### Binomial Tree Option Pricing

```Binomial Tree Option Pricing
A Three Step Process:
1) Construct a Stock Price Binomial Tree
2) Value the Option at Time of Expiry
3) Value the Option Through Backward Induction
Three Guys, One Method
• John Cox, Stephen Ross, and Mark Rubinstein published a
paper detailing their method in 1979
• “Option Pricing: A Simplified Approach”
• A discrete, numerical alternative to the Black Scholes PDE
using relatively simple techniques
• No Calculus required!
What is a Binomial Tree?
• Say we have a stock price at t0, S0. In the Binomial Method, the
price can go either up or down. At t1 (after one time interval),
the price can either be an “up” price or a “down” price. These
prices can each go either up or down over the course of the next
time interval.
• You can see that the possible prices quickly “branch” out over
time, thus the term Binomial “Tree.”
• By making the number of time intervals
between t0 and time of expiry T very large,
we will get many possible stock prices at T
and we will have a better approximation of
the Brownian Random Walk, which is a
time continuous model.
The Up and Down Ratios
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In order to get from S0 to Su, we have to multiply S0 by what’s called the up
ratio, labeled u. Similarly, to get from S0 to Sd, we have to multiply S0 by the
down ratio, labeled d. These factors are constant throughout the tree.
Recombining Feature: If the stock takes an up move followed by a down move,
it’ll arrive at the same price had the stock taken a down move followed by an
up move. Order does not matter.
Consequence: The Stock Price after m up moves and k down moves, regardless
of the order they happened, would equal S0umdk
u and d depend on two things: volatility of the stock and the length of a time
interval. Cox, Ross, and Rubinstein chose the up and down ratios to be these:
Because d is the reciprocal of u, u*d = 1. Therefore, if S0 takes an up move
followed by a down move or vice versa, the price will return to S0.
We will use these formulas for u and d to model a Stock Price Binomial Tree.
Transition Probabilities
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If the probability of S0 rising to Su is p, then the probability of S0 falling to Sd
must be 1-p, since one of those two outcomes must happen in this model.
We can say that the expected price at t1 is the probability of the up move
happening times the up price plus the probability of the down move happening
times the down price.
We want the Binomial Method to be risk neutral. A riskless asset should grow
by a factor of
after delta t, with r as the riskless interest rate.
Set the risk neutral expected value of S equal to the binomial expected value of
S.
• Solving for p yields:
• This is the risk-neutral transition probability of an up move.
• Remember that u and d only depend on the volatility and the length
of the time interval, so this probability only depends on volatility,
the length of a time interval, and the riskless interest rate. All of
these will remain constant throughout our binomial tree, so this
probability will remain constant throughout the tree as well.
Step 1: Constructing a Stock
Price Tree
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Let’s say we want to price a European call option with a strike of \$100. The
initial stock price is \$100, volatility is 30%, time to expiry is one year, and the
riskless interest rate is 5%. Let’s build our stock price tree with four time
intervals.
(where node * is located m up
moves and k down moves from S0)
T
Step 2: Valuing the Option at
Time of Expiry
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While we don’t know the value of the option before time of expiry, we do know
the value of the option at the time of expiry: it’s simply the payoff of the option.
For our example, the value of the European call at T is ST – K if ST > K and 0
otherwise.
We will construct a second binomial tree identical in structure to our Stock
Price Tree, which will serve as our Option Value Tree. We can fill in the nodes
at t4 = T.
How Do I Value the Option at
Earlier Nodes?
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Let’s build a portfolio that contains delta shares of a stock and a riskless bond
that matures to B after one time period.
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We want to choose delta and B such that the value of the portfolio is equivalent
to the value of an option on that stock, depending on the direction the stock
goes. We’re replicating the payout of the option with our portfolio, so we want:
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Solving for delta and B, we get:
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The portfolio, with these values of delta and B, replicates the value of the
option V on stock S. So at t0:
Substitute in our values for delta and B :
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We finally have a way to find the value of option at earlier nodes. We know S0,
the riskless interest rate, the length of the time interval, and the value of the
option at later nodes (specifically at T). Further manipulation brings the
riskless probability into play:
Step 3: Valuing the Option
Through Backward Induction
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Recall our example of the European call from before.
S0 = 100, K = 100, σ = 30%, T = 1 year, n = 4, r = 5%
Stock Price Tree
Option Value Tree
Observe: each sub-piece of the Binomial Tree is its own one-period tree
Let’s calculate p, and then we’ll be able to determine “V0.”
= e-.05(.25)(.5043*82.21+(1 - .5043)*34.99)
= \$58.07
Now we can fill in that node on our option value tree.
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We continue the process by filling in the rest of the nodes at t3. Once we’ve
done that, the nodes at t3 become the the Vu’s and Vd’s for the “V0’s” at t2. We
continue working backwards until we have a value for the option at t0, the
actual V0, which is the price of the option and what we’ve been looking for.
The rest of the option value tree fills in like this:
Compare to Black-Scholes
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The Binomial Tree Method gave us a price of \$13.53 on that European call
option. The Black Scholes PDE gave that same option a price of \$14.23.
Why the discrepancy? Remember, this is a time-discrete approximation to the
Black-Scholes method. We only used four time intervals and came within less
than a dollar of the Black-Scholes method.
Increasing the number of time intervals (and thus making each time interval
shorter in length), would increase the method’s accuracy because our model
would then be a better approximation of the time continuous model.
Pricing an American Option
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With the Binomial Method, we can easily adapt a European option to an
American option.
When using backwards induction to fill in the nodes on the option value tree,
compare the value that you get by using the formula from before to the value of
early exercise at that respective node. The actual value at that node is the
greater of the two.
Value of Option at a Node = max (Binomial Value, Exercise Value)
Interestingly enough, in our example from before, the exercise value is lower
than the binomial value at every node. Therefore, the price of an American
option with those same parameters is still \$13.53.
Here’s an example where the American price will be different than the
European price:
We want to price an American put option with S0=50, K=52, T=2 years,
σ=22.31%, r=5%, and n=2. So Δt= 1 year, u=1.25, d=.8, and p=.5584
Stock Price Tree
Value at Node “a” = max (
=\$0.84
Value at Node “b” = max (
= \$12
Value at Node “c” = V0 = max (
= max(5.49, 2) = \$5.49
Option Value Tree
, max(K – Sa, 0)) = max(0.84, 0)
, max(K – Sb, 0)) = max (9.46, 12)
, max(K – Sc, 0))
Modifications to the General
Cox-Ross-Rubinstein Method
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Dividends: If a stock pays a dividend as a percent of the stock price at a certain
date, that can be reflected in our stock price tree by taking what would be the
stock price at that date and multiplying it by the percent of that value not being
paid out. We can model our stock price tree to a variety of situations.
Flexible Trees: In the Cox-Ross-Rubinstein model, Binomial Trees called
“Standard” Trees were used. The biggest difference between Standard Trees
and “Flexible” Trees is that the volatility need not be constant in the latter. This
allows us to more accurately model our Stock Price Tree because volatility
does generally change quite a bit, but it comes at a cost: the values for u, d, and
p must be recalculated at every time step.
Pros and Cons
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Pros:
 It uses relatively simple Mathematics.
 It can be used to price American and Bermudan options.
 It can be implemented in computer programs.
 It can be adapted to various kinds of stock features (like dividends)
Cons:
 Being discrete, it does not produce exact answers.
 By hand, it would take a long time to price an option using a lot of time
intervals.
 At least with the Cox-Ross-Rubinstein Model, it must use a constant
volatility, a downside that the Black-Scholes PDE has as well.
Sources
•
Black-Scholes and Beyond: Option Pricing Models
by Neil A. Chriss
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Introduction to Futures & Options Markets
by John C. Hull
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The Concepts and Practice of Mathematical Finance
by Mark Joshi
•
Option Pricing: A Simplified Approach
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by John C. Cox, Stephen A. Ross, and Mark Rubinstein