OPTIONS
The buyer of an option buys the right to buy or sell as asset in the future at a predetermined price.
E.g. you can buy an option that gives you the right to buy shares in the future at a specific price.
Exchange of money takes place at the expiration of the option.
The counterparty is always the exchange through which you buy the option.
There are options on various assets and quantities such as stocks, bonds, currencies, stock market
indices, weather quantities, energy, metals and agricultural products.
CALL OPTIONS
The buyer of a call option has the right (but not the obligation) to buy the underlying asset at a
predetermined price at a specific time in the future.
Example: you buy today a call option on Microsoft shares. The call refers to 100 shares, expires on
the 20th of December, has an exercise (strike) price of $45 and a price of $3,20.
The call refers to 100 shares so the value of the option is 100 x 3,20 = $320. This is how much
money you give to buy the option. This gives you the right to buy on the 20th of December 100
Microsoft shares at $45 each.
Case 1. The price of the share at expiration is $44. You do not exercise your right so you don’t buy
the shares.
Case 2. The price of the share at expiration is $48. You exercise your right and buy the shares. You
give 100 x 45 = $4.500 and buy 100 shares worth 100 x 48 = $4.800. So, you make a profit of
$300.
The seller (writer) of the call option has the obligation to sell the shares at $45 if the buyer
exercises his right. For this reason, the writer of the option receives a premium which in the above
example is $3,20 per share.
PUT OPTIONS
The buyer of a put option has the right (but not the obligation) to sell the underlying asset at a
predetermined price at a specific time in the future.
Example: you buy today a put option on Microsoft shares. The call refers to 100 shares, expires on
the 20th of December, has an exercise price of $45 and a price of $2,80.
The call refers to 100 shares so the value of the option is 100 x 2,80 = $280. This is how much
money you give to buy the option. This gives you the right to sell on the 20th of December 100
Microsoft shares at $45 each.
Case 1. The price of the share at expiration is $44. You exercise your right and sell the shares. You
receive 100 x 45 = $4.500 and sell 100 shares worth 100 x 44 = $4.400. So, you make a profit of
$100.
Case 2. The price of the share at expiration is $48. You do not exercise your right so you don’t sell
the shares.
The seller (writer) of the put option has the obligation to buy the shares at $45 if the buyer
exercises his right. For this reason, the writer of the option receives a premium which in the above
example is $2,80 per share.
Option writers have to deposit collateral in a bank account called margin account.
European Options can be exercised only at expiration.
American options can be exercised at any time until expiration.
OPTION PRICE LIMITS
The down limit of the price of a call option is its profit if it is exercised immediately.
E.g. Share price = €6 and exercise price = €4
The lowest price for the call is 6 – 4 = €2.
Suppose the price was €1,80. You can buy a call and exercise it immediately. You would have a
risk free profit of 6 – 4 – 1,80 = €0,20.
The upper limit for the price of call is the share price. If the price of a call is equal to the
underlying share price you can buy the share and sell the call.
Call value
Upper limit
Down limit
Call price
Share price
Exercise price
OPTION VALUATION – THE BIONOMIAL MODEL
Suppose the price of a share is €10 and you expect that in 6 months the price of the share can be
€13 or €8.
You want to value a 6 month call option with an exercise price of €10. The 6-month risk free rate
is 5% annualized.
Suppose you buy 0,6 shares and you borrow the present value of €4,80. In 6 months, depending on
the price of the share you will have:
Share price
13 euros
Option
€3
Portfolio
(0,6x13)-4,8= €3
So, the call must have the same price as the portfolio.
8 euros
€0
(0,6x8)-4,8= €0
Call = value of 0,6 shares – P.V. loan = (0,6 x 10) – (4,8 / 1,025) = 1,317 euros.
The number of shares you must buy to construct the portfolio is called option delta and is
calculated as:
Range of call value
delta =
Range of share value
3-0
=
13-8
=0,6
EUROPEAN OPTION VALUATION – BLACK & SCHOLES MODEL
Black and Scholes applied the binomial model rationale but for continuous share prices.
The model is:
C0 = P0 N(d1) - E e-rt N(d2)
where:
2
ln( P / E )  (r  
)t
2
0
d1 =
 t
and
2
ln( P / E )  (r  
)t
2
0
d2 =
 t
C0 is the value of the call
P0 is the present share price
Ε is the exercise price
σ is the standard deviation of the share price returns
t is the time to expiration in years
r is the risk free rate
N(d1) and N(d2) are the probabilities to have deviations equal to d1 and d2 under a standardized
normal distribution
The model shows that:
When the share price exceeds the strike price, the value of the call approaches the value of the
share minus the present value of the strike price.
The higher the volatility of the share price, the higher the value of the call.
The longer the time to expiration, the higher the call price.
The risk free rate is positively correlated to the value of the call.
The relationship between strike price and value of the call is inverse.
THE GREEKS
The first derivative of the price of the call with respect to the share price is called delta.
The first derivative of the value of the call with respect to the time to expiration is called theta.
The first derivative of delta with respect to the price of the share is called gamma.
The first derivative of the price of the call with respect to the volatility of the share price is called
vega.
The first derivative of the value of the call with respect to the risk free rate is called rho.
HEDGING
Using options you can ensure that the value of your portfolio will not drop below a certain amount.
Suppose you have €A and you want to build a portfolio with a terminal value at least €F.
To achieve that, you invest €X in the risk free rate and €(A-X) in shares. You buy N shares at a price
of €S each, and put options for N shares with strike price K and price P.
The final value of your portfolio will be:
F = X (1+r)T + K [(A - X) / (S + P)] = X (1+r)T + NK
if the share price at the end of the investment period is less than the strike price, or
F = X (1+r)T + NST
if the value of the share price is higher than the strike price at the end of the investment.
The amount you should invest in the risk free rate is:
X = [F (S + P) – K A] / [(S + P) (1 + r)T - K]
Example
Suppose you have €10.000 and you want to invest it for 3 years. The risk free rate is 9% and the
price of XYZ’s share is €3,50. At the end of the investment period you want to have at least €11.000.
To build the desired portfolio you buy put options on XYZ’s share with a strike price of:
3,50 x (1 + 10%) = €3,85
Suppose the put option price is €0,2166.
You should invest in the risk free rate:
11000(3,50 + 0,2166) - (3,85)(10000)
X=
[(3,50+0,2166)(1,09)3] - 3,85
=€2473,87
So, you must buy N=(A-X)/(S+P) = (10.000 – 2.473,87) / (3,50 + 0,2166) = 2.025 shares of XYZ
and put options for these shares.
Case 1. In 3 years, the price of XYZ is lower than 3,85. You exercise the put options and sell the
shares for €3,85 each.
The value of your portfolio will be:
F = [2.473,87 (1,09)3] + [2.025 x 3,85]= €11.000
Case 2. The price of XYZ in 3 years is higher than €3,85, e.g it is €4,00.
The value of your portfolio will be:
F = [2473,87 (1,09)3] + [2.025 x 4,00]= €11.303,74
So, the minimum value of your portfolio is €11.000 while there is no upper limit.
The same result can be achieved in another way, if we consider that:
S + P = C + K(1 + r)-T
where S is the price of a share, P is the price of a put, C is the price of a call, K is an amount
invested in the risk free rate and r is the risk free rate.
Therefore, buying calls and lending money is the same as buying shares and put options.
In the above example, we can invest 11.000 / 1,093 = €8.494,02 in the risk free rate and use the
remaining 11.000 – 8.494,02 = €2.505,98 to buy call options on the share.
Delta hedging
The delta of a share is equal to N(d1)
Suppose we have sold call options on a share with delta = 0,6368. To hedge our position we must
buy 0,6368 shares for each call we sold.
Delta changes as the price of a share changes so, the hedged position needs constant rebalancing.
Exotic options
Asian options are similar to the European options but their terminal value depends on the average
price of the underlying asset during the life of the option.
In lookback options the exercise price is the lowest (highest) price of the underlying asset during
the life of the call (put) option.
With shout options the buyer of the option can set the strike price equal to the current price of the
underlying asset ant time he wants.
Barrier options can expire early if the price of the underlying asset reaches a certain predetermined
price.