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An Introduction to Derivative Securities
Introduction
As the name is meant to imply, derivative securities are financial instruments that derive their value from another “underlying” asset.
In this sense they can be seen as “side bets” between two investors as to what will happen to the value of the underlying asset. The characterization of them as a bet implies pure
speculation on the part of the investors. The main use of derivatives is actually in hedging risk.
Being side bets, they are in “zero net supply.”
Options
Our concentration here will be on option contracts, in particular call option contracts.
Option contracts are financial contracts that give their owner the right (not the obligation) to buy (call) or sell (put) the underlying asset (commonly a stock or a bond) on (European) or before (American) a specific date (expiration date) for a fixed price (exercise price).
Call Options
A call option gives the owner the right to buy an asset at a fixed price on or before a given date.
Definitions Call price Stock price Exercise price Time to expiration American vs European In/out/at the money
Long Position in a Call
Suppose that a particular call option can be exercised 6 months from now at the exercise price of $20. What will be the value of a long position in the call at expiration if: The call is in the money – e.g. ST = $40?
It is out of the money – e.g. ST = $15? Under what condition would we say the call
is at the money?
Put Options
Ownership of a put option on an underlying asset provides the right to sell that asset for a fixed price on or before the expiration date.
Suppose a particular put option can be exercised one year from now at the exercise price of $15. What is the value of the put at expiration if: The put is in the money – e.g. ST = $8? It is out of the money – e.g. ST = $35?
Short Positions in Options
For every buyer there is a seller. An investor who writes a call on common stock
promises to deliver shares of that stock if required to do so by the option holder. The seller is obligated to do so. What are the possible payoffs at expiration?
An investor who writes a put agrees to purchase shares of that stock if the put holder should so request (exercise). What are the possible payoffs at expiration?
Combinations of Options
“Betting on volatility.” What if you don’t disagree with the market
on the current price but you think it is more volatile than other investors. Can you take a position that will provide positive returns if you are right?
Can you bet against volatility? There exist a myriad of other possibilities,
let’s look at a particular relationship of great interest.
Combinations – Cont…
Example A: Buy a put and buy a share of the underlying stock. (E = $30) What is the value of this position at expiration?
ST = $20 ST = $30 ST = $70
put value
share value
portfolio value
Combinations – Cont…
Example B: Buy a call. (E = $30) What is the value of this position at expiration?
ST = $20 ST = $30 ST = $70
call value
Combinations – cont…
Look at examples A and B. What is the difference between them?
How does the difference between them change as the stock price changes?
A ST = $20 ST = $30 ST = $70
call value (B) $0 $0 $40
portfolio A value $30 $30 $70
Difference (A – B)
Put-Call Parity
A portfolio long a put with exercise price E and expiration T and long the underlying stock has exactly the same payoff across all possible states as a portfolio long a call with exercise E and expiration T and long a zero coupon bond with face value E and maturity T.
This means their current prices must be equal. Why?
C + PV(E) = P + S
Example
Consider two European options, both have exercise price $25 and expire in one year, and both are written on AIM Inc. stock. One is a call and one a put.
AIM stock price is currently $24 and in one year will be either $38 or $14.
Strategy 1: Buy the call and a bond with face value $25 maturing in 1 year (r = 10%)
Strategy 2: Buy a put and buy a share of AIM stock.
What are the possible payoffs in one year?
Example – cont…
Strategy 1 ST = $14 ST = $38
Call
Bond
Portfolio
Strategy 2 ST = $14 ST = $38
Put
Stock
Portfolio
A Slightly Different Example
Suppose you buy the call today, invest in the bond, sell the put and short the stock, how much would you pay for that portfolio?
ST = $14 ST = $38
Call
Bond
Short Put
Short Stock
Portfolio
A Final Example
Suppose you desperately want to buy a put option on AIM Inc. but there is no one who wants to write a put? Is there a way you can satisfy your cravings?
Binomial Option Pricing Model
Can we price a one year call on AIM Inc. stock? Use the same approach as in deriving the put-
call parity relation. If we can find a portfolio of AIM stock and a
bond that mimics the payoff on the call we can price the call. (Assume rf = 10%.)
That portfolio and the call must have the same price. Why?
We can then price the portfolio since we know the current price of the stock and the bond.
Binomial Model
Recall the payoff on the call is $0 if the stock price goes down to $14 and is $13 if the stock price rises to $38. This is a change of $13 from “bad” to “good”
outcome. One share of stock however has a change of
$24 across outcomes. What if we buy 13/24ths of a share? The payoff on this position is $7.58 if the stock
price goes down and $20.58 it goes up. The position costs $13.
Binomial Model
Notice that the value of our position now changes by $13 for an up versus a down move in stock price.
The only problem is that the payoff does not match what the call payoff.
This is easily corrected however if we could subtract $7.58 from each outcome on our position in the stock.
We can do that by borrowing so we have to payback $7.58 at the expiration of the call.
Binomial Model
A portfolio that is long 13/24ths of a share of stock and borrows $6.89 ($7.58/(1.1)) has a payoff of $0 ($7.58 - $7.58) if the stock price falls to $14 and a payoff of $13 ($20.58 - $7.58) if the stock price rises. This perfectly mimics the call.
Thus the cost of the portfolio must be the same as the call price.
C = $13 (13/24$24) – $6.89 = $6.11
