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Date Submitted: 10/06/2011 04:34 AM
Chapter 1
Review
1.1 Finance
Calculating Returns
Suppose you observe historical prices, Pt and Pt1.
The simple return is rt = PtPt1
Pt1
. Simple return is used in calculating holding period
return.
The log return or continuously compounded return is rc
t = ln( Pt
Pt1
), and erc
t = 1 + rt.
Question 1: Suppose the log return is 10% a year. What are the 2-year log return and
2-year simple return?
Capital Asset Pricing Model (CAPM)
Suppose that the return of the ith stock is generated by the model
ri = i + irM + "i:
The total risk of the stock i can be decomposed into components of systemic and diver-
siable risks as
2
ri = 2
i 2
M + 2
"i :
1
Only systematic risk is rewarded since diversiable risk can be eliminated when holding
a diversied portfolio, which gives
E(ri) = rf + i(E(rM) rf ):
Alternative explanation of CAPM: investors do not like the assets positively correlated
with the overall market, so they require a higher returns for those assets.
Question 2: Is the expected return of a stock with negative lower or higher than risk-free
rate? Why?
Black-Scholes Model
The payo of a European call cT at maturity date T is
cT = max(ST K; 0);
where ST is the underlying asset price at the maturity T, and K is the strike price.
Suppose the logarithm of the underlying asset return follows normal distribution, under
no-arbitrage condition, the Black-Scholes options pricing formula tells us the relation
between ct and St at any time t before the maturity date is
ct = StN(d1) Kerf (Tt)N(d2);
where
d1 =
lnSt
K + (rf + 1
22)(T t)
p
T t
;
d2 =
lnSt
K + (rf 1
22)(T t)
p
T t
;
and N(x) is dened by
N(x) =
1
p
2
Z x
1
e1
2u2
du:
Question 3: Why does the risk-free rate rather than the expected return of the underlying
stock enter the formula? What is the so-call risk-neutral world?
2
1.2 Maths
1.2.1 Calculus
Dierentiation
For a function f(x), the rst order derivative is dened as...