OPTION AND OPTION PRICING
Kelvin Harrison Nzowa
CPA(T), Bcom Finance, MFA-OG (UDSM)
Sunday, January 23, 2022
Options Basics and Option pricing
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Basic concept of option
Terminologies used in describing options
Uses of option
Types of options
Call and Put options and their payoffs
How options are traded and settled
Option valuation and pricing
Implied volatility
Index options
Sunday, January 23,
2022
Options Basics and Option pricing
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This topic is aimed
understanding of:
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at
providing
an
Meaning and types of options
Intrinsic value and time value of an option
Put and call options – payoff
Basic concepts of option
Pricing models
Sunday, January 23, 2022
Options Basics and Option pricing
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
Option is
the financial derivative that
represent a contract sold by one party(option
writer) to another party(option holder).

The contract offers the right but not the
obligation, to buy(call) or sell(put) a security
or another financial asset at an agreed- upon
price during a certain period of time.
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Options Basics and Option pricing
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Strike price: Is the predetermined price at the
time of buying/writing of an option at which it
can be exercised.
Strike Date/Maturity Date: Is the valid
date/time when the option can be exercised.
American option; may be exercised anytime
up to the expiration date inclusive.
European option: may be exercised only on
the expiration date.
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Options Basics and Option pricing
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Call option: Gives a right but no obligation to buy the underlying asset
at predetermined price within specified interval of time.
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Put option: Gives a right but no obligation to sell the underlying asset
at pre-determined price within a specified interval of time.
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Option Buyer/Holder: Is the person who obtain the right but no
obligation to buy or sell the option. The right is obtained at expense of
premium.
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Writer or seller: one who confers the right and undertake the
obligation to the holder.
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Premium: while conferring a right to a holder, who is under no
obligation to perform, the writer is entitled to charge a fee upfront.
This upfront amount is called premium, This is paid by the holder to
the writer of an option.
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Options Basics and Option pricing
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Source of additional income for the portfolio of
an investor with a large portfolio of securities.
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Leverage: ability to control more securities than
could be done with realistic margin requirement.
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Change the risk complexion of a portfolio of
securities. A manager can undertake the quantity
of risk that he feels appropriate at a point of time,
also they give flexibility on deciding the amount
of risk one is willing to assume.
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Options Basics and Option pricing
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Call option
Gives a right but not obligation to buy an
underlying asset at a pre-determined price
within the specified time.
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Put option
Gives a right but not obligation to sell an
underlying asset at a predetermined price within
the specified time .
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Sunday, January 23, 2022
Options Basics and Option pricing
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Call option contract is profitable to exercise if
the spot price of the underlying asset has gone
beyond the strike price.
Put option contract is worth executing if the
strike price is greater than the price of
underlying asset.
Options are zero-sum game; the gain of the
holder is the loss of the writer and vice versa
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Options Basics and Option pricing
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When the underlying stock moves up in price call do well
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Options Basics and Option pricing
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When the underlying stock drops in price put do well
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Options Basics and Option pricing
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Depending upon the pay off of the options, we have
the following moneyness conditions:
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At-The-Money will always result in no cash flow
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At any time In-The-Money options are those which
if exercised would result in positive cash flow to the
holder.
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Similarly Out-Of-Money options would result in
cash outflow if exercised.
Sunday, January 23, 2022
Options Basics and Option pricing
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For Call options:
Spot price = strike price (At The Money)
Spot price > strike price (In The Money)
Spot price < strike price (Out Of Money)
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For Put Options:
Spot price = strike price (At The Money)
Strike price > Spot price (In The Money)
Strike price < spot price (Out Of Money)
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Options Basics and Option pricing
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The profit potential of puts and call
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Market price behaviour of the underlying
equity is most important variable that
derive any significant moves in the price
of the option and which in turn determine
the option profit potential.
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2022
Options Basics and Option pricing
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The valuation of the option i.e. the premium
payable by the holder is dependent upon
several factors; viewed in different ways.
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Therefore the price of an option has two
components called the intrinsic value and time
value.
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2022
Options Basics and Option pricing
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Intrinsic value is the money the holder would get
upon exercise.
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The fundamental value of put or call depend
ultimately on exercise price and prevailing
market price of the underlying asset.
V of a Call=(MP-SPC) x 100
Where:
V= Fundamental value
MP= Market price
SPC= Strike price on call option
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Options Basics and Option pricing
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V of a put = (SPP-MP) X 100
Where:
SPP is strike price of a put option
MP is the market price
NOTE:
The put has value so long as market price of underlying asset is
less than the strike price stipulated on the put
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Options Basics and Option pricing
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The value attached to the chances that strike price will
be priced in times to come before expiry of the option
contract.
TV of an Option = Actual price- intrinsic value
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Time value is the excess of actual value over intrincsic
value, it can be at least zero but not negative.
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A rational investor would pay this value over its
current exercise value based on the probability it will
incre in value before expiry.
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Options Basics and Option pricing
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The following are some key variables that affects option
price:
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Current stock price
Exercise price
Volatility
Interest rates
Cash dividends
Time to expiration
Sunday, January 23, 2022
Options Basics and Option pricing
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As the current stock price goes up, the higher
the probability that the call will be in the
money.
As a result, the call price will increase. The
effect will be in the opposite direction for a
put.
As the stock price goes up, there is a lower
probability that the put will be in the money.
So the put price will decrease.
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Options Basics and Option pricing
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The higher the exercise price, the lower the
probability that the call will be in the money.
 The call prices will decrease as the exercise
prices increase.
 For the put, the effect runs in the opposite
direction.
 A higher exercise price means that there is higher
probability that the put will be in the money.
 So the put price increases as the exercise price
increases.
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Sunday, January 23, 2022
Options Basics and Option pricing
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Both the call and put will increase in price as
the underlying asset becomes more volatile.
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The buyer of the option receives full benefit of
favourable outcomes but avoids the
unfavourable ones (option price value has zero
value).
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Options Basics and Option pricing
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The higher the interest rate, the lower the
present value of the exercise price.
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As a result, the value of the call will increase.
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The decrease in the present value of the
exercise price will adversely affect the price of
the put option.
Sunday, January 23, 2022
Options Basics and Option pricing
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On ex-dividend dates, the stock price will fall
by the amount of the dividend.
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Therefore, the higher the dividends, the lower
the value of a call relative to the stock.
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As more dividends are paid out, the stock
price will jump down on the ex-date which is
exactly what you are looking for with a put.
Sunday, January 23, 2022
Options Basics and Option pricing
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Logically, chances of achieving the desired outcome
improve if there is more time available in hand.
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Farther the expiration date, more valuable is the option,
because the chances of approaching the strike price
become higher.
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If the time available is less, then the option is less
valuable.
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Both Put and Call options benefits from long time of
expiration.
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Options Basics and Option pricing
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FOLLOWING IS THE SUMMARY OF THE VARIOUS FACTORS THAT
DETERMINE THE OPTION PRICE
VARIABLE
(Increasing)
CALL OPTION
PUT OPTION
Price of underlying asset
INCREASE
DECREASE
Strike price
DECREASE
INCREASE
Time for Expiration
INCREASE
INCREASE
Volatility
INCREASE
INCREASE
Risk-free Rate
INCREASE
DECREASE
Dividends
DECREASE
INCREASE
Sunday, January 23, 2022
Options Basics and Option pricing
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There exists a relationship in the prices of call
and put known by the name of put call parity,
which is derived on the principle of no
arbitrage.
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For the same underlying asset, exercise price
and time to expiry, the call price would exceed
the put price by differential of spot price and
the present value of exercise price.
Sunday, January 23, 2022
Options Basics and Option pricing
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Let derive an important relationship between p
and c with consideration of the following
portfolio;
Portfolio A: one European call option plus an
amount of cash equal to Xe-rT
Portfolio C: one European put plus one share
Both are worth max(ST, K) at expiration of
options.
Sunday, January 23, 2022
Options Basics and Option pricing
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Because options are European, portfolios therefore
must have identical values today.
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The above relationship is known as put -call parity
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This means that, the value of European call with a
certain X and T can be deduced from the value of
European put with the same X and T and vice versa
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Options Basics and Option pricing
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Suppose that the stock price is $31, the exercise
price is $30, the risk-free interest rate is 10% per
annum, the 3-month European call option is $3,
and the price of a 3-month European put option
is $2.25. in this case,
Sunday, January 23, 2022
Options Basics and Option pricing
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Portfolio C is overpriced relative to
portfolio A, this means there exist
arbitrage opportunities.
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The correct arbitrage strategy is to buy
securities in portfolio A and short
securities in portfolio C.
Sunday, January 23, 2022
Options Basics and Option pricing
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American options provide for early exercise of
the option.
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It is never optimal to exercise the American
call prior to maturity because one loses the
time value.
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Options Basics and Option pricing
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In case a stock pays dividend of “q” then
put call parity would stand modified with
the spot price adjusted for present value of
dividend. Since the stock price falls by the
value of dividend on the ex-dividend date
it is analogous to say that current price of
the stock be adjusted for the present value
of the dividend.
Sunday, January 23, 2022
Options Basics and Option pricing
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In case options on currencies the
underlying asset would be foreign
currency. The underlying asset, the foreign
currency in question is like a dividend
paying asset that yields risk-free interest
rate in foreign currency.
Sunday, January 23, 2022
Options Basics and Option pricing
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In this presentation we will discuss two
models responsible for option pricing.
 The determination of option value has
raised concern to many scholars.
 Currently we have several models which
are used in the pricing of options.
 Most of them are restricted to some
factors in order to bear the pricing logic.
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Sunday, January 23, 2022
Options Basics and Option pricing
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• The binomial option pricing model is an option
valuation method developed by Cox in 1979.It is
very simple model that uses iterative procedure
to price option ,allowing the specification of
nodes, or point in time during the time span
between the valuation date and option expiration
date.
• Is a numerical method that estimates the value of
option. It is a very flexible model to value options
that are not regularly traded or are specifically
traded/negotiated between contracting parties.
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Options Basics and Option pricing
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• The theoretical fair value is then compared to
actual price and Level whether the option is
underpriced, overpriced or correctly priced. It
based on the use of more computation
procedure.
• It is called binomial because it assume that
during the most period of time share price will
go to only one of two values.
Sunday, January 23, 2022
Options Basics and Option pricing
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Assume that the CRDB stock sells at So = 300.
We also assume an investment horizon of one
year and at the end of investment period S1
the stock price can either be 25% up i.e. TZS
375 or 20% down i.e. TZS 240.
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Both prices cannot move in both direction
instantly and thus we cant assume both
movements.
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Options Basics and Option pricing
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at s1
s1 = 375
so = 300
s1 = 240
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Options Basics and Option pricing
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Let now attempt to value ATM European call
with strike price X of TZS 300 and time to
maturity of one year.
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Remember that the maximum value accepted is
(s-x) and 0.
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So if stock price moves to 375 the payoff would
be 375-300 = 75 and if the price of the stock goes
down to 240 the call expires worthless.
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Options Basics and Option pricing
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One naïve approach to value the call is to find the
expected value of stock and find the expected
value of the call at t = 1.
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The value of the call is then discounted at t= 0
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Expected value of the underlying stock is
obtained through probabilities of both branches
of the binomial model.
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Remember: if p(upward movement) = p then
p(downward movement) = 1-p.
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Options Basics and Option pricing
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Results from the above method will differ
depending on whether the investor is an optimist,
a realist and a pessimist.
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Optimist investor will increase value on the
probability of upward movement.
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A realist will place equal weight on either
movement.
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A pessimistic investor believes in downturn and
hence probability for downturn is higher.
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Options Basics and Option pricing
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Expected value of stock (at t=1)
0.9×375+0.1×240= TZS 361.5
lower probability for
downward movement
Higher probability for optimists
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Options Basics and Option pricing
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Expected value of the call (at t = 1)
361.5 -300 =TZS 61.5
Exercise price
Expected value of the stock
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Options Basics and Option pricing
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value of the call today = 61.5/1.08=TZS 56.94
NOTE:
 Any arbitrary selection of the discount rate
would result to arbitrary pricing of the call
option.
 For the investor above we have assumed the
probabilities are fairly certain and appropriate
rate of discount at risk free rate of 8%
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Options Basics and Option pricing
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 There are only two possible values, one
up (upside move) and another down
(downside move) from the current price.
 The underlying asset does not pay any
dividends.
 All returns arrive through capital gains.
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Options Basics and Option pricing
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The rate of interest(r) is constant through out
the life of the option.
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Investor are risk neutral i.e. investors are
indifferent toward risk.
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Market are frictionless that mean there are no
taxes and no transaction cost.
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Options Basics and Option pricing
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Binomial option pricing model is useful for
valuing America option in which the option
owner has right to exercise the option ant
time up till expiration.
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Binomial option modes is useful for pricing
Bermudan option which can be exercised at
various point during the life of the option.
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Options Basics and Option pricing
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One major limitation of the binomial
option pricing model is it is slow speed
and complex.
Complexity of the computation is
increased twofold in multi-period binomial
option pricing model.
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Options Basics and Option pricing
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 The model was developed by Myron Scholes
and Fischer Black in 1973 by using complex
mathematics and differential calculus.
 It is analytical model for valuing options on
non-dividends paying stock. The theory is
now called Black -Scholes Model (BSM).
Sunday, January 23, 2022
Options Basics and Option pricing
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While pricing options, the intrinsic value is
always apparent, but estimating the time value of
the option poses problem.
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The time value of the money reflects the
possibility of it becoming ITM before maturity
date, as well as the extent to which it can be ITM
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The Black-Scholes model is the single important
development in the theories for option pricing
and it forms the backbone of modern option
pricing.
Sunday, January 23, 2022
Options Basics and Option pricing
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Black – Scholes Model of option price for European
Call on non- dividend paying stock is given by
C = S0N(d1) – Xe-rt N(d2)
c- is a call premium
s0- spot price of underlying asset
x- exercise price
r- annual risk free interest rate expressed as decimal
t- time remaining for expiration in years
N(d1) and N(d2) are cumulative normal distribution functions at
d1 and d2 respectively
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Options Basics and Option pricing
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The values of d1 and d2 are given by
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Options Basics and Option pricing
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Example: Use the Black Scholes model to price
the CRDB stock price of TZS 125.94, an
exercise price of TZS 125 and time to
expiration of 0.0959 and risk free rate of 4.56
and volatility of 0.83.
In calculating the value of this call by using the
BSM we go through five steps;
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Options Basics and Option pricing
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So = 125.94, X = 125, rc = 0.0446, σ = 0.83, T= 0.0959
Step one: compute d1
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Options Basics and Option pricing
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Step two: Compute d2
d2 = 0.1743 -0.83√0.0959
= - 0.0827
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Options Basics and Option pricing
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Step 3: Look up N(d1)
N(0.17) = 0.5675
Step 4: Look up N(d2)
N(-0.08) = 1- N(0.08) = 1 – 0.5319 = 0.4681
Step 5: Plug into formula for C
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Options Basics and Option pricing
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Options Basics and Option pricing
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 There
are no transaction cost and no taxes
 The risk from interest rate is constant
 The market operate continuously
 The share prices are continuous; there are no jump
in share prices.
 The share pay no dividends
 The option is of European type ,that is option that
can be exercised only at maturity
 Share can be sold short without penalty and short
seller receive the full proceeds from the transaction
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Options Basics and Option pricing
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The valuation of European put can be found by
inserting the Black- Scholes call price into the
put call parity model below;
P=-PsN-d1 + [Px / ert ]N-d2
Example:
Assume the stock price is TZS 40, the put exercise price is TZS
40, expiration date is 4-m, the continuous risk-free interest rate is
12% and the standard deviation of continuous compound stock
return is 30% per year .
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Options Basics and Option pricing
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Solution:
D1 = [ln(40/40) + 0.12 (1/3 + 0.09/2)]/ 0.3√1/3
D1 = 0.26
Nd1= 0.3974
D2=0.26 – 0.3√1/3 =0.09
Nd2 =0.4641
Therefore Pp is TZS 1.94
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Options Basics and Option pricing
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