu.

496

OPTIONS

Note that this is how one would value the entire value of Cuu. But we only
want the value contributed to Cuu by the path segment CuuCuud. So, we need
to link up the path segment CuuCuud to the path segment CuCuu in terms of
the risk-neutral probabilities.
The subsequent path segment CuuCuud only contributes (1–p′)*Cuud/r to Cuu ,
as demonstrated in the second previous paragraph, and it is this value that we
are interested in.
Remember that the full value of Cuu does not matter, only the contribution
to its value made by the path segment CuuCuud is relevant. We computed this
contribution as (1–p′)*Cuud/r. Applying p′ to this contributed quantity, rather
than to the full Cuu, and discounting by r, we obtain [p′*[((1–p′)*Cuud)/r]]/r
or [p′*(1–p′)*Cuud]/r2. This represents the contribution to Cu made by the ultimate
endpoint of the path, Cuud.
Finally, we look at the contribution to the time 0 option value made by the
last path segment of Path 1, C0Cu, and that is p′Cu/r. Of course, the whole
of Cu is not relevant for our purposes. Only [p′*(1–p′)*Cuud]/r2 is relevant, as
just demonstrated. This corresponds to the time 0 value of [(p′*[p′*(1–p′)*
Cuud])/r2]/r or [p′*p′*(1–p′)*Cuud]/r3.
Thus, we have calculated that part of the value of Cuud which is contributed
to the time-zero option value C0 by Path 1 as [p′*p′*(1–p′)*Cuud]/r3. Another
equivalent way to write this quantity is as [(p′)2*(1–p′)*Cuud]/r3.
This is useful, as we will see. Notice that it is the discounted, risk-neutral,
present value of the product of Cuud times the joint probability of its occurring
at the end of Path 1.
n

CONCEPT CHECK 8

a. Run through the path contribution analysis for Path 2 to obtain the
contribution made by Cuud to C0 along Path 2.

n

CONCEPT CHECK 9

a. Run through the path contribution analysis for Path 3 to obtain the
contribution made by Cuud to C0 along Path 3.

OPTION PRICING IN DISCRETE TIME, PART 2

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To summarize, each path segment of each of the three paths terminating at
Cuud makes a contribution to the option price at time 0 in accordance with the
risk-neutral path probability assigned to it.
If we look at the path segments of path 2, for example, C0Cu has risk-neutral
probability p′, because it corresponds to an up move in the underlying stock
from C0. CuCud has risk-neutral probability (1–p′), because it corresponds to
a down move in the underlying stock from Cu. Finally, CudCuud also has riskneutral probability p′, because it corresponds to an up move in the underlying
stock from Cud. Thus the (joint) path probability for path 2 is p′*(1–p′)*p′.
Similarly, Path 1 has risk-neutral path probability p′*p′*(1–p′) and Path 3
has risk-neutral path probability (1–p′)*p′*p′. Note that all these three paths
have the same joint risk-neutral probability (p′2)*(1–p′). So, the sum of their
path probabilities is just 3*(p′2) *(1–p′).
The contribution to C0 of the terminal option price Cuud , made through
all the paths terminating at Cuud , is its risk-neutral discounted value which is
simply equal to [3*(p′2)*(1–p′)Cuud ]/r3.
If we perform this counting exercise for the remaining three outcomes of
the option price process, Cuuu, Cudd and Cddd, discount by r3, and add them
up, we will end up with the BOPM, N=3 value of C0. To accomplish this,
we reproduce the relevant information from Chapter 13 in Table 14.5.
TABLE 14.5

Path Structure of the BOPM, N=3

Terminal Stock
Price

Terminal Option
Price

Number of Option Price
Paths Terminating in
Terminal Option Price

Risk-Neutral Path
Probability

Suuu

Cuuu

C (3, 3)=1

p′ 3

Suud

Cuud

C (3, 2)=3

p′2 *(1–p′)

Sudd

Cudd

C (3, 1)=3

p′*(1–p′)2

Sddd

Cddd

C (3, 0)=1

(1–p′)3

Then, Table 14.6 gives the contribution to C0 made by all the paths. Outside
the brackets on the left are the number of paths terminating in the given
terminal option value. Inside the bracket is the joint risk-neutral probability
of arriving at the given terminal option value times that terminal option value.
Outside the brackets on the right, is the compounded risk-free rate.

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OPTIONS

TABLE 14.6

Value Contributions to C0 made by the Paths of the
BOPM, N=3

Terminal Option Price

Contribution to C0

Cuuu

1*

Cuud

3*

Cudd

p ′ (1− p ′)2C
3 * * 3 udd
r

Cddd

3*

p ′3 *Cuuu
r3
p ′2 *(1− p ′)Cuud
r3

(1− p ′)3Cddd
r3

When we add all these contributions up we obtain the European option price
at time 0,

p′3 * C uuu
p′2 * (1 − p′) * C uud
p′ * (1 − p′)2 * C udd
C0 = 1 *
+ 3*
+ 3*
r3
r3
r3
(1 − p′)3 * C ddd
+ 1*
r3
p′3 * C uuu + 3 * p′2 * (1 − p′) * C uud + 3 * p′ * (1 − p′)2 * C udd + (1 − p′)3 * C ddd
=
r3
But, this is exactly the same as the (BOPM, N=3) pricing formula, as it
must be,

C0 =
n

j =0,1, 2 , 3

C ( 3,j ) (p′) j (1 − p′)3− jC u jd 3− j
r3

CONCEPT CHECK 10

a. Verify the last statement that,

p′3 * C uuu + 3 * p′2 * (1 − p′) * C uud + 3 * p′ * (1 − p′)2 * C udd + (1 − p′)3 * C ddd
r3
∑ j=0,1,2,3C( 3,j ) (p′) j (1 − p′)3− jCu jd 3− j
=
r3

OPTION PRICING IN DISCRETE TIME, PART 2

499

14.5.2 Proof of the BOPM Model for general N
Now we can give an easy proof of the BOPM for any N based on the summary
Figure 14.10. It’s a simple counting exercise based on the path analysis we
accomplished.
FIGURE 14.10 Summary of Stock Price Evolution (N-Period Binomial

Process)

Terminal Stock Price

Number of Price Paths
Terminating in Terminal
Stock Price

Risk-Neutral Path
Probability

SN = ujdN – jS0

C(N, j)

(p')j(1 – p')N – j

1. For each terminal option price Cu jdN–j corresponding to j up moves and
N–j down moves in the underlying stock price, there are C(N,j) paths
terminating at Cu jdN–j .
2. The risk-neutral path joint probability of each one of these paths is
(p′)j(1–p′)N-j.
3. The contribution of Cu jdN–j to C0 is the number of paths times the risk-neutral
path probability times the terminal value Cu jdN–j discounted by rN. That is,

C (N ,j ) * (p′) j (1 − p′)N − j * C u j d N − j
rN
4. Now, add up all these contributions across all the paths corresponding to
j=0, 1,…,N to obtain the multi-period BOPM for any N where we factored
out rN because it is common to all terms in the sum,

C0 =

j =0 ,1,…N

C(N ,j ) * (p′) j * (1 − p′)N − j * C u j d N − j
rN

(BOPM, all N)

500

OPTIONS

We know that this is the correct option price, because it is exactly the same
one that would be generated by dynamic replication. There can’t be anything too
new in a complete model. However, there are some new ideas in the proof.
It’s part of the martingale approach to pricing derivatives which we turn to
in Chapter 15.

n

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OPTION PRICING IN DISCRETE TIME, PART 2

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1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
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501

KEY CONCEPTS
Modeling Time and Uncertainty in the BOPM, N>1.
Stock Price Behavior, N=2.
Option Price Behavior, N=2.
Hedging a European Call Option, N=2.
Step 1, Parameterization.
Step 2, Deﬁning the Hedge Ratio and the Dollar Bond Position.
Step 3, Constructing the Replicating Portfolio.
Replication under Scenario 1, over Period 2.
Replication in the Up State.
Replication in the Down State.
Solving the Two Equations for ⌬u and Bu.
The Complete Hedging Program for the BOPM, N=2.
Implementation of the BOPM for N=2.
The BOPM, N>1 as a RNVR Formula.
Multi-period BOPM, N>1: A Path Integral Approach.
Thinking of the BOPM in Terms of Paths.
Proof of the BOPM Model for general N.
END OF CHAPTER EXERCISES FOR CHAPTER 14

1. The current price of a share of RJR Nabisco Co. common stock is \$100.
The stock price can only go up by 10% with probability 0.5, or it can go
down by 5% with probability 0.5 in each period.
Consider a two-period European call option written on RJR Nabisco
Co. stock with current value C0. The call option is written so that it is
currently At-the-Money and the riskless rate per period is 7%.
a. Refer to the Binomial tree for the underlying asset price in Chapter
13, End of Chapter Exercise 2, part a.
b. Calculate the current price of the European call option according to
the two-period binomial option pricing model.
2. Consider Figure 11.4 from Chapter 11, reproduced below. Answer the
following questions:
a. What is the monotonicity property?
b. What is the convexity property?

502

OPTIONS

c. Are a. and b. rational option properties?
d. Pick a stock price, on the horizontal axis around E. Go directly up to
the European option price and draw a tangent to the option price graph.
What does the slope of that tangent line represent?
e. What happens to the slope of the tangent line in d. as the stock price
changes from St=0 to above E?
f. Interpret the result of e. in terms of the replicating portfolio.
FIGURE 11.4 A European Call Option Price, C E (St ,␶,E ), for a Given

Maturity ␶=T–t

CE(St, τ, E)

0

_r τ

St

St =e

_rτ

St –e

E

E

St

3. (Constructing the Replicating Portfolio under Scenario 2)
First, recall Scenario 2, which is that at the end of the ﬁrst period the stock
price has decreased to Sd and the option price to Cd.
Step 1, Parameterization, is the same in that u, d, and r are the same
constants and dFor Step 2, deﬁning the hedge ratio and the dollar bond position, let
⌬d be the number of shares to hold in the replicating portfolio per share
of stock underlying the option. The subscript d helps us remember that
we are constructing the hedge based on a starting stock price equal to Sd .
Let Bd be the dollar position in riskless bonds needed to replicate the
option over period 2. Note that Bd<0 means we borrow at the riskless
rate.
Let Cd be the current value of the European call option when the stock
price is Sd. Cd depends on the current value of the underlying stock, Sd ,

OPTION PRICING IN DISCRETE TIME, PART 2

503

the number of periods to expiration, N–1=1, and the exercise price E. So
we can write Cd=C(Sd,1;E). This is what we are trying to determine.
Now we let time pass, and see what happens to the option’s value as
the stock price either goes up or goes down over the second period. If
the stock price goes up, the option price will also go up. If the stock price
goes down, then the option price will also go down.
Let Cdu be the value of the option at time N=2 if the stock price goes
up over the second period. Using our notation Cdu=Cdu(udS0,0;E), which
represents the value of the option based on the new stock price udS0, no
time to expiration since N–2=0, and exercise price E.
Let Cdd be the value of the option at time N=2 if the stock price goes
down over the second period. Using our notation again, Cdd=Cdd(d 2S0,0;E),
which represents the value of the option based on the new stock price
d 2S0, no time to expiration since N–2=0, and exercise price E.
We know that Cdu=IVdu=MAX[0,udS0–E] and that Cdd=IVdd=MAX
[0,d 2S0–E], where IVdu means intrinsic value if the stock price goes up,
and IVud means intrinsic value if the stock price goes down over the second
period.
The schematic is given in a. and is what we call the ‘natural’ or actual
call option under Scenario 2. This is the object to be synthesized by the
replicating portfolio.
a. Fill in the question marks in all of the ﬁgures below.

?
Cd
?
Next, we represent the replicating portfolio,

?
∆dSd
?

Bd

?

504

OPTIONS

Replication must be done for both the up state and the down state.
Up state means that the stock price goes up over the second period.
Down state means that the stock price goes down over the second
period.
A. Replication in the Up State
b. The call option’s payoff if the stock price goes up over the second
period= _______ .
c. The replicating portfolio’s payoff=payoff to _______ stocks and
\$_______ in bonds= _______+_______.
d. Replication in the up state over the second period means that
_______+_______ = _______.
(up)
B. Replication in the Down State
e. The call option’s payoff if the stock price goes down over the second
period=_______ .
f. The replicating portfolio’s payoff=payoff to _____ stocks and \$_____
in bonds=_____ _____ .
g. Replication in the down state means that
_______+_______=_______ .
(down)
We now have two equations in two unknowns which must hold
simultaneously in exercise 4.
4. (This exercise is a continuation of exercise 3.)
C. Solving the Two Equations for ⌬d and Bd
h. The two equations are,
_______=_______
_______=_______

(up)
(down)

i. To solve for ⌬d, just subtract (down) from (up) to obtain,
_______=_______
which when solved gives the hedge ratio ⌬d ,
⌬d=_______
j. Interpret the hedge ratio ⌬d

(⌬d)