Risk Management

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-

siable risks as

2

ri = 2

i 2

M + 2

"i :

1

Only systematic risk is rewarded since diversiable risk can be eliminated when holding

a diversied 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 dened 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 dened as...