The document discusses pricing of options. It defines what an option is, the two main types of options - calls and puts, and some key terminology used related to options. It then discusses the major factors that affect the pricing of an option, including the spot price of the underlying asset, time until expiration, volatility, distance between the strike price and spot price, and risk-free rate of return. The document also discusses intrinsic and time value that make up an option's premium price. It concludes by covering the Black-Scholes model for pricing options theoretically based on these factors.

1. Pricing of
options
Keyuri Halari (M-15-18)
Nikhil Kadam (M-15-27)
Piyush Patil (M-15-44)

2. What is an option ?
 An option is a contract that gives the
buyer the right, but not the obligation,
to buy or sell an underlying asset at a
specific price on or before a certain
date.
 An option, just like a stock or bond, is
a security.

3. Options are of 2 types:
 A CALL gives the holder the right to buy
an asset at a certain price within a specific
period of time. Calls are similar to having
a long position on a stock. Buyers of calls
hope that the stock will increase
substantially before the option expires.
 A PUT gives the holder the right to sell an
asset at a certain price within a specific
period of time. Puts are very similar to
having a short position on a stock. Buyers
of puts hope that the price of the stock will
fall before the option expires.

4. Terminologies used
 Strike price
 Premium
 In-the-money
 Out-the-money
 At-the-money

5. Pricing of an Option
 The price, or cost, of an option is an amount of money
known as the premium. The buyer pays this premium to
the seller in exchange for the right granted by the
option.
 For example, a buyer might pay a seller for the right to
purchase 100 shares of stock XYZ at a strike price of
Rs. 600 on or before December 22. If the position
becomes profitable, the buyer will decide to exercise
the option; if it does not become profitable, the buyer
will let the option expire worthless. The buyer pays the
premium so that he or she has the "option" or the
choice to exercise or allow the option to expire
worthless.
 Premiums are priced per share.

6. Factors affecting pricing of an
Option
There are 5 major factors that affect the
pricing of an option:
 SPOT PRICE OF UNDERLYING
ASSET
 TIME
 IMPACT OF VOLATILITY
 DISTANCE OF STRIKE PRICE FROM
SPOT PRICE (Market movement or
price movement of an underlying
asset)
 RISK FREE RATE OF RETURN

7. SPOT PRICE
 A spot price is the current price in the
marketplace at which a given asset
such as a security, commodity or
currency can be bought or sold for
immediate delivery. While spot prices
are specific to both time and place, in a
global economy the spot price of most
securities or commodities tends to be
fairly uniform worldwide.

8. TIME
 The longer an option has until
expiration, the greater the chance that
it will end up in-the-money, or
profitable. As expiration approaches,
the option's time value decreases.
 The longer the time until expiration,
the higher the option price
 The shorter the time until expiration,
the lower the option price

9. IMPACT OF VOLATILITY
 Volatility in market also affects the
price of options
◦ If the market is volatile the price will be
high
◦ If the market is less volatile the price will
be low

10. DISTANCE OF STRIKE FROM
SPOT
 The distance of the Strike price from
the spot price will also affect the
option price
◦ For a Call Option If the distance is more of
strike price from the spot price in the
money, the Price increases.
◦ For a Call Option If the distance is more of
strike price from the spot price out of the
money the Price decreases.

11. RISK FREE RATE OF
RETURN
 The risk-free rate of return is the
theoretical rate of return of an investment with
zero risk. The risk-free rate represents the
interest an investor would expect from an
absolutely risk-free investment over a specified
period of time.
 In theory, the risk-free rate is the minimum
return an investor expects for any investment
because he will not accept additional risk
unless the potential rate of return is greater
than the risk-free rate.

12. Valuation of option
 Valuation i.e., the premium payable is
dependent upon several factors. Two
most important components
◦ Intrinsic Value
◦ Time Value

13.  Intrinsic value- The value attached to
the option if it is exercised now is
called the intrinsic value of the
option, the difference between spot
price and exercise price will determine
this value.
 Time Value- Any premium that is in
excess of the options intrinsic value is
referred to as time value of the
option. It is a difference of actual price
and intrinsic value.

14. Boundary Conditions for Call
Option Pricing
 Maximum price of the option cannot
exceed the price of the asset itself.
◦ Price <= Spot price
 Similarly the minimum price the call
option would sell for is the intrinsic
value of the option adjusted for the
present value of the option
◦ Price >= Spot – Xe-rt

15. Boundary Conditions for Put
option pricing
 The Maximum price of the put option
cannot be more than its exercise price
◦ Price <= Strike price * e-rt
 The minimum price that a put option
would sell for is its intrinsic value
◦ Price >= Strike price-rt – Spot price

16. Pricing Models :The Black-
Scholes Model
The Black-Scholes model is used to calculate a theoretical call
price (ignoring dividends paid during the life of the option) using
the five key determinants of an option's price: stock price, strike
price, volatility, time to expiration, and short-term (risk free)
interest rate.
The original formula for calculating the theoretical option price is
as follows:
Call Price= 𝑆𝑁 𝑑1 − 𝑋𝑒−𝑟𝑡
𝑁 (𝑑2)
Where:- 𝑑1 =
ln
𝑆
𝑋
+ 𝑟+
𝜎2
2
𝑡
𝜎√𝑡
𝑑2 =
ln
𝑆
𝑋
+ 𝑟−
𝑣2
2
𝑡
𝜎√𝑡
or 𝑑2 = 𝑑1 − 𝜎√𝑡

17. The variables are:
S = stock price
X = strike price
t = time remaining until expiration, expressed as a
percent of a year
r = current continuously compounded risk-free
interest rate
𝜎 = annual volatility of stock price (the standard
deviation of the short-term returns over one year).
ln = natural logarithm
N(x) = standard normal cumulative distribution
function
e = the exponential function

18. Advantages and Limitations
Advantage: The main advantage of the Black-Scholes
model is speed -- it lets you calculate a very large
number of option prices in a very short time.
Limitation: The Black-Scholes model has one major
limitation: it cannot be used to accurately price
options with an American-style exercise as it only
calculates the option price at one point in time

19.  http://www.business-
standard.com/article/markets/sebi-
may-allow-options-in-select-
commodities-116050301004_1.html