Principles of Option Pricing

7/31/2019 Principles of Option Pricing

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Principles of Option

PricingBy: Ajay Mishra

JSSGIW faculty of Management, Bhopal


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Principles of Option Pricing

Arbitrage opportunitiesare quickly eliminated

by investors.(lets see examples)


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Situation (1) Suppose in a game you draw a ball from a box

known to contain three black and three white

balls. If you draw a black ball you receive nothing.

If you draw a white ball you receive Rs. 10.

Will you Play?


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Situation (1) As no entree fee is mentioned , most people

will play.

You incur no cash outlay up front and havethe opportunity to earn Rs. 10.

Of course this opportunity is too good, but no

one will offer you such option withoutcharging any entry fee.

(lets play another one)


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Situation (2) Now suppose that a fair fee to play Game I is

Rs.4

And for a game II the person offers you to payRs. 20 if you draw a white ball and nothing if

you draw a black ball.

Will the entry fee be higher or lower?


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Situation (2) If u draw a black ball you receive the same

payoff as in game I but if you draw a white

ball you receive a higher payoff.

You should be willing to pay more to play

game II because these payoffs dominate thoseof game I.


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From these simple games and opportunities it

is easy to see some basic principles of how

rational people behave when they faced withrisky situations.

The collective behaviour of rational investors

operates in an identical manner to determinethe fundamental principles ofoption pricing.


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Basic Terminology in Option Pricing

The following symbols are used further:

S = Stock price today

X = Exercise Price. T = Time to expiration

r = Risk free rate

S = Stock price at options expiration; after

the passage of a period of time of legth T


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Basic Terminology in Option Pricing C(S,T,X) = Price of a call option in which the

stock price is S, the time to expiration is T,

and the exercise price is X.

P(S,T,X) = Price of a put option in which the

stock price is S the time to expiration is T andthe exercise price is X.


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Basic Terminology in Option Pricing Ca(S,T,X) = American Call

Ce(S,T,X) = European Call

If there is no a or e subscript, the call canbe either American or a European Call.

In the case where two options differ only by

exercise price, the notation C(S,T,X1) andC(S,T,X2) will identify the prices of the calls

with X1 less than X2.


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Basic Terminology in Option Pricing

Always a the subscript of the lower exercise

price is smaller than the higher exercise

price. In the case where two options differ only by

time to expiration will be T1 and T2, where

T1< T2. Identical adjustments will be made for put

option prices.


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Time to expiration

The time to expiration is expressed as a decimal

friction of a year.

For example if the current date is April 9 and theoptions expiration date is July 18.

We count the number of days between these two

dates.

That would be : April= 21, May =31, June = 30 andJuly = 18 . Total 100 days.

The time to expiration would be 100/365 = 0.274


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Dividend

For most of the examples we shall assume

that the stock pays no dividends.

If during the life of the option, the stock paysa dividends of D1, D2.. And so forth, then

we can make a simple adjustment and obtain a

similar adjustment and obtain similar results. To do so we simply subtract the present value

of the dividends.


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Risk free rate of return r

It is the rate earned on a riskless investment.

An example of such an investment is a treasury bill.

T-bills pay interest not through coupons but byselling at a discount.

The T-bill is purchased at less than face value.

The difference between the purchase price and the

face value is called the discount.

The discount is the profit earned by the bill holder.


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The rate of return on a T-bill of comparable

maturity would be a proxy for the risk free of

return. All T-bills mature on Thursday because most

exchanged traded option expire on Fridays.

There is always a T-bill maturing the daybefore expiration.


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Bid and Ask discounts for several T bills for thebusiness day of May 14 of particular year are as

follows.

Maturity Bid Ask

5/20 4.45 4.37

6/17 4.41 4.37

7/15 4.47 4.43

Bid and Ask figures are the discount quoted

by dealers trading in T-bills


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With first T-bill case if we take expiration

date May 21.

To find the T-bill rate we use the average ofthe bid and ask discount. Which is

(4.45+4.37)/2=4.41

Then we find the discount from par value as-4.41(7/360)= 0.08575, using the fact that the

option has seven days until the maturity

Thus the price is 100-0.08575= 99.91425


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The yield on our T-bill is based on theassumption of buying it at 99.91425 and

holding for seven days, at which time it will

be worth of 100.

This is a return of

(100=99.91425)/99.91425=1.000858

Suppose we repeat this transaction everyseven days for a full year, the return would

be: ((1.000858)^365/7)-1 = 0.0457

Which can be taken as risk free rate of return.


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Principles of call option pricing

Minimum Value of a call

Maximum Value of Call

Value of a Call at Expiration.


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Minimum Value of a Call

CALL PUT

ExercisePrice

May June July May June July

120 8.75 15.4 20.9 2.75 9.25 13.65

125 5.75 13.5 18.6 4.6 11.5 16.6

130 3.6 11.35 16.4 7.35 14.25 19.65

The below given is one example option data for a

stock for the date may 14. Try to find out the intinsic

value and time values for the call.

Assuming the current stock price is 125.94 andExpirations : May 21, June 18, July 16


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Minimum value of a Call

A Call is an instrument with limited liability.

If the call holder sees that it is advantageous

to exercise he it, the call will be exercised. If exercising it will decrease the call holders

wealth he will not exercise it.

The option cannot have negative value,because the holder cannot be forced to

exercise it. Therefore, C(S,T,X)>= 0


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For an American Call it will be

Ca(S,T,X)>= Max(0, S-X)

Max(0, S-X) means take the maximum value of twoarguments, zero or S-X

The minimum value of an option is called its

intrinsic value some time referred to as parity value,

parity or exercise value.

Intrinsic value, which is positive for in the money

calls and zero for out of the money calls.


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Intrinsic Values and Time Values of a given Call

TimeValue

Exercise

Price

Intrinsic

Value

May June July

120 5.94 2.81 9.46 14.96

125 0.94 4.81 12.56 17.66

130 0.00 3.60 11.35 16.40


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To check the intrinsic value rule we take the

June 120 Call.

The stock price is 125.94 and the exerciseprice is 120.

Taking Max(0, 125.94-120) = 5.94

Now what would happen if the call werepriced at less than 5.94 , say Rs. 3


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An option trader could buy the call for Rs.3,

Exercise it

You can purchase the stock for Rs. 120 and then sell

the stock for Rs. 125.94. This arbitrage would provide a risk less profit of

Rs.2.94

All investors would do it, which would drive up the

option price When the price of the option reached Rs.5.94, the

transaction would no longer be profitable.


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What if the exercise price exceeds the stock

price ? Do it with exercise price = 130

Max(0, 125.94-130) = 0 Then minimum value be zero

Now check at all the given calls.


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The calls with an exercise price of Rs. 125 have a

minimum values Max(0,125.94-125) = 0.94 and are

priced at no less than 0.94.

The calls with an exercise price of 130 have a

minimum value of Max(0, 125.94-130) = 0

Al those option obviously have nonnegative values


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Minimum Value of a call

The intrinsic value concept applies only to an

American call, because a European call can be

exercised only the expiration day. The price of an American call normally exceeds

its intrinsic value.

The difference between the price and theintrinsic value is called the time value or

speculative value of the call.

Which is defined as Ca(S,T,X)-Max(0,S-X)


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Minimum value of a call

The time value refers what traders are willing

to pay for the uncertainty of the underlying

stock. Time values increase with the time of

expiration


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Minimum value of a European call

(a) European

Call

Price

Stock Price (S)

The call price lies in a shaded area . The European call price lies in the entire area.


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Minimum value of a American call

(b) American

Call

Price

Stock Price (S)

X

Max(0,S-X)

The American call price lies in a smaller area. This does not mean that the American call

price is less than the European call price but only that its range of possible values is narrower.


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Maximum Value of a call

C(S,T,X)S

The most one can expect to gain from the call

is the stocks value less than the exerciseprice. Even if the exercise price were zero,

No one would pay more for the call than for

the stock. However, one call that is worth the stock

price is one with an infinite maturity.


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Minimum and maximum values of a

European call

(a) European

Call

Price

Stock Price (S)


0

S


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Minimum and maximum values of an

American call

(b) American

Call

Price

Stock Price (S)


0 X

Max(0,S-X)


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Value of a Call at Expiration

The prospect of future stock price increases isirrelevant to the price of the expiring option,

which will be simply its intrinsic value.

At expiration an American option and aEuropean option are identical Instruments.

Therefore this rule holds for both the options.

C(St,0,X) = Max (0,St-X


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The value of a Call at Expiration.

CallPrice

Stock Price at Expiration (St)

X

Max(0,St-X)

0

*Because of the transaction cost of exercising the option, it could be worthslightly less than the intrinsic value.

C(ST,0,X)


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Effect of time to Expiration.

Consider two American Calls that differ only

in their time to expiration so their price will

be as follows :

(1) Ca(S,T1X)

(2) Ca(S,T2X) (T2 is greater than T1 )

Now think which of these two option will

have a greater value?


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Effect of time to Expiration.

Suppose that today is the expiration day ofthe shorterlived option. The stock price is

the value of the expiring option is :

Max (0, ST1-X).

The second option has a time to expiration of

T2-T1.

Its minimum value is MAX(0,St1-X). Thus

when the shorter lived option expires, its

value is the minimum value of the longer

lived one.


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References:

Don M. Chance, Derivatives and Risk

Management Basics, India edition.

John C. Hull and Sankarshan Basu, Options,Futures, and Other Derivatives seventh

edition.

http://content.icicidirect.com/learning/university.htm

Principles of Option Pricing

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