It define the option pricing model in simple way

1. .
Prepared by:
Vishal Jain
MBA 3rd Sem.
Option Pricing

2. The amount that an investor must
pay for an option contract. This price
is based upon factors such as the
underlying security as well as the
left until the options expiration date.
The twopopular models are:
1. The Binomial Model
2. Black – Scholes Model

3. The model assumes,
 The price of asset can only go up or
go down in fixed amounts in
discrete time.
 There is no arbitrage between the
option and the replicating portfolio
composed of underlying asset and
risk-less asset.

4.  Current stock price = S
 Next Year values = uS or dS
 Bamount can be borrowed at ‘r’.
Interest factor is (1+r) = R
 d< R < u(no risk free arbitrage
possible)
 E is the exerciseprice

5. Depending on the change in stock
value, option value will be
Cu = Max (uS – E, 0)
Cd = Max (dS – E, 0)

6. .
S
Su
Sd
Su2
Sud
Sd2

7. The Binomial Model converges to the
Black- Scholes model as the number
of time periods increases.

8.  1820s – Scottish scientist Robert Brown
observed motion of suspended particles in
water.
 Early 19th century – Albert Einstein used
Brownian motion to explain movements of
molecules, many research papers.
 1900 – French scholar, Louis Bachelier wrote
dissertation on option pricing and developed a
model strikingly similar to BSM.
 1951 – Japanese mathematician Kiyoshi Ito
developed Ito’s Lemma that was used in option
pricing.

9.  Fischer Black and Myron Scholes worked
in Finance Faculty at MIT Published paper
in 1973. They were later joined by Robert
Merton.
 Fischer left academia in 1983, died in
1995 at 57.
 1997 – Scholes and Merton got Nobel
Prize

10. Assumptions:
 The underlying stock pays no dividends.
 It is a European option.
 The stock price is continuous and is
distributed lognormally.
 There are no transaction costs and taxes.
 No restrictions or penalty on short selling
 The risk free rate is known and is
constant over the life of the option.

11. C0= S0N(d1) – E/ert N(d2) where,
C0 = Present equilibrium value of call option
S0 = Current stock price
E = Exercise price
e = Base of natural logarithm
r = Continuously compounded risk free
interest rate
t = length of time in years to expiration
N(*) = Cumulative probability distribution
function of a standardized normal
distribution

12. C = S N(d 1)– Ke-rtN(d2) where,
C = Present equilibrium value of call option
S = Current stock price
K = Exercise price
e = Base of natural logarithm
r = Continuously compounded risk free interest rate
t = length of time in years to expiration
N(*) = Cumulative probability distribution function of a
standardized normal distribution

13. ln(S0/E) + (r + ½σ2)t
d1=
σ √t
ln(S0/E) + (r - ½σ2)t
d2=
σ √t
where lnis the naturallogarithm

14. ln(S/K e- rt)
d1= + 0.5 σ√t
σ√t
d2= d1- σ √t
where lnis the natural logarithm

15. .
Call
Stock
Put
Risk-Free
Bond
Black-Scholes
Call Option
Pricing Model
Put-Call
Parity
Black-Scholes
Put Option
Pricing Model

16.  Find the Standard Deviation of the
continuously compounded asset value change
and the square root of the time leftto
expiration
 Calculate ratio of the current asset value to
the present value of the exercise price
 Consult the table giving %age relationship
between the value of the Call Option and the
stock price corresponding tothe value in
steps 1and 2
 Value of Put Option = Value of Call Option +
PV of exercise price – StockPrice

17.  To price European options on dividend
paying options and American options on
non-dividend paying stocks (Robert
Merton, 1973 and Clifford Smith, 1976).
 American call options on dividend-paying
stocks (Richard Roll, 1977; Robert
Whaley, 1981; and Richard Geske and
Richard Roll, 1984)