1. .
Option pricing

2. Options Pricing
Presented by
Rajesh Kumar
Sr. Lecturer (Fin.)- Satya College of Engg. &
Tech., Palwal.
Visiting Faculty (Project finance)- STC, Oriental Bank
of Commerce, Sec. 18, Noida.
Visiting Faculty (Derivatives)- IIBS, Noida Branch.
{Scholar Student of YMCA University, MBA-Fin.
(CMS, Jamia Millia Islamia), BA (Eco.
Hons.), AMFI, NCFM (A Grade), A Certified Trainer

3. Derivatives
Derivatives: A derivative is a standardized financial
contract whose value is determined from the
underlying assets like security, share price Index,
exchange rate, commodity, oil price, etc.
Major components :
• Futures
• Options
• Forex
• Swaps
• Commodity, etc.

4. Options
Options: An option is a legal contract which gives the
holder the right to buy (or not) or sell (or not) a
specified amount of underlying asset at a fixed price
within a specified period of time.
Types of options:
Call options: A call option is a legal contract which gives
the holder the right to buy or not to buy an asset for a
predetermined price on or before a specified date.
Put options: A put option is a legal contract which gives
the holder the right to sell or not to sell an asset for a
predetermined price on or before a specified date.

5. Options
Situations for Call options and Put options:
(a) A Call option may face three situations:
1. In-The-Money (ITM), when S0 ˃ E, or
2. Out-of-The-Money (OTM), when S0 ˃ E, or
3. At-The-Money (ATM), when So = E
(b) A put option may face three situations:
1. In-The-Money (ITM), when S0 ˃ E, or
2. Out-of-The-Money (OTM), when S0 ˃ E, or
3. At-The-Money (ATM), when So = E

6. Option Pricing
The price (value) or the premium of an option is
made up of two components: (IV & TV)
1. Intrinsic value (or parity value) : The IV refers to
the amount by which it is in money if it is In-The-
Money. An option which is OTM or ATM has zero
Intrinsic Value.
Intrinsic value of a call option: Max (0, S0 - E)
{Excess of Stock price (S0) over the Exercise price(E)}
Intrinsic value of a put option: Max (0, E - S0)
{Excess of Exercise price (E) over Stock price (S0)}

7. Option Pricing
2. Time Value (TV): The TV of an option is the
difference between the premium of the option
and the intrinsic value of the option.
Time value of a call option: C – {Max (0, S0 – E)}
Time value of a put option: P – {Max (0, E – S0)}

8. Option Pricing
Hence,
P = IV + T V
Where,
P = price or premium of an option,
IV = Intrinsic value and
TV = Time value.

9. Option Pricing
Factors affecting the price of an option:
• Exercise price (E)
• Time to expiry (t)
• Volatility (σ)
• Interest rate (r)
• Dividend

10. Black & Scholes (B&S) Model:
Propounded by Fisher Black and Myron Scholes
in 1973.
Uses of the B&S Model:
• Valuation of Call & Put options
• Valuation of Index
• Hedging etc.

11. B&S Model:
(1) C = S0 × N(d1) – Ee-ert × N(d2)
(2) P = C + Ee-rt – S0
Or,
P = E-rt × N(-d2) – S0 × N(-d1)
Where,
d1 = [ln(S0/E) + t(r + 0.5σ2)] ÷ σ √ t
d2 = [ln(S0/E) + t(r – 0.5σ2)] ÷ σ √ t
Or,
d2 = d1 - σ √ t

12. B&S Model:
(1) C = S0 × N(d1) – Ee-ert × N(d2)
(2) P = C + Ee-rt – S0
Where,
C = Current value of Call option,
P = Current value of Put option,
r = Continuous compounded risk-free rate of
interest,
S0 = Current price of the stock,
E = Exercise price of the option

13. B&S Model:
(1) C = S0 × N(d1) – Ee-ert × N(d2)
(2) P = C + Ee-rt – S0
Where,
t = Time remaining before the expiration date
(expressed as a fraction of a year),
σ = Standard deviation of the continuously
compounded annual rate of return,
ln(S0/E) = Natural logarithm of (S0/E),
N(d) = Value of the cumulative normal distribution.

14. B&S Model:
Assumptions:
(1) The option being valued is a European style
option, with no possibility of an early exercise.
(2) There are no transaction costs and there are no
taxes.
(3) The risk-free interest rate is known and constant
over the life of the option.
(4) The distribution of the possible share prices (or
index levels) at the end of a period of time is log
normal or, in other words, a share’s continuously
compounded rate of return follows a log normal
distribution.

15. B&S Model:
(5) The underlying security pays no dividends
during the life of the option.
(6) The market is an efficient one.
(7) The volatility of the underlying instrument
(Standard Deviation, σ) is known and is constant
over the life of the option.

16. B&S Model:
Required inputs:
(1) Current price of the stock (S0)
(2) Exercise price of the option (E)
(3) Time remaining before expiration of the option
(t)
(4) Risk free rate of interest (r)
(5) Standard deviation of the continuously
compounded annual rate of return (σ)

17. B&S Model:
Steps involve for the calculation of standard
deviation (σ):
Step1. calculate the price relative for each week.
Step2. Find natural logarithms of each of the PR.
Step3. calculate the standard deviation (σ).
Step4. convert the weekly standard deviation to
a yearly standard deviation by multiplying it
by the √52.

18. .
Q.No.1. consider the following information with
regard to a call and put option on the stock of ABC
Company:
Current price of the share, S0 = Rs 120
Exercise price of the option, E = Rs 115
Time period to expiration = 3 months.
Standard deviation of the distribution of
continuously compounded rates of return, σ = 0.6
Continuously compounded risk free interest rate, r =
10% pa.
With these inputs, calculate the value of the call and
put option using the Black and Scholes formula.

19. Adjustment for Dividend
Step 1. calculate the present value of Dividend D0 = De-rt
Step 2. adjust the present value of dividend (D0) with
Current market price of the stock (S0) i.e.,
Price of stock (After dividend) = S0 – D0
Step 3. take this value in B&S Model in the place of S0
Q.No. 2. Reconsider the Q.NO.1. suppose a dividend of
Rs. 2.50 will de received from the underlying share 40
days from today. Take all other inputs from Q.NO.1.
and calculate the value of call and put options.

20. .
‘Thanks
&
Have a Challenging Day
Today’