Lecture 9

Question 1

Three desirable economic properties of the stochastic process for stock prices

Answer 1

1. Price process should be consistent with weak form of market efficiency, present stock price impounds all information contained in a record of past prices
2. The price process should also be scale independent
3. Because of limited liability stock prices can never go below zero

Question 2

The Black-Merton-Scholes formula – Notation

r

Answer 2

the continuously compounded risk less interest rate

Question 3

The Black-Merton-Scholes formula – Notation

C

Answer 3

the current value of a European call

Question 4

The Black-Merton-Scholes formula – Notation

S

Answer 4

the current price of the stock; the underlying asset

Question 5

The Black-Merton-Scholes formula – Notation

K

Answer 5

the exercise price of the call

Question 6

The Black-Merton-Scholes formula – Notation

T

Answer 6

the time remaining before the expiration date expressed as a fraction of a year

Question 7

The Black-Merton-Scholes formula – Notation

σ

Answer 7

the standard deviation of the continuously compounded annual rate of return of the stock

Question 8

The Black-Merton-Scholes formula – Notation

ln(S/K)

Answer 8

the natural logarithm of S/K e = 2.7183

Question 9

Assumptions behind the Black-Merton - Scholes Formula

Answer 9

1. The stochastic process for the stock price is lognormal with constant parameters µ and σ.
2. Short selling of securities with full use of proceeds is permitted.
3. There are no transaction costs or taxes. All securities are perfectly divisible.
4. No dividends during the life of the derivative security
5. Security trading is continuous
6. The risk free continuously compounded interest rate is constant and the same for all maturities
7. There are No Risk Free Arbitrage Opportunities.

Question 10

Value of a European Put today

Answer 10

P = K e^(-rt) (1-N(d2)) - S e-yt (1-N(d1))

Question 11

N(d1)

Answer 11

the hedge ratio

the fraction of one share that you invest in in the underlying stock in the replicating portfolio

Question 12

N(d2):

Answer 12

fraction of the present value of the exercise price that you borrow at the risk free interest rate
- Probability that the call will end up in-the-money at date of maturity using the Martingale probability p

Question 13

Delta (∆)

Answer 13

European call: Delta call / Delta Stock price

European put: Delta put/ Delta Stock price

Question 14

Theta (Θ)

Answer 14

European call: Delta call / delta time

European put: Delta put/ Delta time

Question 15

Gamma (Γ)

Answer 15

European call: Delta hedge ratio / delta Stock price

European put: Delta hedge ratio / delta Stock price

Question 16

Lambda (Λ)

Vega

Answer 16

European call: Delta call / delta Sigma

European put: Delta put / delta Sigma

Question 17

Rho (ρ)

Answer 17

European call: Delta call/ delta interest rate

European put: Delta put/ delta interest rate

Question 18

Omega (Ω) (Elasticity of an

option/leverage)

Answer 18

European call: (Stock price * delta call)/ (Call price * delta stock price)
European put: (Stock price * delta put)/ (Put price * delta stock price)

Question 19

Cash-or-nothing call price with payoff $1

Answer 19

C=e^(-rT) N(d2)

Question 20

Asset-or-nothing call price:

Answer 20

C=S*N(d1)

Question 21

Cash-or-nothing put price with payoff $1:

Answer 21

P=e-rTN(-d2)

Question 22

Asset-or-nothing put price:

Answer 22

P=S*N(-d1)

Question 23

Financial option VS Real Option (Notation)

Answer 23

Stock Price = Current Market Value of Asset
Strike Price = UpFront Investment Required
Expiration Date = Final Decision Date
Risk Free Rate = Risk Free Rate
Volatility of Stock = Volatility of Asset Value
Dividends = FCF Lost from Delay

Question 24

Key Insights from Real Options

Answer 24

Out-of-the-money real options have value
In-the-money real options need not be exercised immediately
Waiting is valuable
Delay investment expenses as much as possible
Create value by exploiting real options