FINANCIAL ENGINEERING - WEEK 4
ARBITRAGE-FREE PRICING IN ONE-PERIOD
FINITE MODELS
MUZAFFER AKAT
The simplest financial models involving random evolution of prices are those in which there
are only two trading times and the prices of the basic securities a
FINANCIAL ENGINEERING - WEEK 3
FORWARD CONTRACTS AND PUT-CALL PARITY
MUZAFFER AKAT
Recall that a forward contract is an agreement between two parties made at some time
concerning the sale of an asset at a future time T , called the delivery time, deliver
FERM 505
Dec 26, 2016
Financial Engineering I
Fall 2016
60 minutes
Sample Midterm - Solutions
Name:
Write your answers clearly in the spaces provided. You may use the back of a page for
additional space; please indicate clearly when you do so. Show all wo
FERM 505
Dec 26, 2016
Financial Engineering I
Fall 2016
60 minutes
Sample Midterm
Name:
Write your answers clearly in the spaces provided. You may use the back of a page for
additional space; please indicate clearly when you do so. Show all work. No credi
FERM 505 - Assignment 0 - Solutions
1. Let T > 0 be given. Let C denote a European call option (on a stock S) with exercise
date T and strike price Kc = $50.00. Let P denote a European put option on the same
stock with exercise date T and strike price Kp
FERM505 - Assignment 1
Due on April 2nd.
1. Consider a simple financial model with two times, t = 0 and t = 1, and a single stock
S that pays no dividends. There is also a bank with one-period interest rate r = 20%.
The stock price at t = 1 will be either
FERM505 - Homework 4
1. Consider a one-period binomial model with r = .1, u = 1.2, d = .9 and S0 = 50.
(a) Find the risk-neutral probabilities p and q
(b) Find the initial price V0 and the number of shares of stock 0 in the replicating
portfolio for a sta
FERM 505 - Homework 2
All interest rates should be interpreted as annualized rates.
1. A 3-year coupon bond was issued two years ago today (i.e. the bond matures one year
from today. The face value of the bond is F = $1, 000 and the bond pays coupons twic
FERM 505
Dec 26, 2016
Financial Engineering I
Fall 2016
60 minutes
Midterm
Name:
Write your answers clearly in the spaces provided. You may use the back of a page for
additional space; please indicate clearly when you do so. Show all work. No credit will
FERM505 - Homework 1 - Solutions
1. If the stock goes up to $40 at t = 1,then the value of the put option to the holder is
P1 = $0. If the stock goes down to $25 then the value of the option at t = 1 is P1 = $5.
Let us replicate the long position. Suppose
FERM 505
Dec 26, 2016
Financial Engineering I
Fall 2016
60 minutes
Midterm - Answer Key
Name:
Write your answers clearly in the spaces provided. You may use the back of a page for
additional space; please indicate clearly when you do so. Show all work. No
FERM505 - Homework 3
Muzaffer Akat
1. Keller Zabel is offering a new financial instrument called a happy call. It has a payoff
function at time T equal to max(.5ST , ST K), where ST is the stock price at T and
K is a fixed strike price.
Suppose that the c
FERM505 - Homework 2 - Solutions
All interest rates should be interpreted as annualized rates.
1. (a) Each coupon payment is C = F q[m]
= 1, 000 .08
= 40 dollars. Arbitrage-free
m
2
price of the bond is the total present value of the of the future payment
FERM505 - Homework 4 - Solutions
1. (a) The risk neutral probabilities are p =
1+rd
ud
=
1+.1.9
1.2.9
= 23 . So q = 1 p = 13 .
(b) Let X = (X0 , 0 ) be the strategy that replicates the option V . Then by the
pricing equation
1
E(V1 )
1+r
1
p(uS0 K)+ + q(
FERM 505 - Homework 3 - Solutions
1. Notice that we can rewrite the final payoff of the happy call as
max(.5ST , ST K) = .5ST + max(0, .5ST K)
= .5ST + .5 max(0, ST 2K)
It is clear that we can replicate the happy call with maturity T and stike K with
half
FERM505 - Assignment 0
Muzaffer Akat
You do not have to turn in.
1. Let T > 0 be given. Let C denote a European call option (on a stock S) with exercise
date T and strike price Kc = $50.00. Let P denote a European put option on the same
stock with exercis
Before Black schole
Bachelier doctoral dissertation in 1900 contains the formula
S- K
S- K
K- S
C ( S , T ) = SN
- KN
+s T n
s T
s T
s T
Sprenkle (1961) has
C ( S , T ) =e rT SN ( d1 ) - (1 - A) KN ( d 2 )
where \rho is the average growth rate and
Chapter2
MechanicsofFutures
Markets
Options, Futures, and Other Derivatives, 8th
Edition,
1
FuturesContracts
Available on a wide range of assets
Exchange traded
Specifications need to be defined:
Where it can be delivered, &
What can be delivered,
When it
Chapter 14
The Black-Scholes-Merton
Model
1
The Stock Price Assumption
Consider a stock whose price is S
In a short period of time of length t, the
return on the stock is normally distributed:
S
t , 2 t
S
where is expected return and is
volatility
2
The
Chapter 18
The Greek Letters
1
Example
A bank has sold for $300,000 a European call
option on 100,000 shares of a non-dividend
paying stock
S0 = 49, K = 50, r = 5%, = 20%,
T = 20 weeks, = 13%
The Black-Scholes-Merton value of the option is
$240,000
How do
FERM531 - Homework 5 Solutions
1. A The buyer will not have a net prot unless the net stock price
exceeds $34. The other statements are true. At $30 the option will be
exercised, but the writer will only lose money in a net sense when the
stock price exce
Chapter 11
Trading Strategies Involving
Options
1
Strategies to be Considered
Bond plus option to create principal
protected note
Stock plus option
Two or more options of the same type (a
spread)
Two or more options of different types (a
combination)
2
Pr
FERM531 - Homework 4 Solutions
1. C A put option is out of the money if S > X and in the money if
S < X. The other statements are true.
2. C American and European options both give the holder the right to
exercise the option at expiration. An American opt
FERM531 - Homework 5
You do not have to turn it in.
1. A call option sells for $4 on a $25 stock with a strike price of $30.
Which of the following statements is least accurate?
(a) At expiration, the buyer of the call will not make a prot unless
the stoc