9A. Other Binomial Trees
Essential Elements of Lattice Definition
In order to construct a binomial lattice, we must define 3 elements: an up-price multiplier u, a down-price
multiplier d and a probability p that the price of the asset (stock in our curren
9. Binomial Lattice Pricing Method
One-Period Binomial Tree
Consider a 1-year call on AAPL with S = 105.99 last January. At that time, AAPL = 1.70% and the risk-free
rate was r = 0.17%. Exchange traded 1-year American options were available a year ago at
11. Market Maker P/L, -Hedging, -Hedging
Option Return (P/L) and Attribution
Let us now update our 01.18.2015 AAPL 105.00-STR call position with 02.20.2015 data.
The current stock price S has increased substantially to 129.49 from 105.99. This increase in
13. Exotic Options
Assumptions Regarding Exercise
All of the options that we have studied so far (European and American style, calls and puts) have conformed
to a set of standard assumptions regarding exercise. We generally refer to these options as stand
14. Lognormality
Normally Distributed Random Variables
As we have discussed previously, if X is a normally distributed random variable with X N (, 2 ), then the
normal density function (x; , ) describes the probability that X will take on a particular val
10. Higher Order Greeks
As discussed previously, the Black-Scholes framework contains several assumptions regarding the stability of the
pricing inputs. We assume that asset prices St with 0 t T will evolve stochastically and distribute themselves
in the
12. Black-Scholes PDE and Finite Differences Models
Option Value Prior to Expiration
Suppose a European call option has reached its expiration date. Using our AAPL 105.00-STR call from January
2015, if we look at the value of the option across a range of
15. Monte Carlo Simulation Model
Option Price as a Discounted Expected Value
To value an option or any other claim on future cash flows, we first establish an expectation regarding the
size and timing of those cash flows. The expectation may be easy to de
12A. Black-Scholes PDE, Heat Equation and Black-Scholes Formula
(Math 7590)
Black-Scholes Equation, Terminal Condition, Boundary Conditions for a European Call
The purpose of this section is to demonstrate that the Black-Scholes Formula is the closed form
17. Black-Scholes Equation Revisitied
Differential Equations for Claims on Assets with Uncertain Future Payoffs (Black-Scholes)
In section 11, we used a specific market maker portfolio to derive the Black-Scholes equation. Here, we
generalize that example
Math 4590/7590 SP2016
Homework 3 Due 05.13.2016
Template: Math 4590_7590_SP2016_HW3_Template.xlsx
Model Summary
1. Begin with your completed Homework 2. Save this as Homework 3.
2. Add worksheets MC European, MC Asian, BS Asian Geo, Jump Workar
dy / dt = f[t, y] = 3 y + t ^ 2
y[a] = 1
Clear[f, t, y, , a, b]
f[t_, y_] = 3 y + t ^ 2;
= 1;
a = 0;
b = 10;
solveit = DSolve[cfw_y [t] = f[t, y[t], y[a] = , y[t], t]
y[t_] = y[t] /. solveit1
Plot[y[t], cfw_t, a, b]
Set:write : Tag Times in
dy
dt
is Prot
In[8]:=
Solve[cfw_a + b + c + d + e 0, - a + c + 2 d + 3 e 1, a + c + 4 d + 9 e 0,
- a + c + 8 d + 27 e 0, a + c + 16 d + 81 e 0, cfw_a, b, c, d, e]
Out[8]=
In[5]:=
a -
1
4
,b-
5
6
,c
3
2
,d-
1
2
,e
1
12
list = cfw_x0 - h, f[x0 - h], cfw_x0, f[x0],
cfw_x0