Assignment 8 (Nov. 25, 2015)
1. Q: Which of the following can be estimated for an American option by constructing a single
binomial tree: delta, gamma, vega, theta, rho?
A: Delta, gamma, and theta can be determined from a single binomial tree. Vega is
det

Homework
1)
Assignment 1
Sep. 09, 2015
Answer the following questions:
a) Explain carefully the difference between hedging, speculation, and arbitrage.
b) What is the difference between the over-the-counter (OTC) market and the exchange-traded
market?

Technical Note No. 17*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
The Process for the Short Rate in an HJM Term Structure Model
This note considers the relationship between the HJM model in Chapter 31 and the
models of the short rat

Technical Note No. 16*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
Construction of an Interest Rate Tree with
Non-Constant Time Steps and Non-Constant Parameters
Consider a one-factor model of the form
df (r) = [(t) a(t)f (r)] dt + (

Technical Note No. 15*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
Valuing Options on Coupon-Bearing Bonds in a One-Factor Interest Rate Model
Jamshidian shows that the prices of options on coupon-bearing bonds can be obtained
from t

Technical Note No. 14*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
The HullWhite Two Factor Model
As explained in Section 30.3 Hull and White have proposed a model where the riskneutral process for the short rate, r, is
df (r) = [(t)

Technical Note No. 12*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
The Calculation of the Cumulative Non-Central Chi Square Distribution
We present an algorithm proposed by Ding (1992).1 Suppose that the non-centrality
parameter is v

Technical Note No. 10*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
The CornishFisher expansion to estimate VaR
As shown in equation (21.7) of the book, i s and ij s can be dened so that P for
a portfolio containing options is approxi

Technical Note No. 9*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
Generalized Tree Building Procedure
This note describes a general procedure for constructing a trinomial tree for a variable,
x, in the situation where
1. There are no

Technical Note No. 7*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
Dierential Equation for Price of a Derivative
on a Futures Price
Suppose that the futures price F follows the process
dF = F dt + F dz
(1)
where dz is a Wiener process

Technical Note No. 5*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
Calculation of Cumulative Probability in Bivariate Normal Distribution
Dene M (a, b; ) as the cumulative probability in a standardized bivariate normal
distribution th

Technical Note No. 4*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
Exact Procedure for Valuing
American Calls on Dividend-Paying Stocks
The Roll, Geske, and Whaley formula for the value of an American call option on a
stock paying a s

Technical Note No. 3*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
Warrant Valuation When Value of Equity Plus Warrants Is Lognormal
This note describes how warrants (or other stock options) can be valued if there is
a single warrant

Homework
Assignment 3
Sep. 16, 2015
1) It is July 30, 2009. The cheapest-to-deliver bond in a September 2009 Treasury bond
futures contract is a 13% coupon bond, and delivery is expected to be made on September 30,
2009. Coupon payments on the bond are

Homework
Assignment 4
Sep. 18, 2015
1) A European call option and put option on a stock both have a strike price of $20 and an
expiration date in three months. Both sell for $3. The risk-free interest rate is 10% per annum,
the current stock price is $

Homework
Assignment 5
Sep. 23, 2015
1) A company's cash position, measured in millions of dollars, follows a generalized Wiener
process with a drift rate of 0.5 per quarter and a variance rate of 4.0 per quarter. How high
does the company's initial cas

Assignment 7 (Oct. 30, 2015)
1. Q: Calculate the delta of an at-the-money six-month European call option on a
non-dividend-paying stock when the risk-free interest rate is 10% per annum and the stock
price volatility is 25% per annum.
A: In this case S0 =

Assignment 6 (Sep. 30, 2015)
1. Q: Show that the Black-Scholes formulas for call and put options satisfy put-call parity.
A: From the Black-Scholes equations
p+S0=Ke-rtN(-d2)-S0N(-d1)+S0
Because 1-N(-d1)=N(d1) this is
Ke-rtN(-d2)+S0N(d1)
Also:
c+Ke-rT=S0N

Assignment 5 (Sep. 23, 2015)
1.
Q: A company's cash position, measured in millions of dollars, follows a generalized Wiener
process with a drift rate of 0.5 per quarter and a variance rate of 4.0 per quarter. How high does
the company's initial cash posit

Assignment 4 (Sep. 18, 2015)
1.
Q: A European call option and put option on a stock both have a strike price of $20 and an
expiration date in three months. Both sell for $3. The risk-free interest rate is 10% per annum, the
current stock price is $19, and

Assignment 3 (Sep 16, 2015)
1.
Q: It is July 30, 2009. The cheapest-to-deliver bond in a September 2009 Treasury
bond futures contract is a 13% coupon bond, and delivery is expected to be made on
September 30, 2009. Coupon payments on the bond are made on

Assignment 2 (Sep.11, 2015)
1
Q: A company has a $20 million portfolio with a beta of 1.2. It would like to use futures contracts
on the S&P 500 to hedge its risk. The index future is currently standing at 1080, and each contract
is for delivery of $250 t

Assignment 1 (Sep. 9, 2015)
1.
a)
Q: Explain carefully the difference between hedging, speculation, and arbitrage.
A: A trader is hedging when he has an exposure to the price of an asset and takes a position in a
derivative to offset the exposure. In a sp

Homework
Assignment 9
Dec 16, 2015
1. The spread between the yield on a three-year corporate bond and the yield on a similar
risk-free bond is 50 basis points. The recovery rate is 30%.
a. Estimate the average default intensity per year over the three-

Homework
Assignment 8
Nov. 25, 2015
1) Which of the following can be estimated for an American option by constructing a single
binomial tree: delta, gamma, vega, theta, rho?
2) A one-year American put option on a non-dividend-paying stock has an exerci

Homework
Assignment 7
Oct. 30, 2015
1) Calculate the delta of an at-the-money six-month European call option on a
non-dividend-paying stock when the risk-free interest rate is 10% per annum and the stock
price volatility is 25% per annum.
2) A fund man

Homework
Assignment 6
Sep. 30, 2015
1) Show that the Black-Scholes formulas for call and put options satisfy put-call parity.
2) Consider an option on a non-dividend-paying stock when the stock price is $30, the
exercise price is $29, the risk-free int

Homework
1)
Assignment 2
Sep. 11, 2015
A company has a $20 million portfolio with a beta of 1.2. It would like to use futures
contracts on the S&P 500 to hedge its risk. The index future is currently standing at 1080, and
each contract is for delivery

Technical Note No. 2*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
Properties of Lognormal Distribution
A variable V has a lognormal distribution if X = ln(V ) has a normal distribution.
Suppose that X is (m, s2 ); that is, it has a n

Technical Note No. 1*
Options, Futures, and Other Derivatives, Eighth Edition
John Hull
Convexity Adjustments to Eurodollar Futures
In the Ho-Lee model the risk-neutral process for the short rate in teh traditional
risk-neutral world is
dr = (t)dt + dz
wh