# call put parity

Posted on Jan 14, 2019 in Economy

#### Options

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.

Vocabulary:

• 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).

examples:

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.