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1

Three desirable economic properties of the stochastic process for stock prices

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

2

The Black-Merton-Scholes formula – Notation
r

the continuously compounded risk less interest rate

3

The Black-Merton-Scholes formula – Notation
C

the current value of a European call

4

The Black-Merton-Scholes formula – Notation
S

the current price of the stock; the underlying asset

5

The Black-Merton-Scholes formula – Notation
K

the exercise price of the call

6

The Black-Merton-Scholes formula – Notation
T

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

7

The Black-Merton-Scholes formula – Notation
σ

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

8

The Black-Merton-Scholes formula – Notation
ln(S/K)

the natural logarithm of S/K e = 2.7183

9

Assumptions behind the Black-Merton - Scholes Formula

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.

10

Value of a European Put today

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

11

N(d1)

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

12

N(d2):

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

13

Delta (∆)

European call: Delta call / Delta Stock price
European put: Delta put/ Delta Stock price

14

Theta (Θ)

European call: Delta call / delta time
European put: Delta put/ Delta time

15

Gamma (Γ)

European call: Delta hedge ratio / delta Stock price
European put: Delta hedge ratio / delta Stock price

16

Lambda (Λ)
(Vega)

European call: Delta call / delta Sigma
European put: Delta put / delta Sigma

17

Rho (ρ)

European call: Delta call/ delta interest rate
European put: Delta put/ delta interest rate

18

Omega (Ω) (Elasticity of an
option/leverage)

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

19

Cash-or-nothing call price with payoff $1

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

20

Asset-or-nothing call price:

C=S*N(d1)

21

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

P=e-rTN(-d2)

22

Asset-or-nothing put price:

P=S*N(-d1)

23

Financial option VS Real Option (Notation)

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

24

Key Insights from Real Options

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