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WARNING: CONTAINS MATHS

A put is a type of forward in which the purchaser (also the buyer, the party whom is long, the party who owns a long position) has the right, but not the obligation, to sell an asset for a predetermined price at a predetermined date in the future. Buying a put is essentially buying a long forward which you have the right (or option) to not execute the contract upon maturity. The agreed upon price is referred to as the strike price which is always priced at the time at which the put is purchased. In essence: a put is a bet that the underlying asset will deccrease in price by the maturity date of the forward, allowing the buyer to purchase the asset at the spot price and then immediately sell at the strike price for a measurable profit.

t = time, generally noted in years, or fractions thereof
X = the strike price, agreed upon at t=0
P = the price of the put, reflecting the premium due to secure the right to cancel the contract
Q = Quantity of asset to be delivered at t=T, agreed to at t=0
St = Price of the asset at t=t
e = The natural number, e
r = The interest rate for the contract period specified at t=0
max = the mathematical function by which the greatest of presented arguments (separated by commas) is selected

Future Position Payoff Profit
Long max(0,(X - St)) * Q { max(0,(X - St)) * Q } - { P * (1 + r ) * Q }
Short - { max(0,(X - St)) * Q } { P * (1 + r ) * Q } - { max(0,(X - St)) * Q }

The table of payoffs and profits illustrates how the buyer, or long position, has the option of executing the put and will do so only if the spot price of the underlying asset is less than the strike price. If this is true, the buyer wins the bet, and the seller (the writer, the owner of the short position, the party whom is short) loses. Options are a zero-sum game. In truth, the wager of a put is somewhat mitigated by the premium reflected in the price of the put, but this only decreases the gain/loss of the two parties and does not affect the zero-sum nature of the transaction.

Graphically...

```P/O (\$)
|
|
|      \     ¦
|       \    ¦
|        \   ¦
|         \  ¦
|          \ ¦
+-----------\¦--------- St
|            ¦
|            ¦
|            ¦
|

THE PAYOFF OF A LONG PUT

∏ (\$)
|
|
|    \       ¦
|     \      ¦
|      \     ¦
|       \    ¦
|        \   ¦
+---------\--¦--------- St
|          \ ¦
|           \¦--------
|            ¦
|            ¦
|

THE PROFIT OF A LONG PUT

P/O (\$)
|
|
|
|
|            ¦
|            ¦
|            ¦
+-----------/¦--------- St
|          / ¦
|         /  ¦
|        /   ¦
|

THE PAYOFF OF A SHORT PUT

∏ (\$)
|
|
|
|
|            ¦__________
|           /¦
|          / ¦
+---------/--¦--------- St
|        /   ¦
|       /    ¦
|      /     ¦
|

THE PROFIT OF A SHORT PUT
```

The final piece to the put puzzle lies in the valuation of the premium to be charged by the writer of the put. Generally the price of the put P will be increased by two factors: the increasing duration of the call, and the decreasing delta between the initial asset price and the agreed upon strike price. There are also premium factors which consider the type of put you intend to purchase. A European style put may only be executed or cancelled on its maturity date, whereas an American style put may be executed on any day between the contract date and maturity date. Since it is more likely that the spot price could exceed the strike price over a range of dates, when compared to one specific date, the premium on an American put must always exceed that of a similarly struck European put. Luckily, there are also two formulae of rather advanced arithmatic which can help solve the valuation of a call. Because puts abide by Put-Call Parity, we are able to solve for the price of the premium P just as soon as we have accounted for a "basis of time" in our calculations.

interest compounding Valuation method
Discrete Binomial Option Pricing
Continuous Black-Scholes

McDonald, Robert L. "Derivatives Markets". 2006. Pearson Education, Inc.