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)
(v)
(vi)
The price of a stock is 40.
The stock pays n
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
r
Sharpe ratio =
Greek for portfolio = sum of the Gree
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 and subtract S h
delta-hedger
Boyle-Emanuel periodic va
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 Brownian motion, then S (t )/S (0) is lognormal. However
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 option, the formula becomes
V = S0 e T N (d 1 ) c S0 e
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 gentle introduction to delta hedging. I recommend reading it
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 determine the discounting rate for
an option if risk-ne
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 here about calls is equally true about puts, equation (9.4
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 annual continuously compounded expected rate of return is
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 =
=
= 1.1111
1p
0.5
30Q L = 50
With a little additional wor
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 total rate of return
on an asset, t , S (t ) its volatil
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 the Black-Scholes equation if you wish, since I dont s
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 expressed as
d X ( t ) = t , X ( t ) d t + t , X ( t ) dZ ( t
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 the price movement of the underlying asset, and
2. A way
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 alternative method is the empirical method. In the context of mo
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 of the section
This lesson combines two disparate topi
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-Scholes formula is a closed-form formula for evaluating the v
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. Evaluating the goodness of the lognormal models t.
8.1
Estimat
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 the introduction, page xi, for a discussion about how yo
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 maturity value 1000 is 336. The short term continuousl
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 N (d 1 )
where
d1 =
ln(F /K ) + 0.52 T
T
d2 = d1 T
and
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) > 25.
23.2. Calculate the expected value of
23.3.
5
0
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 = S (0)a e [a (r )+0.5a (a 1)
2
]T
(22.2)
P
F0,T S a = e
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) The continuous dividend rate of the stock is 0.02.
(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) = 60.
Under the risk-neutral process, the 50th percentile
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 stock pays no dividends.
(iii) The price of the stock fo
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
(18.1)
Ornstein-Uhlenbeck Process
d X ( t ) = X ( t )
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 has discrete dividends, the volatility of the stock is
de
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 motion
16.1. Which of the following are required for a sto
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 Exercises Below
X = X + E[ Y ] Y
Var(X ) = Var(X ) + Var(Y )