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# Maple's Magic Money Tree

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## Rebecca Spencer-Strong

on 11 March 2015

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#### Transcript of Maple's Magic Money Tree

And now for the Grand Finale...
The results!
Tesla
Results from Maple Procedure
[41.55, 11.05, 4.41, 1.28, .03]
Maple's Magic Money Tree
To construct a trinomial model in Maple that uses backwards induction to produce a list of call prices for a given list of strike prices
The goal was to be able to calibrate our model to approximate market data
The reason to construct a model like this is to identify mispriced options
Purpose
S0-initial stock price
u and d are our factors to get to next step
r is the interest rate
We have 2 risk-neutral probabilities, p and q, with which the model is arbitrage-free
Use backwards induction to price initial call prices

Formula:

Review of Binomial Model

Takes in an S0 and the amount of steps until expiration
Computes the first step so that we have 3 branches
Puts that in a list with the initial stock price and their corresponding time intervals

[[[0,S0]],[[1,u*S0],[1,m*S0],[1,d*S0]]]

Then it takes the end of each branch and multiplies each by u,m, and d
Each of these values is put into a list with its corresponding time step then is added to our overall list

Maple’s Building of the Tree

Initial plan:
To compute growth rate (mu) and variance of the return (sigma-squared) from historical stock data

Assumption: m = 1

Using the tree model

We set conditions for our model
u=1+r+a
m=1+r
d=1+r-a

Formulas for the growth rate and the variance (in terms of a)
Arbitrage-free
Looked at N period model

New Plan
Calculations (solving for the growth rate)

Solving for
a
using Variance of the return

More Calculations

We set it equal to sigma squared in order to input the formula into Maple

We have our value for sigma which was computed from the stock data
Sigma allows us to solve for a
Once we have a value for a we can know u,m,d since it is in terms of r and a
Then Maple uses the computed u,m,d values to build the tree
This tree will represent different expected stock prices at a particular time N

Growing the real tree

Once we have built our tree, we can enter a strike price and have Maple compute the values of the call prices for every stock price at time N.

A separate procedure takes those values and and adds them to their corresponding location in the tree.

Before:

[[1,16], [1,4], [1,2]]

After (Strike price 8):

[[1,16,8], [1,4,0], [1,2,0]]

Appending the call prices

Risk-neutral probabilities

Similar to the binomial model we have 2 equations but here we have 3 unknowns.

The purpose of the parameter t
Set p3=t

Interval of t
Conditions of t create appropriate risk-neutral probabilities

Using the t parameter

Formula:

Choose strike prices
Pick a t in the domain
Execute equation
=> [C1,C2,C3,C4,C5]
Then repeat with different t values

Backwards induction (cutting down the tree)

5 initial call prices for each t value
Take those 5 values and perform

End up with one value (y-axis coordinate corresponding to the specific t value)

Least Squares
The repetition of the procedure

Least Squares Error Plot

Reason for using Newton’s Method (approximation for graph)
Newton’s Method

Maple’s application of Newton’s Method
Plug in t to solve for the risk-neutral probabilities
Use probabilities to complete backwards induction for our strike prices
Results in final call prices which are the time-zero values
Use of optimal t

Proof by J.C. Penney

Rebecca Spencer-Strong
Alexandra Van Neste

tutorial.math.lamar.edu
We know we can approximate derivatives
Observations
Symmetry
List of Strike Prices=>[P1,P2,P3,P4,P5]
P3 closest price to current stock
P1 and P5 need to be within certain range of P3
Symmetry around P3
P2 and P4 should be in between their neighbors
AT&T
Results from Maple Procedure
Nike
Results from Maple Procedure
[4.71, 2.11, 1.40, 0.66, 0.05]
Newtons Method (solving f'(t)=0):
Derivative Approximations
Our function:
Our aim is to make these as close to the real values as possible or at least consistent so that it can be used as a predictive tool
[1.57, 0.57, 0.09, 0.02, 0.01]
[4.05, 0.98, 0.63, 0.12, 0.01]
Directions of Further Inquiry
Reducing the list size by not having repeats occur
A method for determining the right "window" when choosing strike prices
Possible different approximation method
with Dr. David Handron
Thank you.
A special thanks to Dr. David Handron
and Dr. Deb Brandon
...(for being awesome!)
Starbucks
Results from Maple Procedure
Questions?
estimated t-value corresponding to the minimum
Issues:
-Volatility of stocks makes it hard to estimate values
-Model would not necessarily be arbitrage-free
Newtons Method (solving f'(t)=0):
Circle and line
Not necessarily arbitrage-free
magoosh.com
Calculated:
Quoted:
Graph of error doesn't have a minimum within the domain of t
Full transcript