mayerbinomial tree paper

7/28/2019 MayerBinomial Tree Paper

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Erik Mayer

Pricing Options with Binomial Trees

When someone invests in a financial derivative, or more specifically, an option, that

individual is taking a risk. That risk could pay off in a substantial way for the investor, but it

could also result in significant losses. Being able to determine the price of these derivatives, what

can be agreed upon as a fair price to enter into the risk, allows this particular market to work.

The Black-Scholes Partial Differential Equation (PDE) emerged as great way to price

derivatives. From the partial differential equation, one could derive analytical formulas for

options, including the European Call. However, deriving exact formulas for American options,

which unlike European options can be exercised at any time before the time of expiry on the

option, became a problem. John Cox, Stephen Ross, and Mark Rubinstein, three financial

economists turned to a discrete model known as Binomial Trees in order to solve that problem.

In 1979, Cox, Ross, and Rubinstein published a paper called Option Pricing: A

Simplified Approach, which detailed their binomial tree method (Chriss 220). It is a discrete,

numerical method, but it uses the same assumptions as the continuous Black-Scholes model,

namely the random walk of a stock price, which says that the movements of the stock are random

except for a basic upward drift that is dependent of the riskless interest rate (Chriss 232). This

assumption will work quite well with binomial trees, as we will see in a moment. An attractive

aspect to keep in mind is that the binomial tree method uses relatively simple mathematics when

compared to the Black-Scholes PDE. There is absolutely no calculus required to use this method.

Before saying any more, it needs to be established what a binomial tree actually is. So

here is one:


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This binomial tree models the possible movements of a stock price over two time periods.

At t0, the initial time, the initial stock price, S0, can move either up or down. So at t1, there are

two different nodes representing two possible stock prices, Su and Sd (the subscript u stands for

up and the subscript dstands for down). Both Su and Sd can move either up or down. If the

stock price were to continue after t2, each price at t2 would move either up or down. This

branching of up and down from each node would continue for however many time periods the

one making the tree wanted. Notice how when Su went down and when Sd went up, their paths

merged into one node, marked Sud. This is a recombining feature ofbinomial trees, one that

has to do with the associativity of multiplication, but we will delve into this a little more deeply

in a moment (Chriss 222).

As more and more time periods are added to the tree, the more and more possible stock

prices there are at the end of the tree, which for our purposes indicates a time of expiry on an

option (hence the capital T in the diagram above). After two time periods, there are only three

possible stock prices at the time of expiry, which is not very many and is not a very good

simulation of the random walk assumption, but by increasing the number of time steps within the

allotted time to expiry (which in turn means that each time period is shorter), the tree would have

more and more possible stock prices at the time of expiry. If one were to theoretically increase


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the number of time periods to infinity (which would mean that each time period would approach

a length of zero), the discrete binomial tree model would converge to the continuous random

walk model (Chriss 302).

The next point that needs to be addressed is to determine how much the stock price

should go up for an up move and how much it should go down for a down move. It turns out that

for any stock price at a node, its corresponding up price is found by multiplying the stock price

by what we call an up factor, labeled u, and the corresponding down price is found by

multiplying the stock price by a down factor, labeled d(Chriss 223). These factors will be

constant throughout a given binomial tree. So if we start with an initial stock price, S0, we

multiply that price by u to get Su. Lets say Su takes a down move. Su would be multiplied by d.

So that would be S0*u*d in total. What we can say by looking at this is that at every node, the

stock price is equal to the initial stock price multiplied by varying amounts of up and down

factors.

This also explains why the recombining feature is present in binomial trees. The price at

any node is calculated by multiplying S0 by constants. Since multiplication is associative, the

order in which we multiply varying us and ds to S0 does not matter. So an up move followed by

a down move will yield the same stock price as a down move followed by an up move. That is


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why the branches conveniently merge back together with regularity (Chriss 226). With all of this

in mind, we can easily calculate the stock price at any node so long as we know S0, u, and d:

(where node * is m up moves and kdown moves from S0)

Cox, Ross, and Rubinstein determined what u and dshould be by letting the amount of

time steps on their binomial tree model approach infinity and trying to match the result of that

with the Black-Scholes model (Cox 20). In order for the two models to match, the three

economists used these values foru and d:

-- where represents the volatility of the underlying asset and t represents the length of

each time period, which is calculated by dividing the time to expiry for the option, T, by the

desired number of time periods, n. Remember that these values determine how much the stock

price at any given node will change, either up or down. So it makes sense that these factors

depend on the volatility, which by definition says how volatile or prone to change the asset is,

and the length of each time interval (because with a larger time interval, the stock is likely to

change more than with a relatively small time interval). It is also interesting to note that dis

equal to the reciprocal ofu. What this means is that when u is multiplied by d, the product is one.

So if an initial stock price takes an up move followed by a down move, the moves cancel each

other out and what is left is the initial stock price (Chriss 234). This will become evident when

we begin to work with examples.

Before we start with an example, however, one more attribute of these binomial trees

need to be addressed. We want to determine the probabilities attached to these up and down


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moves. We will say that the probability of an up move isp, which would make the probability of

a down move 1p (since there are only two possible for any given node, the sum of the

probability of the up move and the probability of the down move must equal 1).

Now imagine that we are standing at t0and want to determine the expected value of the

stock at t1. The expected value of the stock at the subsequent time interval is kind of like an

educated guess at what the stock price might be, on average. Only instead of a simple average

of the stock price outcomes, we will take an average of Su and Sd weighted by their respective

probabilities (Hull 245).

Recall that we want the binomial tree method to be risk neutral in order for it to follow

the random walk assumption. So this means that on average, the stock price should grow by a

factor of e rt aftert, where ris the riskless interest rate because of the upward drift built into

the random walk assumption (Hull 245). Now we set this riskless growth expression equal to the

binomial expected value assumption:

The only parameter whose value we do not know how to calculate here isp, so we can

solve for it:


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This is the risk-neutral transition probability for an up move. Notice how all actual stock

prices are cancelled out when solving forp, and all remaining input parameters are constant

throughout our binomial tree. This means thatp will be constant throughout the tree as well. It

turns out that an up move usually has a greater chance of happening than a down move, unless

the volatility is very high and the riskless interest rate is low. This has to do with the upward drift

assumption.

Now we have all the tools we need in order to build a stock price tree. We can finally get

to the point of all of this, which is to price options using binomial trees. If we are given a stock

price and an option on that stock, the first step we would take toward pricing the option would be

to construct a stock price tree. So lets try an example. We will start with a European Call and

then modify our approach for American options. For the purposes of this example, the number of

time intervals will be four, but in reality, the more time intervals we include, the better our

answer will be.

S0 = $100 K (Strike Price) = $100 = .30 r = .05 T = 1 year n = 4

Using this information, here is the stock price tree we would model:

T


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We do not know the value of the European Call at t0 (the price of the option), nor do we

know the value of the call at t1, t2, or t3. However, since t4 is the time of expiry, the key insight is

that we do in fact know the value of the call at t4, since the value is simply the payoff of the

option (which for the European Call is STK if ST > K, and 0 otherwise). For each node on the

stock price tree at the time of expiry, we can attach a corresponding option value (Wikipedia

BOPM). So now we will build an Option Value Tree, one identical in structure to the

corresponding stock price tree, where we will keep track of the option values for each node,

starting with the values at T.

We have successfully figured out how to value an option at its time of expiry, but our

goal is to value the option at t0. So how do we value the option at preceding time steps? The

answer lies in building a portfolio containing delta shares of the stock in question and a riskless

bond that matures toB after one time period (Chriss 275). We choose delta andB so that

depending on the direction the stock price moves, the value of our portfolio replicates the value

of the option (note that the delta, or, in this context is not the same as in the context of a

change in, like a change in time. Its simply a representation of an amount).


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Portfolio Value Binomial Tree Option Value Binomial Tree

In order to accomplish this, well set the up value of our portfolio equal to the up

value of the option, and the down value of our portfolio equal to the down value of the

option.

Su + B = Vu

Now we have two equations with two unknowns (delta andB), so we can solve for those

unknowns. After some manipulation, here are the values fordelta andB.

Since our portfolio is replicating the value of the option with these values ofdelta andB,

we can draw the connection between the portfolio and the option at t0:

Plugging in our values fordelta andB:

This can be simplified into something that should look rather familiar (Chriss 277):


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V0, for a one time-step tree, is the expected value of the option at t1 (the weighted

average of both outcomes), discounted by the riskless growth factor. So in order to calculate the

value of an option at t0 for a one-step tree, we need to know both possible values at t1. Now here

is the key insight: we know the values of the option at the time of expiry for our example from

above, so we can use this formula. Apply this formula to each one-step subset of our option

value tree by treating the t3values as the unknown V0 and the values at T as the Vus and Vds.

From there, we will be able to work backwards through the option value tree until we arrive at a

value for the actual V0. Lets do this with our example from above.

Notice how Im treating upper-rightmost portion of the tree as its own one-step tree,

designating values of the option that we know at the time of expiry as Vu and Vd, and the

unknown value of the option at a preceding steps node as V0. Now we just use the formula to

find this V0.

= e-.05(.25)(.5043*82.21+(1 - .5043)*34.99)

= $58.07


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Now we can fill in this node on our option value tree.

Next, you would simply go to the next node at t3, treat it as V0, treat $34.99 as Vu

and $0 as Vd,and calculate that V0. You would continue in this fashion until all nodes at t3

had been filled in. At that point, you would treat the nodes at t3as the Vus and Vds and the

nodes at t2as the V0s. You would continue working backwards until you arrived at a price for

the option at t0 (Hull 358). In more general terms, we can write the formula like this:

Here is the rest of the option value tree filled in, complete with a price at t0:


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So using a four time-step binomial tree, we calculated a price of $13.53 for this European

Call option. The price of this option calculated by the Black-Scholes Partial Differential

Equation is $14.23. The discrepancy comes from the fact that this is a time-discrete

approximation to the Black-Scholes model. We only used four time-intervals, which allowed for

only five possible stock prices at the time of expiry. Remember that this model converges to the

Black-Scholes model as we allow the number of time-steps to approach infinity. So our answer

would get closer to the Black-Scholes answer if we built our binomial tree with more time steps

because then our time-discrete model would be a better approximation of a time-continuous

model (Joshi 169).

This method for pricing European options can easily be adapted to American options,

which can be exercised at any time prior to expiry. While pricing nodes through this process of

backwards induction, you look at the value you get by using the formula that we derived and

you compare it to the value of early exercise at that node, which is simply the payout. The

greater of the two is the actual value at that node (Hull 251). Perhaps some readers figured out

that for a strictly European option one only needs to calculate the stock prices at time of expiry in

order to price the option, but for American options, we need the entire stock price tree to move

forward. If we were to change our example of the European call into an American call, here is

how we would do it:


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The first two steps are the same: construct the stock price tree (the tree on the left), and

value the option at time of expiry (the tree on the right).

So we proceed with backwards induction. Start with the upper-rightmost subsection of

the option value tree. The formula for V0 gives us a value of $58.07, just like before. Now we

compare that to the value of early exercise at that node. Look at the stock price tree at that node

and determine how much the option would be worth if one were to exercise it at that point.

max (156.83100, 0) = $56.83

Now you determine which of those two values is greater:

max (58.07, 56.83) = $58.07

So the value at that node for the American call is $58.07, as it was before. You would do

this one extra step for all the nodes in the option value tree in order to get the American price. It

turns out that for this American call, the value of early exercise is never greater than the

binomial value given by the formula, so the price of the American call is $13.53, the same

price as the European call. The reason for this is the upward drift that was built into this binomial


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tree method. Because the value of a call rises when the stock price rises, it is not optimal to

exercise an American call option early. Recall the fact that the probability of an up move is

usually greater than a down move. But if you look at an American put, whose value will rise

when the stock price drops, it will be optimal at some points to exercise early because of the

upward drift (Hull 251). Here is a quick example:

American put option with S0=50, K=52, T=2 years,

=22.31%, r=5%, and n=2. So t= 1 year, u=1.25, d=.8, and p=.5584

Stock Price Tree Option Value Tree

Value at Node a = max (e-r*t(p*Vu + (1p)*Vd) , max(KSa, 0)) = max(0.84, 0)

=$0.84Value at Node b = max (e-r*t(p*Vu + (1p)*Vd) , max(KS

b, 0)) = max (9.46, 12)= $12

Value at Node c = V0 = max (e-r*t(p*Vu + (1p)*Vd) , max(KS

c, 0)) = max(5.49, 2)= $5.49

So the price according to a two time-step binomial tree of this American put is $5.49.

Notice how at node b the value of early exercise was greater than the binomial value given by

the formula, so that affected the calculation of node c. Again, it needs to be stressed that our

answer would be better if we built our binomial trees with more time steps. Using a lot of time

steps would make this difficult to do by hand, but luckily the binomial tree method can be

implemented into computer programs likeExcelandMaple.


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European and American options are not the only kind of options that binomial trees can

price. In fact, there are an abundance of exotic options in the world that can be applied to the

basic binomial tree framework. We will look in particular at the up-and-out barrier call. Barrier

options have a stock price level that acts as a point either where the option becomes active or

where the option becomes worthless. So for an up-an-out barrier call, if the stock goes above a

certain price (or barrier) at any point during the course of the option, the option becomes

worthless, even if the stock goes back below the barrier by the time of expiry (Wikipedia

Barrier). We will rework the example of the European call from before into an up-and-out barrier

call. Here are the parameters, now with a barrier level:

S0 = $100 K = $100 = .30 r = .05 T = 1 n = 4 t = .25u = 1.1618 d = 0.8607 p = 0.5043 B = $150

So if the stock price goes above $150 at any point during the lifespan of the option, the

option becomes worthless. The stock price tree is constructed exactly as it was before, only now

with an indication of thebarriers placement.

T

With European and American options, the next step we would take would be to value the

option at the time of expiry. At first glance, it may seem that we would still be able to do that

with the barrier option, but such a course of action would be foolhardy indeed. Remember that if

the stock price goes above the barrier at any point during the course of the option, the options

B = $150


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value becomes $0. If you look at the stock price tree, you can see that there is a path on the tree

that takes three up moves to $156.83, which is above the barrier, and then takes a down move to

$134.99, which is below the barrier. If the stock takes this path, the value of the option is

worthless. If we value the option at the time of expiry, the $134.99 node would be $34.99, but

this does not hold true for all possible paths ending at the $134.99 node. The barrier option and

many other exotic options arepath-dependent, meaning that the payoff of the option is directly

related to what kind of course the stock price takes (Chriss 303). There are options with varying

degrees of path-dependency. An Asian option, for example, whose payoff is calculated using the

average stock price over the lifespan of the option, has what is known as a very strong path-

dependency (Joshi 209).

At this point, it may seem like we are stuck. But before giving up, let us take note of a

few things. We know every possible path that the stock price can take on our tree. The binomial

tree is a discrete model. There are a limited number of possible paths and if we are ambitious

enough, we can find all of them. We also know the payoff of the option for any individual path.

Finally, we know the total probability of each possible path. Using these three pieces of

information, we will be able to work around the problem of path-dependency and price the

barrier option.

In order to find the total probability of a certain path, one would need to multiply the

individual probabilities for up and down moves on that path together. So, for instance, if a path

takes two up moves, a down move, and then an up move, the total probability would bep*p*(1

p)*p. And as was the case before, since multiplication is associative, the order in which the

moves happen does not matter and we can say that if we call the total probability Q,

Q = pm(1p)k(where m is the number of up moves the stock price takes and k is the number of down moves it takes)


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As for the different paths, start by looking at a one-step binomial tree. There are two

different paths on that one-step tree: up and down. If we extend that tree into a two-step tree,

both nodes at the end of the one-step tree will extend both up and down and the number of paths

doubles to four. If we keep going in this fashion, we will see that for an n time step binomial

tree, there are 2n different possible paths on that tree (Volpert). Knowing exactly how many

paths we should be looking for on some larger trees will be very helpful.

So how do these pieces of information allow us to value an option? The answer lies in

looking at the formula we used for valuing an option at a certain node and generalizing it. Here is

what I mean:

In words, the options value at a certain point is the average of the different outcomes

weighted by their respective probabilities, discounted by the amount of time that has passed. So

change Vuto Outcome 1, O1, and Vdto Outcome 2, O2. Thenp is the probability attached to

O1, so we call it Q1, and (1p) is the probability attached to O2, so we call it Q2. But there does

not necessarily have to be only two outcomes for the logic of this formula to hold. There could

be as many outcomes as we want. So let us apply that logic to the different paths on the binomial

tree. An n time-step tree has 2n different paths, each with an attached probability, and 2n different

payoff outcomes. And now that we are talking about the entire tree instead of a one-step

subsection, the tbecomes T, the entire time to expiry. Additionally, the generic V,

representing the option value at any node, firmly becomes V0. With these changes, the formula

looks like this:


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Now we have an alternate way to price an option on binomial trees, one that makes use of

all the different paths on the binomial tree. Using this formula, we can price any path-dependent

option on binomial trees, including our example of the up-and-out barrier.

Now we return to our example of the up-and-out barrier call using a four time-step

binomial tree, which had a barrier of $150 and a strike price at $100. There are 24 = 16 different

paths on this tree that we will need to find. For each of those paths, the total probability Q = pm(1

p)k, where p = 0.5043 (recall this figure from a previous calculation).

T

The payoff of each path will look like this:

Payoff of a Path = max(0, STK) unless the stock rises above $150, in which case payoff is $0

So now we are ready to find the sixteen paths and value the barrier option:

B = $150


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One way that we can check to see if we have found all of the correct paths is to add up

the probability of each path. If the sum is something other than 1, then something is wrong. It

must be 1 because we know for certain that the stock price has to take one of these paths. So we

can do this:

Now that we are sure that we have all the correct paths, we can calculate the price of the

barrier option at t0.

=

= $6.35

Using binomial trees, we calculated the price of this option as $6.35. The actual price of

this option is $5.13 (SITMO). Before we address the discrepancy, however, it should be pointed

out that only two paths got knocked out by the barrier, the ones labeled 1 and 2 above. If this

was just a regular call, without a barrier, those payoffs would have been $82.21 and $34.99

respectively. Without a barrier, its the same European Call that we calculated above, so if we

plug these values into the equation instead of the zeroes, we should get the same value of $13.53


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that we got before. Sure enough, doing this yields a value of$13.52 (the pennys difference can

be attributed to round-off error during the backwards induction process). It is comforting to

know that the backwards induction process and this path process are in agreement.

As for the difference between our calculated value for this barrier of $6.35 and the actual

price of $5.13, we said before that the binomial tree approximation should converge to the actual

price as we let the number of time steps approach infinity. I took a Maple worksheet that does

this process of finding all the different paths on a binomial tree and modified it to calculate the

barrier options value, so we can see how long this process of convergence takes (Volpert). The

following graph has n, the number of time-steps built into the tree, on the x-axis, and the barrier

option prices on the y-axis. The red curve follows the binomial tree calculations of the barrier

price, while the horizontal green line represents the actual barrier price of $5.13.

The graph takes n out to 18. Remember that an 18 time-step tree has 218 different paths,

which equals 262,144 different paths. Each of those paths contains (n + 1) different stock prices,

or 19 in this instance. The point is that as the number of time intervals increases, the amount of

information that Maple has to keep track of using this procedure is startling. Calculating the price

of this barrier with 18 time steps took Maple more than a few minutes. If you try to make Maple

do much more than 18 time steps, the program will get overloaded. And the discouraging aspect


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of all of this is that it appears as if the method is not converging very quickly or with a clear

pattern. It seems to get close to $5.13 at one juncture, and then shoots up at the very next

juncture. Part of the problem with using binomial trees for pricing barrier options is the prices

sensitivity to a barrier slightly above or below a line of nodes. Recall that the up and down

factors as Cox, Ross, and Rubinstein defined them are reciprocals of one another, meaning that

an up move followed by a down move has the same effect of no moves at all. If you look at the

stock price trees from above, you will see stock prices repeating horizontally every two time-

steps. That is what is meant by a line of nodes.

The barrier for this example was set at $150 and the resultant barrier option price was

$6.35. However, a barrier as high as $156.82 and a barrier as low as $135.00 yield the same price

of $6.35. Thats a range of over 20 dollars where the price as deemed by this four step binomial

tree is constant. This is because in that range, the exact same paths are being knocked out.

However, if the barrier is ever so slightly above $156.83, the price jumps to $8.46. If the barrier

is placed slightly $134.99, the price plummets to $0 because all potential non-zero payoff paths


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have been knocked out. The actual price of the barrier option smoothly changes as the barrier

level changes. There are no large stretches of stagnancy and no sudden and severe jumps. What

does this have to do with the increasing of the number of time intervals? Well, when you rebuild

a tree with more time intervals, the values ofu and dchange (because remember, they depended

on the length of a time interval), which changes the stock prices at each node. If the barrier was

slightly below a line of nodes on a four step-tree, and on a five-step tree the new u and dmake

it so that the barrier is slightly above the same corresponding line of nodes, the price will jump

just as it did when we moved the barrier around.

On the left is the stock price tree for this barrier call constructed with five time intervals

instead of four, and on the right is the stock price tree built with six time intervals. The horizontal

black lines indicate the barrier position of $150, while the horizontal green lines highlight lines

of interest. For the five time-step tree, the barrier is still above that particular line of nodes and

still only two different possible paths are being knocked out. Combine that with the fact that

there are now 32 total paths on this tree and there are two different nodes at time of expiry that

can result in a non-zero payoff, and its clear that the price is going to shoot up. To be exact, the

price is $10.33 on this tree (SITMO). For the six time-step tree, there are now 64 total possible


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paths, and now the barrier is below one more line of nodes than it was before. Additionally,

there is only one node at time of expiry that can result in a non-zero payoff. So the barrier option

price as calculated by this tree is $5.86 (SITMO). The price will continue to jump around in this

manner as we increase the number of time intervals. The stock price tree would have to be much

more dense with stock prices before this behavior would settle down, and unfortunately, that

requires being able to find all the paths on trees with a large amount of time intervals, a task that

is too mammoth in scope even for Maple to undertake.

These lines of nodes occur as a result of the up and down factors being reciprocals of

one another. However, up and down factors as chosen by Cox, Ross, and Rubinstein are not the

only up and down factors used when modeling binomial trees. In fact, John C. Hull and Alan

White modified the Cox-Ross-Rubinstein model by using their own up and down factors.

A crucial difference with these up and down factors is that they are no reciprocals of one

another, so the resulting stock price tree that we build will not have those lines ofnodes

(Volpert). We can see if it makes a difference in the convergence for this example. I modified the

Maple procedure that I used before with these White-Hull up and down factors, and then graphed

the behavior of the barrier options price, just as before. The green curve represents the behavior

with the Cox-Ross-Rubinstein up and down factors, the red curve represents the behavior with

the White-Hull up and down factors, and the horizontal orange line is the actual price for this

option, at $5.13.


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Surprisingly enough, aside from prices at n = 5 time intervals, which jumped in opposite

directions, the change in up and down factors did close to nothing to change the convergence

behavior of the barrier option price. The White-Hull factors accentuate the upward drift more

than the Cox-Ross-Rubinstein factors, so by looking at the input parameters for this example,

one can see that perhaps the reason these two outcomes followed each other so closely is that the

riskless interest rate was only 5%, while the volatility was 30%. If the interest rate was higher,

its probable that the White-Hull factors would have skewed the tree upward a little more.

Regardless, the White-Hull up and down factors are a good example of the versatility of

the binomial tree method. People can modify the binomial trees to suit different purposes. For

example, if you want to price an option on an asset that pays out dividends at certain times, that

can be incorporated into the binomial trees (Hull 365). And there are always economists trying to

make the model more realistic. One shortcoming of both the Black-Scholes Partial Differential

Equation and the Cox-Ross-Rubinstein model is the constant historical volatility input. In the

real world, however, it can be shown that volatility is constantly moving up and down. Neil A.

Chriss, in his bookBlack-Scholes and Beyond, talks about modeling binomial trees without

constant volatility with a type of binomial tree he calls the Flexible Tree (Chriss 221). Some


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economists have actually favored a trinomial tree instead of a binomial tree because they feel

that it is a more realistic model of how a stock price behaves (Joshi 170). The point is that the

basic idea of pricing options with binomial trees can be adapted and manipulated in a large

number of ways for a large number of reasons. That is part of the models appeal.

So even though the process of pricing path-dependent options with binomial trees

requires keeping track of a lot more information than European options, for example, the idea

that such a diverse and large array of options can theoretically work with binomial trees is

fascinating. The processing power of todays computers may make it so that pricingpath-

dependent options with a binomial tree is impractical when you can use a more efficient method

called Monte-Carlo Simulations, but one can never know what the future of technology will

hold. Perhaps one day, easily accessible computers will be powerful enough to practically

compute binomial trees containing hundreds of thousands of paths. Regardless of what happens,

however, what we have with the binomial tree method is a versatile, conceptually simple way to

price a large assortment of options. And on that we can hang out hats.


25/25

Sources

Chriss, Neil A.Black-Scholes and Beyond: Option Pricing Models. New York: McGraw-Hill,

1997.

Hull, John C.Introduction to Futures and Options Markets. Englewood Cliffs, NJ: Prentice Hall,

1995.

Cox, John C., Stephen A. Ross, and Mark Rubinstein. "Option Pricing: A Simplified

Approach."Journal of Financial Economics (1979). Web. 20 Oct. 2010.

.

Joshi, Mark S. The Concepts and Practice of Mathematical Finance. Cambridge, U.K.:

Cambridge UP, 2003

"Binomial Options Pricing Model." Wikipedia. Web. 20 Oct. 2010.

.

Volpert, Klaus. Valuing Derivatives via the Binomial Tree Method (Cox-Ross-Rubinstein).

SITMO - Financial Engineering. Web. 13 Nov. 2010. .

"Barrier Option." Wikipedia. Web. 13 Nov. 2010. .

mayerbinomial tree paper

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