From James’s Page

Actuary: Derivative

Derivatives Basics

These are financial instruments whose prices are based on prices of other things.

Setup: the buyer and seller of an underlying asset enter into a contract in which the buyer agrees to buy the underlying asset from the seller at a price (called different things in different kinds of contracts) some time in the future, on the expiration date when the contract is settled.

The three basic kinds of contracts they can enter are

• Forward contracts -- At the expiration date the buyer is obligated to buy the underlying asset at the forward price, and the seller is obligated to sell. There are no premiums, or initial payments.
  • long forward: This is the position of the buyer of the asset. When the actual price of the asset on the experation date (spot price at expiration) is higher than the forward price, the buyer gains. When the spot price is less than the forward price, the buyer loses. The buyer can have unlimited gain (because the asset may grow to have arbitrarily high value), but can only lose at most the forward price (when the spot price is 0). The payoff for a long forward contract is

• short forward: This is the position of the seller of the asset. When the spot price at expiration is lower than the forward price, the seller gains. When the spot price is higher than the forward price, the seller loses. The seller can gain at most the forward price (when spot price is 0), but can lose an unlimited amount (because the seller may have to give up an asset that can have arbitrarily high value). The payoff for a short forward contract is

• Call options -- At the expiration date the buyer has two choices: buy the asset (exercise the call option) at the strike price (or exercise price), or do nothing. So the buyer has the right, but not the obligation, to buy the asset. The seller offsets this disadvantage by charging a premium.
  • purchased call (long call) -- This is the position of the buyer of the asset. When the spot price at expiration is sufficiently higher than the strike price, the buyer gains, but only when the difference is more than the future value of the premium. The purchaser of the call option can have unlimited gain, but can only lose an amount up to the future value of the premium (because if the spot value at expiration is too low, the buyer can choose to do nothing). The payoff of a purchased call option is

The profit of a purchased call option is

• written call (short call) -- This is the position of the seller of the asset. Since the spot price at expiration can be infinitely higher than the strike price, the writer of the call option can have unlimited loss. However, the writer of the call option can only gain an amount up to the future value of the premium (because if the spot value at expiration is too low, the purchaser of the put can choose to do nothing). The payoff of a written call option is

The profit of a written call option is

• Put options-- At the expiration date the seller can exercise the put option at the strike price or do nothing. So the seller has the right, but not the obligation, to buy the asset. The buyer offsets this disadvantage by charging a premium.
  • purchased put (long put) -- This is the position of the seller of the asset. Note that the seller of the aseet purchases the put option (by paying the premium). When the spot price at expiration is sufficiently lower than the strike price, the purchaser of the put option gains from selling the asset, but only when the difference is more than the future value of the premium. The purchaser of the put option can gain at most an amount of the strike price minus the future value of the premium (when the strike price is 0), and can lose an amount up to the future value of the premium (because if the spot value of the asset is too high, the put pruchaser can choose to do nothing and keep the asset). The payoff of a purchased put option is

The profit of a purchased put option is

• written put (short put) -- This is the position of the buyer of the asset. If the spot price is too high, the seller won't sell the asset and the writer of the put gets to keep the premium. If the spot price is sufficiently low, the purchaser of the put will exercise the option and the writer of the put option can lose an amount up to the strike price minus the future value of teh premium. The payoff of a written call option is

The profit of a written call option is