call put parity


Kinds of options: Calls and puts are the two simplest forms of option. For this reason they are often referred to as vanilla because of the ubiquity of that flavor. 
There are many, many more kinds of options, some of which will be described and examined later on. Other terms used to describe contracts with some dependence on a more fundamental asset are derivatives or contingent claims.

• Premium: The amount paid for the contract initially. How to find this value is the subject of much of this book. 
• Underlying (asset): The financial instrument on which the option value depends. Stocks, commodities, currencies and indices are going to be denoted by S. The option payoff is defined as some function of the underlying asset at expiry.
• Strike (price) or exercise price: The amount for which the underlying can be bought (call) or sold (put). This will be denoted by K. 
• Maturity time or expiration (date) or expiry (date): Date on which the option can be exercised or date on which the option ceases to exist or give the holder any rights. This will be denoted by T. 
• Intrinsic value: The payoff that would be received if the underlying is at its current level when the option expires. 
•Time value: Any value that the option has above its intrinsic value. The uncertainty surrounding the future value of the underlying asset means that the option value is generally different from the intrinsic value. 
• In the money: An option with positive intrinsic value. A call option when the asset price is above the strike, a put option when the asset price is below the strike. 
• Out of the money: An option with no intrinsic value, only time value. A call option when the asset price is below the strike, a put option when the asset price is above the strike. 
• At the money: A call or put with a strike that is close to the current asset level. 

Effect of parameters on option pricing

If the underlying increases the price,

Variable             Call                 Put

S0                         +                   –

K                           –                    +

T                          ?                     ?

Volatitlity              +                    +

r                           +                     –

Speculation example:

On March 14 2018 the price of OurCompany Inc. stock was 90.38 euros, and the cost of a 100 call option with maturity September 21 2018 was 5 euros .

Assume we expected this price to rise between then and September 21 2018. 

One strategy was to buy 1 asset. The final profit of this strategy was (106.8-90.38)*100/90.38=18.17% of the initial investment.

Another possibility was to buy a call option. The final profit of this strategy would have been (106.8-100-5)*100/5=36%. 

Speculation and gearing This is an example of gearing or leverage. The out-of-themoney option has a high gearing, a possible high payoff for a small investment.

The downside of this leverage is that the call option is more likely than not to expire completely worthless and you will lose all of your investment. If OurCompany Inc. remains at 90.38 euros, then the stock investment has the same value but the call option experiences a 100% loss. 

Call-put parity

Consider a call option with strike K and time to maturity T. Consider a put option (on the same underlying) with the same K and T. 
Imagine that we buy the call option and you write the put option The payoff of the corresponding portfolio is given by: max(ST -K,0)-max(K-ST ,0)=ST -K the portfolio of a long call and a short put gives me exactly the same payoff as a long asset, short cash position. The equality of these cashflows is independent of the future behavior of the stock.
The price of a call is some quantity c. The price of a put is some quantity that we denote p. The price of an asset is S. 
On the other hand, to lock in a payment of E at time T involves a cash flow of Ke−r(T−t) at time t. 
Then, the previous equality implies that: c-p=S-Ke−r(T−t). 
1. Assume that S0 = 100. Suppose that a European call option with K = 100 and T = 1/12 is priced at 3.062, while the a European put option with the same K and maturity is priced at 2.652. The volatility is equal to 2%. What is the corresponding (continuously compounded) interest rate? This can be deduced from the call-put parity. 3.062 − 2.652 = 100 − 100e −r/12 Then r = 4.9%.
2. Assume that you want to construct a strategy with a payoff equal to ST . This strategy can include a call, a put and a risk-free investment. How can you construct this strategy? What is the price of this strategy? Buy one at-the-money call and sell one at-the-money put and invest a quantity equal to S0e −rT at risk-free rate. The price of this strategy has to be S0.