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

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