BMGT 444 Project 1 Fall 2014
Dilip B. Madan
Robert H. Smith School of Business
October 12 2014
The rst project consists of the following 4 questions.
1. Let f (0; t1 ); f (0; t2 ) be forward prices for an asset with no intermediate
income and delivery dat
Solution to Quiz 1
1. The price of a dollar at t, B(t) is using simple interest rate is
B(t) =
1
1 + is t
The continuously compounded rate r has the price at
B(t) = exp( rt):
Hence we must have
exp( rt) =
1
1 + is t
or
rt =
or
r=
ln (1 + is t)
ln(1 + is t
Pricing American Options
Dilip Madan
Department of Finance
Robert H. Smith School of Business
American Call Options
We have seen as a consequence of call lower bounds
that in the absence of dividends there is no early
exercise for an American call option.
Binomial Option Pricing Model
Dilip Madan
Department of Finance
Robert H. Smith School of Business
Call Price on One Period
Tree
Consider the following one period tree for the stock
127:12
100
85:21
The details for the tree construction dier from what
we
Martingale Probability
Dilip Madan
Department of Finance
Robert H. Smith School of Business
Martingales and No
Arbitrage
Suppose we develop a system for simultaneously quoting on the numerous prices of nancial contracts, for
example options of all strikes
Option Price Equalities
Dilip Madan
Department of Finance
Robert H. Smith School of Business
Put Call Parity
The put call parity result for European options on
stocks with no intermediate dividends is
p(S (t); t :; K; T ) = c(S (t); t; K; T )+KB (t; T ) S
The Stock Price Model
Dilip Madan
Department of Finance
Robert H. Smith School of Business
Modeling Stock Price
Movements
We wish to model the motion of the stock price over
the time interval [0; T ] for some maturity T:
We may partition this time interva
Information Content of Option
Prices
Dilip Madan
Department of Finance
Robert H. Smith School of Business
Model Free Results on
Option Prices
We will go on to build a model about stock price
movements from which we shall build the Black Merton Scholes opt
Arbitrage Pricing of Forward
Contracts
Dilip Madan
Department of Finance
Robert H. Smith School of Business
Price of Forward Stock
Forward contract pricing is based when possible on
replicating the exact contract in other markets.
For long forward stock o
Futures And Options Contracts
Dilip Madan
Department of Finance
Robert H. Smith School of Business
Forward Contracts
A forward contract is an obligation to transact at
a future date T for a price agreed upon today say
t < T:
Forward prices like discount c
Hedging at BMS Implied
Volatilities
Dilip Madan
Department of Finance
Robert H. Smith School of Business
Hedging Considerations
Under the BMS assumptions we have seen when deriving the PDE that one may hedge an option eectively by holding the option delta
Black Merton Scholes Option
Pricing Formula
Dilip Madan
Department of Finance
Robert H. Smith School of Business
Black Merton Scholes
Model
Consider an economy with two assets, a money market account earning a continuously compounded interest rate of r wi
Pricing Variance Swaps
Dilip Madan
Department of Finance
Robert H. Smith School of Business
Contract Description
The contract pays at the end of day T the sum of
0
T
X x2
t 252
P@
t=1 T
1
k2A
where
P = notional principal
!
St
xt = log
St 1
St = Market Pri
Black Merton Scholes Formula
Adjusted for Dividends
Dilip Madan
Department of Finance
Robert H. Smith School of Business
A Fixed Dividend
Suppose we have a xed dividend of d1 at time t1 <
T:
The Black Scholes formula is for an asset with no
intermediate d
1. Consider a forward contract on a 1000 dollar face value, 10% coupon
bond with semiannual coupons maturing in two years with a forward delivery
date of 1:75: The yield curve in simple interest rates is at quarterly maturities
given by the rates :025; :0
10/20/14
The Stock Price Model
Dilip Madan
Department of Finance
Robert H. Smith School of Business
Modeling Stock Price
Movements
we want to model the movements in the stock price
We wish to model the motion of the stock price over
the time interval [0;
10/20/14 continued
Binomial Option Pricing Model
Dilip Madan
Department of Finance
Robert H. Smith School of Business
Call Price on One Period
Tree
Consider the following one period tree for the stock
127:12
100
85:21
The details for the tree construction
Futures And Options Rate
Conventions
Dilip Madan
Department of Finance
Robert H. Smith School of Business
The Discount Curve
The discount curve is the basic object announcing
the time value of money in an economy.
We denote it by B (t; T ) and it is the p
BMGT 444
FUTURES CONTRACTS AND OPTIONS
FALL 2014
Instructor: Prof. Dilip B. Madan, VMH 4409, 405-2127
email: dmadan@rhsmith.umd.edu
Time and Location: Mon: 7.00pm-9.40pm in VMH 1307
O ce Hours:
Monday 11.00-12.00
Tuesday 11.00-12.00
Textbook: Derivative S