1. Call Option:
A call option is a financial contract that gives the buyer the right, but not the obligation, to buy a specific asset (such as a stock) at a predetermined price (strike price, X) on or before a specified date (maturity or expiration date). The buyer pays a premium (Co) to the seller for this right.
**Call Option Payoff (Cr)**:
- The payoff for the call option buyer is the amount of money they would gain (or lose) at the expiration of the option contract if they were to exercise it.
- It is calculated by taking the maximum of 0 and the difference between the stock price at maturity (ST) and the strike price (X).
- If the stock price (ST) is higher than the strike price (X), then the option has intrinsic value, and the buyer will profit by exercising it.
- If the stock price (ST) is equal to or lower than the strike price (X), then the option has no intrinsic value, and the buyer will not exercise it, resulting in a zero payoff.
Example:
Let's say the buyer purchased a call option for a specific stock with a strike price (X) of $50. At the expiration date, the stock price (ST) is $60.
Cr = max(0, $60 - $50) = max(0, $10) = $10
The buyer's payoff (Cr) would be $10.
2. **Call Option Profit**:
- The profit for the call option buyer is the actual amount of money the buyer gains or loses after taking into account the premium they paid (Co) to purchase the call option.
- It is calculated by subtracting the call premium (Co) from the call option payoff (Cr).
- If the call option payoff (Cr) is negative (i.e., the stock price at maturity is less than or equal to the strike price), then the buyer's profit will be equal to the negative call premium (Co).
Example (Continuing from the previous example):
Let's assume the buyer paid a premium (Co) of $3 to purchase the call option.
Profit = $10 (Cr) - $3 (Co) = $7
The buyer's profit would be $7.
3. **Payoff for Call Option Seller (-C)**:
- The payoff for the call option seller (writer) is the opposite of the buyer's payoff. It represents the amount the seller would gain (or lose) at the expiration of the option contract.
- It is calculated by taking the negative of the call option buyer's payoff. This is because the seller's gain is the buyer's loss, and vice versa.
- If the stock price at maturity (ST) is higher than the strike price (X), the buyer will exercise the option, resulting in a negative payoff for the seller.
- If the stock price at maturity (ST) is equal to or lower than the strike price (X), the buyer will not exercise the option, and the seller's payoff is zero.
Example (Continuing from the previous example):
Since the stock price (ST) at maturity is $60, which is higher than the strike price (X) of $50:
-C = -max(0, $60 - $50) = -max(0, $10) = -$10
The seller's payoff (-C) would be -$10.
4. **Profit for Call Option Seller**:
- The profit for the call option seller is the amount of money the seller gains or loses after taking into account the premium they received (Co) from selling the call option.
- It is calculated by subtracting the negative payoff (-C) from the call premium (Co) received by the seller.
- If the negative payoff (-C) is negative (i.e., the stock price at maturity is higher than the strike price), then the seller's profit will be equal to the call premium (Co) plus the negative payoff.
Example (Continuing from the previous example):
Assuming the seller received a premium (Co) of $3 for selling the call option:
Profit = $3 (Co) - ($10) (-C) = $3 - $10 = -$7
The seller's profit would be -$7.
In summary, the payoff and profit for a call option buyer and seller depend on the stock price at maturity (ST) compared to the strike price (X), and both parties' outcomes are influenced by the premium (Co) paid or received for the option contract.
B. Put Option:
1. **Put Option Definition**: A put option is a financial contract that gives the buyer the right (but not the obligation) to sell a specific asset (e.g., a stock) at a predetermined price (strike price, X) on or before a specified date (maturity or expiration date). The buyer pays a premium (Po) to the seller for this right.
2. **Put Option Payoff** (PT): The payoff for the put option buyer is the difference between the strike price (X) and the stock price at maturity (ST) if the stock price is lower than the strike price. Otherwise, the payoff is zero.
PT = max(0, X - ST)
**Explanation**: The max function is used here to ensure that the payoff is never negative. If the stock price at maturity (ST) is below the strike price (X), then the difference (X - ST) is positive, resulting in a non-zero payoff. If the stock price is equal to or higher than the strike price, the difference (X - ST) will be negative or zero, and the payoff will be zero.
3. **Put Option Profit**: The profit for the put option buyer is calculated by subtracting the put premium (Po) from the put option payoff (PT).
Profit = PT - Po
**Explanation**: The profit is the net gain or loss the put option buyer experiences. It is calculated by subtracting the premium paid (Po) from the payoff (PT). If the payoff (PT) is negative (stock price at maturity higher than the strike price), the buyer's profit will be negative, representing a loss equal to the premium paid. If the payoff (PT) is positive (stock price at maturity lower than the strike price), the buyer's profit will be positive, and it will be the payoff minus the premium.
4. **Put Option Payoff for the Seller (-P)**: The payoff for the put option seller (writer) is the opposite of the buyer's payoff.
-P = -max(0, X - ST)
**Explanation**: Since the put option seller is taking the opposite side of the trade compared to the buyer, their payoff is the negative of what the buyer receives. If the stock price at maturity (ST) is below the strike price (X), then the difference (X - ST) is positive, resulting in a negative payoff for the seller. If the stock price is equal to or higher than the strike price, the difference (X - ST) will be negative or zero, and the payoff for the seller will be zero.
5. **Put Option Profit for the Seller**: The profit for the put option seller is calculated by subtracting the put premium (Po) from the negative payoff (-P).
Profit = Po + max(0, X - ST)
**Explanation**: The profit for the seller is the premium received (Po) plus the payoff from the option contract. If the stock price at maturity (ST) is below the strike price (X), then the difference (X - ST) is positive, resulting in a negative payoff (-P) for the seller. In this case, the profit will be the premium received plus the negative payoff, which represents the profit the seller makes. If the stock price is equal to or higher than the strike price, the difference (X - ST) will be negative or zero, and the payoff for the seller will be zero, resulting in the profit being the premium received.
**Numerical Example**:
Let's consider a numerical example to illustrate the concepts. Suppose the following details for a put option:
- Strike Price (X) = $50
- Stock Price at Maturity (ST) = $45
- Put Premium (Po) = $3
1. Put Option Payoff (PT):
PT = max(0, $50 - $45) = max(0, $5) = $5
2. Put Option Profit for the Buyer:
Profit = PT - Po = $5 - $3 = $2
3. Put Option Payoff for the Seller (-P):
-P = -max(0, $50 - $45) = -max(0, $5) = -$5
4. Put Option Profit for the Seller:
Profit = Po + max(0, $50 - $45) = $3 + max(0, $5) = $3 - $5 = -$2
In this example, the put option buyer pays a premium of $3 and has a payoff of $5 at maturity, resulting in a profit of $2. On the other hand, the put option seller receives a premium of $3 but has a negative payoff of -$5 at maturity, which leads to a profit of -$2. The profit for the seller is higher because they receive the premium and benefit when the stock price remains above the strike price.
3. Forward Contract:
1. A Forward Contract is a customized agreement: It means that the terms of the forward contract are negotiated between the two parties involved. Unlike standardized futures contracts traded on exchanges, a forward contract allows flexibility in determining the specific details, such as the asset being traded, the delivery date, and the delivery price.
2. Two parties: A forward contract involves two parties - a long position holder (buyer) and a short position holder (seller). The long position holder agrees to buy the underlying asset, while the short position holder agrees to sell the asset.
3. Delivery price (K): Also known as the "strike price" or "forward price," the delivery price is the agreed-upon price at which the underlying asset will be bought or sold on the maturity date. It is determined at the inception of the contract and remains fixed throughout the contract's duration.
4. Maturity date: This is the future date on which the forward contract expires, and the actual exchange of the underlying asset and payment takes place.
5. Long position payoff = S - K: The payoff for the long position holder (buyer) is the difference between the spot price at maturity (S) and the delivery price (K). If the spot price at maturity (S) is higher than the delivery price (K), the long position benefits.
6. Short position payoff = K - S: The payoff for the short position holder (seller) is the opposite of the long position's payoff. It is the difference between the delivery price (K) and the spot price at maturity (S). If the spot price at maturity (S) is higher than the delivery price (K), the short position suffers a loss.
Now, let's illustrate this with a numerical example:
Suppose two parties, Alice and Bob, enter into a forward contract on gold. The contract details are as follows:
- Delivery price (K): $1,800 per ounce
- Maturity date: 1st August 2023
Scenario 1: Spot price at maturity (S) is $2,000 per ounce.
Long position payoff = S - K = $2,000 - $1,800 = $200 per ounce.
Short position payoff = K - S = $1,800 - $2,000 = -$200 per ounce.
In this scenario, Alice, the long position holder, benefits from the higher spot price. She can buy gold at the lower delivery price of $1,800 and immediately sell it in the market at $2,000, making a profit of $200 per ounce. On the other hand, Bob, the short position holder, suffers a loss of $200 per ounce, as he has to sell gold at $1,800 while it is trading at $2,000 in the market.
Scenario 2: Spot price at maturity (S) is $1,600 per ounce.
Long position payoff = S - K = $1,600 - $1,800 = -$200 per ounce.
Short position payoff = K - S = $1,800 - $1,600 = $200 per ounce.
In this scenario, Alice, the long position holder, incurs a loss of $200 per ounce, as she has to buy gold at $1,800 while it is trading at a lower spot price of $1,600 in the market. Conversely, Bob, the short position holder, benefits from the lower spot price. He can sell gold at the higher delivery price of $1,800 and immediately buy it back in the market at $1,600, making a profit of $200 per ounce.
Remember, forward contracts are usually used for hedging purposes or to lock in future prices, but they carry the risk of price movements going against either party.
Remember that these calculations represent the payoffs and profits at the maturity or expiration date of the contract. The profitability of these financial instruments depends on various factors, including market conditions, the asset's price movement, and the decisions made by the option or forward contract holder.
Sure, here are some multiple-choice questions related to options and forward contracts along with their possible answers:
1. Call Option Payoff:
Which formula represents the payoff for the call option buyer (Cr) at maturity?
a) Cr = max(0, ST - X)
b) Cr = max(0, X - ST)
c) Cr = ST - X
d) Cr = X - ST
Answer: a) Cr = max(0, ST - X)
2. Put Option Profit:
The profit for the put option buyer is calculated as:
a) Profit = PT - Po
b) Profit = Po - PT
c) Profit = PT + Po
d) Profit = PT * Po
Answer: a) Profit = PT - Po
3. Forward Contract Payoff:
In a forward contract, the payoff for the short position is calculated as:
a) Short position payoff = S - K
b) Short position payoff = K - S
c) Short position payoff = S + K
d) Short position payoff = K + S
Answer: b) Short position payoff = K - S
4. Option Premium:
The price paid by the option buyer to the option seller for the right to exercise the option is called:
a) Strike price
b) Spot price
c) Option premium
d) Delivery price
Answer: c) Option premium
5. Call Option Seller's Profit:
The profit for the call option seller is calculated as:
a) Profit = Co - Cr
b) Profit = Cr - Co
c) Profit = -Co - (-Cr)
d) Profit = Co + Cr
Answer: c) Profit = -Co - (-Cr)
6. Put Option Payoff:
The formula to calculate the payoff for the put option buyer (PT) at maturity is:
a) PT = max(0, X - ST)
b) PT = max(0, ST - X)
c) PT = ST - X
d) PT = X - ST
Answer: a) PT = max(0, X - ST)
7. Forward Contract Long Position:
In a forward contract, the payoff for the long position is given by:
a) Long position payoff = S - K
b) Long position payoff = K - S
c) Long position payoff = S + K
d) Long position payoff = K + S
Answer: a) Long position payoff = S - K
8. Put Option Seller's Payoff:
The payoff for the put option seller is calculated as:
a) Payoff = max(0, ST - X)
b) Payoff = max(0, X - ST)
c) Payoff = -max(0, X - ST)
d) Payoff = -max(0, ST - X)
Answer: c) Payoff = -max(0, X - ST)
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