Define an option and its terminology
Look at the properties of options
Discuss why it is difficult to price options
introduce the Black-Scholes model for option pricing
Look at investment strategies using options
Do some numerical examples
What is an Option?
Like a future in many respects;
It is a contract between a buyer and seller to trade an asset at a future date BUT.....
... the purchaser of the contract has the right but not the obligation to proceed with the trade;
The purchaser pays something over to the seller for this right today;
The purchaser can be either buying the asset (a call option) or selling the asset (a put option).
Some terminology
Often uses the same terminology as futures;
The expiration date is the final date of the contract;
The time from now until the expiration date is called the time to expiration or time to maturity;
The strike price is the price of the asset when the contract is exercised;
We’ve already seen call (buy) and put (sell) options;
The purchaser of the option is the option holder;
The seller of the option is the option writer;
The option holder pays the option writer some money today for the option. The holder has the right but not obligation to exercise the trade. The writer must do what the holder wants!
American vs European options
A European option is one where the asset is traded only on the expiration date;
This is more similar to a futures contract;
An American option allows the purchaser to “exercise” the option at any time up to the expiration date;
The distinction is important – the value of these options for the same asset and expiration date may be different!
Trading in options
Begun in 1973 in Chicago (call options only)
Now a very large business;
It is regulated, like futures, through an exchange
Examples of Options (1)
Today, I purchase an American option from you to buy 1 million shares in Microsoft with expiration date July 1st 2007 at $28 per share.
I am the option holder and you are the option writer;
The strike price is $28;
Today I pay you something to purchase the option;
I am purchasing a call (I will be buying the asset);
At any time between today and July 1st 2007 I can come to you to buy 1 million shares at $28 each (and you must sell);
Examples of Options (2)
2. I purchase an American option from you to sell 1 million shares in Microsoft on July 1st 2007 at $28 per share
I am the option holder and you are the option writer;
Today I pay you something to purchase the option;
I am purchasing a put (I will be selling the asset);
At any time between today and July 1st 2007 I can sell you 1 million shares at $28 each (and you must buy);
Examples of Options (3)
3. Today, you purchase a European option from me to buy 1 million shares in Microsoft with expiration date July 1st 2007 at $28 per share.
I am the option writer and you are the option holder;
Today you pay me something to purchase the option;
You are purchasing a call (you will buy the asset);
On July 1st 2007 you decide whether to buy from me 1 million shares at $28 each (and I must sell if you decide to buy);
Options and Futures
They have some similarities:
They are all to do with agreeing a trade at some future date;
They are zero-sum games (ignoring transaction costs) – what one side wins the other loses
They are also different:
A futures contract trade will happen (both sides of contract are equally obliged to honour it)
An options contract trade might happen (and it’s the option holder who decides) – so the two sides are not equal
How much should you pay for an option?
The value of a futures contract is easy – it’s to do with the present value of the asset at settlement day;
For options it’s more tricky:
we’re talking about the value of having the option to trade an asset (something that does not exist with a future);
the trade may or may not happen;
whether it happens depends on the future price which is unknown!
The Basics of Option Pricing
Suppose the strike price of an option is S;
At time t let the price of the asset be A(t);
Question: how much is an option for this worth today (P)?
An option is in the money at time t if the option holder will make money by exercising the option now (so P > 0).
So for a call, in the money ↔ A(t) > S
What about a put?
The Basics of Option Pricing (2)
An option is out of the money at time t if the option holder loses money by exercising the option:
For a call, out of the money ↔ A(t) < S
For a put:
At the money is S = A(t) for calls and puts
The Basics of Option Pricing
The value of an option is the sum of two things:
How much money the holder would make today by exercising the option (the intrinsic value);
The “time value” – this is more tricky – this is the extra amount that the holder may make by waiting to exercise
It could be positive or negative;
It’s different for European and American options
So P = IV + TV
The Basics of Option Pricing – Intrinsic value
If you have to decide to exercise a call today:
What do you do if S < A(t)?
What do you do if S > A(t)?
So the intrinsic value can be written:
IV = max(A(t)-S, 0)
Intrinsic Value of a Put
Let’s do the same for a put:
Time value
Suppose today is day “t”
The settlement date is day “T” (> t)
The intrinsic value of a call [put] is max(A(t) - S, 0) [max(S – A(t), 0)]
What might happen between now and settlement?
The price of the asset might increase [decrease], thus raising the intrinsic value of the call [put]
Of course the price might decrease [increase], thus lowering the intrinsic value of the call [put]
Time value (2)
There is a certain probability of each happening
However (see formulae):
The lower bound to IV is ALWAYS 0
The upper bound to IV is infinite (if a call) and S (if a put)
So the IV can increase much more than it can decrease (usually)
As a result the expected change in value of IV is usually positive
Time value (3)
The time value is this additional expected gain in the intrinsic value between now and settlement date;
Unfortunately, there is no easy formula to work out what it should be
We can say some easy things about it:
For an American option, TV > 0 always
It decreases as you get closer to expiration date;
At expiration date, TV = 0 and P = IV
Time value (4)
There is no easy formula for the price of an option because it is difficult to work out what the time value should be
So, the bad news is:
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