11. THE BLACK-SCHOLES FORMULA: APPLICATIONS AND VOLATILITY
252
Exercises
Prot diagrams before maturitycalls
Use the following information for questions 11.1 and 11.2:
You are given:
(i)
(ii)
(iii)
(iv
10.3. WHAT WILL I BE TESTED ON?
225
Table 10.4: Formula Summary for Lesson 10
See Exercises Below
T
call = e
N (d 1 )
put = call e T = e T N (d 1 )
S
=
C
option = stock |
Risk premium for a stock = r
EXERCISES FOR LESSON 12
273
Table 12.1: Formula Summary for Lesson 12
Delta-gamma-theta approximation
1
C (St +h ) = C (St ) + + 2 2 + h
Price movement with no gain or lossSee Exercises Below
to
Add
Lesson 17
Differentials
Reading: Derivatives Markets 20.120.3, Appendix C
17.1
Differentiating
Geometric Brownian motion is a useful model for stock prices. If a stock price S (t ) follows geometric B
14. GAP EXCHANGE, AND OTHER OPTIONS
,
342
The formula for a forward start option can be generalized to a case in which you are offered at time t an
option whose strike price is c St . If it is a call
Lesson 12
Delta Hedging
Reading: Derivatives Markets 13, including Appendix 13.B
There are always questions based on this lesson on the exam, usually more than one.
The textbooks chapter 13 is a gentl
Lesson 5
Risk-Neutral Pricing
Reading: Derivatives Markets 11.2, Appendix 11.A and 11.B
5.1
Pricing with True Probabilities
The title of this lesson is misleading. We will really be discussing how to
Lesson 10
The Black-Scholes Formula: Greeks
Reading: Derivatives Markets 12.3
The Black-Scholes formula, equation (9.3), expresses the value of a call C in terms of six arguments. (Everything said her
7. MODELING STOCK PRICES WITH THE LOGNORMAL DISTRIBUTION
162
Exercises
Stock prices in the lognormal model
7.1. For a nondividend paying stock:
(i) The price of the stock at time t is St .
(ii) The an
5. RISK-NEUTRAL PRICING
122
Thus we can solve for QH and Q L . Subtracting 70 times the rst equation from the second, we get
See Exercises Below
70
1.05
5
7
QL = +
= 0.55556
3 3.15
QL
0.55556
UL =
=
=
Lesson 20
Sharpe Ratio
Reading: Derivatives Markets 20.4
20.1
Calculating and using the Sharpe ratio
The Sharpe ratio measures the return on an asset relative to its volatility. If t , S (t ) is the t
Lesson 19
The Black-Scholes Equation
Reading: Derivatives Markets 21.121.2 (excluding What If the Underlying Asset is Not an Investment Asset
on pages 688690)
Derivation
You may skip the derivation of
Lesson 18
Its Lemma
Reading: Derivatives Markets 20.2, 20.3, 20.6 up to but excluding Multivariate Its Lemma on pages 665
666
An It process is a random process X (t ) whose differential can be express
Lesson 16
Brownian Motion
Reading: Derivatives Markets 20.120.3, Appendix C
We will now study the theoretical background for Black-Scholes pricing. In order to price options, we need
1. A model for th
Lesson 15
Monte Carlo Valuation
Reading: Derivatives Markets 19.119.5
15.1
Introduction
Most derivative instruments cannot be priced with a closed form formula. An alternative is needed.
One alternati
Lesson 11
The Black-Scholes Formula:
Applications and Volatility
Reading: Derivatives Markets 12.412.5; 23.123.2 up to but excluding Exponentially Weighted Moving Average on p. 746 and through the end
Lesson 9
The Black-Scholes Formula
Reading: Derivatives Markets 12.112.2, Appendix 12.A
This lesson is very important. The formula we discuss is used repeatedly throughout the course.
The Black-Schole
Lesson 8
Fitting Stock Prices to a Lognormal
Distribution
Reading: Derivatives Markets 18.5, 18.6
This lesson discusses two topics:
1. Estimating and for a lognormal model for stock prices.
2. Evaluat
Lesson 7
Modeling Stock Prices with the
Lognormal Distribution
Reading: Derivatives Markets 18.118.4
Starting with this lesson, the normal distribution will be used for many of our calculations.
See t
26.5. DELTA-GAMMA APPROXIMATION
?
557
Quiz 26-7 In a Vasicek model with a = 0.5, the price of a 5-year zero-coupon bond with maturity value 1000
is 597 and the price of a 10-year zero-coupon bond with
25. THE BLACK FORMULA FOR BOND OPTIONS
528
Table 25.1: Formula Summary for Lesson 25
Black formula:
C F , P (0, T ), , T = P (0, T ) F N (d 1 ) K N (d 2 )
P F , P (0, T ), , T = P (0, T ) K N (d 2 ) F
23. STOCHASTIC INTEGRATION
494
Exercises
Integration
23.1. You are given
t
t
dZ (s )
0.3 ds +
X (t ) = 22 +
0
0
Determine the expected payoff, as of time 0, of an option that pays 5 at time 2 if X (2)
EXERCISES FOR LESSON 22
475
Table 22.1: Formula Summary for S a
See Exercises Below
Expected value
E S (T )a = S (0)a e [a ()+0.5a (a 1)
2
]T
(22.1)
Forward price and prepaid forward price
F0,T S a =
21. RISK-NEUTRAL PRICING AND PROPORTIONAL PORTFOLIOS
464
Exercises
Risk-neutral process
21.1. The time-t price of a stock is X (t ). X (t ) satises
d ln X (t ) = 0.12 dt + 0.4 dZ (t )
You are given
(i
20. SHARPE RATIO
448
Exercises
Sharpe ratio
20.1. The time-t price of a stock is S (t ). You are given:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
The price of the stock follows geometric Brownian motion.
S (0) = 6
19. THE BLACK-SCHOLES EQUATION
434
Determine the price of the put option using the Black-Scholes equation.
Exercises
19.1. For an option on a stock, you are given
(i) The stock price is 47.
(ii) The s
18. ITS LEMMA
422
Table 18.1: Formula Summary for Its Lemma
Multiplication rules:
d t d t = d t dZ = 0
dZ dZ = dt
dZ dZ = d
See Exercises Below t
Its Lemma
dC (S , t ) = CS dS + 0.5CSS (dS )2 + C t dt
17. DIFFERENTIALS
410
After all, F0,t (S ) = e (r )t S , and after logging this, (r )t becomes an additive term with no variance. SimiP
larly for F0,t (S ) = e t S , t has no variance. If the stock ha
16. BROWNIAN MOTION
400
ANSWER: Were given = 0.15 and = 0.2. By formula (16.3),
Cov S (1), S (3) = 102 e [0.15+0.5(0.2
2
)](4)
2
e 0.2 1 = 100e 0.68 e 0.04 1 = 8.0555
Exercises
Arithmetic Brownian mot
EXERCISES FOR LESSON 15
377
Table 15.1: Formula Summary for Lesson 15
12
i =1 u i
A standard normal random variable may be generated as
For the control variate method,
6, or as N 1 (u i ).
See Exerci