Quantitative Finance - LUMSsuraj.lums.edu.pk/~adnan.khan/classes/classes/QuantFin/QFLecture… ·...
Transcript of Quantitative Finance - LUMSsuraj.lums.edu.pk/~adnan.khan/classes/classes/QuantFin/QFLecture… ·...
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Quantitative Finance
Lecture 9
Black Scholes Framework
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Covered Call
Long Position on the stock
Short position on Out of the Money Call
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Married Put
Long position on the stock
Long position on At the Money Put
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Straddle
Long position on ATM Call
Long positon on ATM put
Both with same expiry and strike price
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Strangle
Long position in OTM Call
Long position in OTM Put
Different strike price same expiry
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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)
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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
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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
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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
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Bounds on Prices
Proof: Recall
Proof: By Put Call parity
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Bounds on Prices
Theorem:
Combining the previous results we have the bounds
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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
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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
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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
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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