# Forwards and Options

• Forward contract: A contract between a buyer and a seller that outlines the quantity, type of the asset or commodity a seller needs to deliver, the delivery logistics, price etc. It is an obligation for the seller to sell and buyer to buy.
1. Expiration date: Time at which contract settles.
2. Underlying asset: Asset / commodity on which the contract is based.
• Stock Index: Average price of a group of stocks. Ex: S&P 500, DJ 3
1. Spot price: Price of the index that particular day.
• Payoff to a contract: Value of the position at expiration
1. Payoff to a long forward: $S_{T} – K$ [Spot price – Forward price]
2. Payoff to a short forward: $K – S_{T}$ [Forward price – Spot price]
3. Payoff graph does not detail how much money was invested for the contracts. It only depicts the position at a certain point in time.
• Funded and Unfunded:
1. A position that has been paid in full is termed as funded.
2. A position for which the payment is deferred is termed as unfunded.
• Profit = Payoff – FV(Investments in the position)
• Bonds shift payoff diagrams vertically but do not change the profit calculations

### Put and Call options

1. Strike Price: Price that the buyer pays for the asset. AKA Exercise price.
2. Expiration: Date when the contract expires.
3. Exercise style:
• American-style option: Right to exercise at any point during the life of the option.
• Bermuda-style option: Right to exercise only at specific periods.
• European-style option: Right to exercise only at expiration.
• Call option: Contract where the buyer has the right but not the obligation to buy.
1. Purchased Call  [Long Call]:
• Payoff = $max [0, S_{T}-K]$                  [$S_{T}$ = Spot price, $K$ = Strike Price]
• Profit at Expiration = $Payoff – FV[P_{C}]$        [$P_{C}$ = Call premium]
• A Purchased call and a Long forward contract are both ways to buy an asset(index).
• If Index rises, forward contract is more profitable since it does not entail paying a premium.
• If Index falls sufficiently, Purchased call option is more profitable since the most the option purchaser loses is the FV of the premium.
• A Purchased call option is thus an Insured Long Forward position.
2. Call option writer  [Short Call]:
• Payoff = -$max [0, S_{T}-K]$                [$S_{T}$ = Spot price, $K$ = Strike Price]
• Profit at Expiration = $Payoff + FV[P_{C}]$         [$P_{C}$ = Call premium]
• Receives the premium for the option from the Call buyer and thus has an obligation to sell the underlying asset in exchange for the strike price should the buyer choose to exercise his option.

• Put option: Contract where seller has the right to sell and not the obligation.
1. Buyer of the put = Seller of index.
2. Purchased Put  [Long Put]:
• Right to sell the index for the strike price
• Payoff = $max [0, K-S_{T}]$                  [ $K$ = Strike Price, $S_{T}$ = Spot price]
• Profit at Expiration = $Payoff – FV[P_{P}]$        [$P_{P}$ = Put premium]
• A Purchased put and a Short forward contract are both ways to sell an asset(index).
• If Index rises sufficiently, put outperforms the short forward
• If index goes down, short forward, which has no premium, has a higher profit than the purchased put.
• A Purchased Put option is thus an Insured Short Forward position.

3. Put option writer  [Short Put]:
• Payoff = -$max [0, K-S_{T}]$                [$S_{T}$ = Spot price, $K$ = Strike Price]
• Profit at Expiration = $Payoff + FV[P_{P}]$         [$P_ {P}$ = Put premium]
• Receives the premium for the option from the Put buyer [Potential seller of the asset] and thus has an obligation to buy the underlying asset in exchange for the strike price should the seller of the asset [Put option buyer] choose to exercise his option.

“Moneyness” of an option

Determines if the option payoff would be positive if the option were to be exercised immediately.

• In-the-money: Positive payoff [But not necessarily positive profit!]
• At-the-money: Strike price approximately equals asset price.
• Out-of-the-money: Negative payoff

Long and Short

Positions long with respect to the index:

1. Long Forward   -> Obligation to buy at a fixed price
2. Purchased call [Long Call] -> Right to buy at a fixed price
3. Written put [Short Put] -> Obligation of the put writer to buy the asset at a fixed price if the option buyer decides to sell

The above three positions benefit from an index price rise.

Positions short with respect to the index:

1. Short Forward   -> Obligation to sell at a fixed price
2. Purchased put [Long Put] -> Right to sell at a fixed price
3. Written call [Short Call] -> Obligation of the call writer to sell the asset if the option buyer decides to sell

The above three positions benefit from an index price fall.

Options as Insurance

• A call option is insurance for an asset we plan to own in the future. Hence it is insurance for a short position.
• A put option is insurance for an asset we already own. Hence it is insurance for long position.