## Call & Put Option Profits and Payoffs

Call and put options have basic formulas for determining the value, profit and break-even point at expiration dependent on whether the investor has bought or sold the option. Using these basic characteristics, more complex option strategies can be evaluated.

### Standard Long and Short Positions

We focus initially on the most fundamental option transactions. That is, buying or selling a single call or put option and holding it to expiration. The value, profit and breakeven at expiration can be determined formulaically for long and short calls and long and short puts. The notation used is as follows:

• c0, cT = price of the call option at time 0 and T
• p0, pT = price of the put option at time 0 and T
• X = exercise price
• S0, ST = price of the underlying at time 0 and T
• Π = profit from the transaction

#### Long Call

• Value at expiration {cT = max(0, ST – X)}
• cT = 0 if ST ≤ X
• cT = ST – X if ST > X
• Profit at expiration
• Π = -c0 if ST ≤ X
• Π = ST – X – c0 if ST > X
• Breakeven {value of ST denoted as ST* where Π = 0}
• ST* = X + c0
• Maximum profit = ∞
• Maximum loss = c0

#### Short Call

• Value at expiration {cT = max(0, X – ST)}
• cT = 0 if ST ≤ X
• cT = X – Sif ST > X
• Profit at expiration
• Π = c0 if ST ≤ X
• Π = X – ST + cif ST > X
• Breakeven {value of ST denoted as ST* where Π = 0}
• ST* = X + c0
• Maximum profit = c0
• Maximum loss = ∞

### Long Put

• Value at expiration {pT = max(0, X – ST)}
• pT = X – ST if ST < X
• pT = 0 if ST ≥ X
• Profit at expiration {Π = pT – p0}
• Π = X – S– p0 if ST < X
• Π = – p0 if ST ≥ X
• Breakeven {value of ST denoted as ST* where Π = 0}
• ST* = X – p0
• Maximum profit = X – p0
• Maximum loss = p0

### Short Put

• Value at expiration {pT = max(0, ST – X)}
• pT = ST – X if ST < X
• pT = 0 if ST ≥ X
• Profit at expiration {Π = -pT + p0}
• Π = ST – X + p0 if ST < X
• Π = p0 if ST ≥ X
• Breakeven {value of ST denoted as ST* where Π = 0}
• ST* = X – p0
• Maximum profit = p0
• Maximum loss = X – p0

### Investor Outcomes

Call options tend to be purchased by investors who hold a bullish view on the underlying, while a bearish view would be expressed by buying a put option. The option seller will have the converse payoff profile to the option buyer and the sum of the positions of buyer and seller is zero. This means the maximum profit and maximum loss are interchanged for the buyer and seller and the breakeven value remains the same.

## Question

If a put option has a premium of \$3 and the exercise price is \$100 and the price of the underlying is \$105, which reflects the value at expiration and the profit to the option seller?

A. pT = \$3; Π = \$0

B. pT = \$0; Π = \$8

C. pT = \$0; Π = \$3

Solution

The put seller is short a put and the exercise price (\$100) is less than the underlying price (\$105) so we have a state where ST ≥ X. Therefore pT = 0 and Π = p0 which means profit = \$3. In the hands of the put buyer (long put),  pT = 0 and Π = – p0 or a loss of \$3. Essentially the option has expired worthless and has cost the buyer the initial premium.

Determine the value at expiration, the profit, maximum profit, maximum loss, breakeven underlying price at expiration, and payoff graph of the strategies of buying and selling calls and puts and determine the potential outcomes for investors using these strategies

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