Quantitative Finance

Lecture 9

Black Scholes Framework

Covered Call

Long Position on the stock

Short position on Out of the Money Call

Married Put

Long position on the stock

Long position on At the Money Put

Straddle

Long position on ATM Call

Long positon on ATM put

Both with same expiry and strike price

Strangle

Long position in OTM Call

Long position in OTM Put

Different strike price same expiry

Put Call Parity

Consider the strategy of writing & selling a put and buying one call with the same strike X and expiry date T

The payoff will be

For no arbitrage opportunity

Independent of S(T)

Put Call Parity

For a stock that pays no dividends the following holds

Proof:

Suppose

At time t=0

Buy one share for S(0)

Buy one put option for

Write and sell a call option for

Invest @ r

At t=T

Close the money market position

Sell the share for X if using the put option or for S(T) if

, this gives a balance

American Options

The prices of American put and call options with the same strike X and expiry time T on a stock paying no dividend satisfy

Proof: Suppose @ t=0: Write and sell a call

Buy a put & a stock

Finance this transaction on the money market

If the option is exercised @t T, then we get X for the share & settle the money market position, ending with

If the option is not exercised at all the @ time T, sell the share for X & settle the money market position ending with

This creates an arbitrage opportunity

Bounds on Prices

&

Proof: By arbitrage argument

Proof: Suppose @t=0: Write and sell a call

Buy a stock and invest the balance in the money market

@t=T

Settle the option by selling the stock for , getting an arbitrage profit of

Bounds on Prices

Proof: Recall

Proof: By Put Call parity

Bounds on Prices

Theorem:

Combining the previous results we have the bounds

Time Value of Options

At time t the intrinsic value of a call option with strike price X is equal to and the intrinsic value of a put option is

The time value of an option is the difference between the time value of an option and its intrinsic value

EXAMPLEStrike Price Intrinsic Value Time Value Option Price

Call Put Call Put Call Put

110 15.2

3

0 3.17 2.84 18.4 2.84

120 5.23 0 6.46 12.2

7

12.2

7

6.46

130 0 5.23 6.78 6.78 6.78 9.64

Why trade in options

Why would one trade in options?

Depending on one’s market view, whether it would rise or fall or, whether it would change or not change, options can be used to hedge risk

We will examine strategies that make use of these different scenarios

Speculating and Gearing

Consider buying a far out of money option, this would usually not cost much

If it expires worthless you lose the small amount you paid for it

If there is a substantial move in the underlying you gain a large profit relative to the initial amount

This is called gearing or leveraging

Example

Suppose a stock is priced at $666 on 14th April , The cost of a 680 call option with expiry 22nd August is $39. If one’s market view is that the stock price will rise sharply (say to $730 by mid August) consider the following Buy 1 stock at $666, it would yield a return

Buy an option for $39, at expiry exercise the option to receive the stock for 680, the return then is