Chapter 12
The Black-Scholes Formula
Question 12.1. You can use the NORMSDIST() function of Microsoft Excel to calculate the values for N(d1) and N (d2). NORMSDIST(z) returns the standard normal cumulative distribution evaluated at z. Here are the interme
Chapter 19
Monte Carlo Valuation
Question 19.1.
The histogram should resemble the uniform density, the mean should be close to 0.5, and the
standard deviation should be close to 1/ 12 = 0.2887.
Question 19.2.
The histogram should be similar to a standard
Chapter 17
Real Options
Question 17.1.
The benet of waiting is to receive possibly higher prot, the cost is forgoing interest. We should
wait if the increase in prot outweighs the interest lost on this (one time) higher prot. We use
continuous growth rate
Table of Contents
Chapter 1
Introduction to Derivatives
1
Part One
Insurance, Hedging, and Simple Strategies
Chapter 2
Chapter 3
Chapter 4
An Introduction to Forwards and Options
Insurance, Collars, and Other Strategies
Introduction to Risk Management
43
Chapter 11
Binomial Option Pricing: II
Question 11.1. a) Early exercise occurs only at strike prices of 70 and 80. The value of the one period binomial European 70 strike call is $23.24, while the value of immediate exercise is 100 70 = 30. The value of t
Chapter 10
Binomial Option Pricing: I
Question 10.1.
Using the formulas given in the main text, we calculate the following values:
a)
for the European call option:
b)
for the European put option:
= 0.5
= 0.5
B = 38.4316
B = 62.4513
price = 11.5684
price =
Chapter 6
Commodity Forwards and Futures
Question 6.1.
The spot price of a widget is $70.00. With a continuously compounded annual risk-free rate of 5%,
we can calculate the annualized lease rates according to the formula:
F0,T = S0 e(r l )T
F0,T
S0
ln
=
Chapter 7
Interest Rate Forwards and Futures
Question 7.1.
Using the bond valuation formulas (7.1), (7.3), (7.6) we obtain the following yields and prices:
Maturity
Zero-Coupon Bond
Yield
Zero Coupon
Bond Price
1
2
3
4
5
0.04000
0.04500
0.04500
0.05000
0.
Chapter 8
Swaps
Question 8.1.
We rst solve for the present value of the cost per two barrels:
$22
$23
= 41.033.
+
1.06 (1.065)2
We then obtain the swap price per barrel by solving:
x
x
= 41.033
+
1.06 (1.065)2
x
= 22.483,
which was to be shown.
Question 8
Chapter 14
Exotic Options: I
Question 14.1. The geometric averages for stocks will always be lower. Question 14.2. The arithmetic average is 5 (three 5's, one 4, and one 6) and the geometric average is (5 4 5 6 5)1/5 = 4.9593. For the next sequence, the a
Chapter 22
Exotic Options: II
Question 22.1.
With a premium of P paid at maturity if ST > K , the COD will have the same value (which will
initially be set to zero) as a regular call minus P cash or nothing call options. That is,
0 = BSCall(S0 , K, , r, T
Chapter 20
Brownian Motion and Its Lemma
Question 20.1.
If y = ln (S) then S = ey and dy =
2
2e2y
a)
dy =
ey
b)
dy =
a
ey
c)
dy =
2
2
dt +
2
2e2y
(S,t)
S
(S,t)2
2S 2
dt +
(S,t)
S dZt ,
ey dZt .
dt +
ey dZt .
d t + dZt .
Question 20.2.
If y = S 2 then
Chapter 24
Interest Rate Models
Question 24.1.
a)
F = P (0, 2) /P (0, 1) = .8495/.9259 = .91749.
b)
Using Blacks Formula,
BSCall (.8495, .9009 .9259, .1, 0, 1, 0) = $0.0418.
c)
(1)
Using put call parity for futures options,
p = c + KP (0, 1) F P (0, 1) =
Chapter 21
The Black-Scholes Equation
Question 21.1.
If V (S, t) = er(T t) then the partial derivatives are VS = VSS = 0 and Vt = rV . Hence Vt +
(r ) SVS + S 2 2 VSS /2 = rV .
Question 21.2.
If V (S, t) = AS a e t then Vt = V , VS = aS a 1 e t = aV /S ,
Chapter 25
Value at Risk
Question 25.1.
Since the price of stock A in h years (Sh ) is log-normal,
1
2 h + hZ < 0
2
=P Z<
h =N
h.
2
2
P (Sh < S0 ) = P
(1)
(2)
Using the parameters and h = 1/365 this is N (.01832) = .4927. If we use h = 1/252 the value
wo
Chapter 18
The Lognormal Distribution
Question 18.1.
The ve standard normals are
+8
15
15
= 1.8074.
7+8
15
= .2582,
+8
11
15
= .7746,
3+8
15
= 1.291,
2+
8
15
= 2.582, and
Question 18.2.
If z is standard normal, + z is N , 2 hence our ve standard normals
Chapter 26
Credit Risk
Question 26.1.
Using formulas in the main text book, we can calculate the true and risk-neutral probability of
bankruptcy to be 0.4256 and 0.5808, respectively. To calculate the credit spread, we need to know
the expected loss given
Chapter 1
Introduction to Derivatives
Question 1.1.
This problem offers different scenarios in which some companies may have an interest to hedge
their exposure to temperatures that are detrimental to their business. In answering the problem, it
is useful
Chapter 3
Insurance, Collars, and Other Strategies
Question 3.1.
This question is a direct application of the Put-Call-Parity (equation (3.1) of the textbook. Mimicking Table 3.1., we have:
S&R Index
900.00
950.00
1000.00
1050.00
1100.00
1150.00
1200.00
S
Chapter 4
Introduction to Risk Management
Question 4.1.
The following table summarizes the unhedged and hedged prot calculations:
Copper price in Total cost
one year
$0.70
$0.90
$0.80
$0.90
$0.90
$0.90
$1.00
$0.90
$1.10
$0.90
$1.20
$0.90
Unhedged prot
$0.
Chapter 2
An Introduction to Forwards and Options
Question 2.1.
The payoff diagram of the stock is just a graph of the stock price as a function of the stock price:
In order to obtain the prot diagram at expiration, we have to nance the initial investment
Source: Datastream
Name
S&P 500 COMPOSITEDOW JONES DIVIDEND YIELDPRICE INDEX
S&P 500 COMPOSITE - INDUSTRIALS - NATIONAL MKT. COMPOSITE - PRICE INDEX
- PRICE INDEX JONES INDUSTRIALS - DIVIDEND YIELD DIV RATE ADJUSTED BUS.MACH.-DIV RATE ADJUSTED DIV $ TO AU
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Appendix 17.A: Calculation of Optimal Time to Drill an Oil Well
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559
$20 and cash ows begin 1 year after the project is started. When should you
invest and what is the value of the option
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Appendix 5.A: Taxes and the Forward Price
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5.19. Suppose the spot $/ exchange rate is 0.008, the 1-year continuously compounded
dollar-denominated rate is 5% and the 1-year continuous
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Appendix 9.B: Algebraic Proofs of Strike-Price Relations
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bound as in the European case with no dividends. The lower bound exists because it
may not be optimal to exercise the call t
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w The Lognormal Distribution
The last equality follows because the integral expression is one: It is the total area under
a normal density with mean + 2 and variance 2 . Thus we obtai
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Appendix 19.A: Formulas for Geometric Average Options
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a. Using 2000 simulations incorporating jumps, simulate the 2-year price
and draw a histogram of continuously compounded return
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Appendix 21.C: Solutions for Prices and Probabilities
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derivatives of this guess and substitute them into equation (21.43). After simplication
(in particular, the Y multiplying every