Long Options Payoff Profiles

The post “Long Options Payoff Profiles” first appeared on Robot Wealth blog.

In this article, we explore the payoff to holding long options positions.

Read the previous parts of this 101 series on options:

• Options are just options – why are they valuable?
• Options 101: Understanding the basics
• The value of an option at expiration

Payoff Profile of a Long Call

So far, we’ve plotted the value of an option at expiration.

This is useful (as we’ll see later), but it doesn’t represent our profit and loss from being long that option.

For that, we need to subtract the amount we paid from the option from the payoff for all values of the underlying asset price.

Example: Long a call option expiring “in the money”

• Pam pays $2 for $100 strike calls on product X.
• At the expiration date, product X is trading in the market at $130.

What is Pam’s net P&L?

We follow this process:

1. Calculate the value of the option
2. Subtract the value we paid for it.

If the product is trading at $130 at expiry, and Pam holds calls with a $100 strike then Pam’s calls are “in the money”. Pam can exercise her call option at $100 and then sell the product in the market at $130. So:

Value of call option at expiration = $130 - $100 = $30

Now we subtract the amount Pam paid for the call option to get her net p&l in the trade.

Net Profit = $30 - $2 = $28

Nice trade, Pam.

Example: Long a call option expiring “out of the money”

Let’s take a similar example…

• Pam pays $2 for $100 strike calls on product X
• At the expiration date, product X is trading in the market at $90

What is Pam’s net P&L now?

We follow this process:

1. Calculate the value of the option
2. Subtract the value we paid for it.

The product is trading below the value of Pam’s call strike. So there’s no point exercising these options. Pam is a smart lady and she wouldn’t pay more than she had to for product X.
So Pam’s options expire “out of the money”.  They expire worthlessly.

Value of call option at expiration = $0

Now, we subtract the amount Pam paid for the call option to get her net P&L in the trade.

Net Profit = $0 - $2 = -$2

It was a loss – but because she bought call options, Pam’s loss was limited to the amount she paid for the calls.

Payoff Profile for Long Call Position

We can plot the P&L of a long options position held to expiration as a function of the price of the underlying asset.

To do this we:

• calculate the value of the option at expiry for a range of underlying asset prices
• subtract the value paid for the option (“the premium”) from each point.

min_price <- 50
max_price <- 150
strike <- 100 
premium <- 2 
call_payoffs <- tibble(price = c(min_price, strike, max_price)) %>%
  mutate(callvalue = case_when(price < strike ~ 0, TRUE ~ price - strike)) %>%
  mutate(payoff = callvalue - premium) 
call_payoffs %>%
  ggplot(aes(x = price, y = payoff)) + 
    geom_line() + 
    ggtitle(paste0('Payoff profile for long $100 call - Premium $2))

We see that:

• Our maximum loss is the amount we paid for the call options
• Our profit is unlimited
• We break even at the price where the value of the call option at expiration is equal to the amount we paid for the calls. This is premium + strike = $102

Now let’s do the same for a Put Option.

Payoff Profile of a Long Put Option

This is exactly the same deal as before.

We calculate the put’s value at expiration at various prices of the underlying asset and then subtract the price we paid for the option.

We just need to remember that a put option is an option on being able to sell the underlying asset at the strike price. So this time around, all the action happens when the underlying asset price is below our strike.

At the risk of being a bit tedious, we’ll run through some examples.

Example: Long a put option expiring “in the money”

• Fabio pays $1 for $95 strike puts on product Z.
• At the expiration date, product Z is trading in the market at $90.

What is Fabios’s net P&L?

We follow this process:

1. Calculate the value of the option
2. Subtract the value we paid for it.

If the product is trading at $90 at expiry, and Fabio holds puts with a $95 strike, then Fabio’s calls are “in the money”. Fabio can exercise his put option and sell product X at $95, then immediately buy back the product in the market at $90. So:

Value of put option at expiration = $95 - $90 = $5

Now we subtract the amount Fabio paid for the put option to get his net p&l in the trade.

Net Profit = $5 - $1 = $4

Nice trade, Fabio.

Example: Long a call option expiring “out of the money”

Let’s take a similar example…

• Fabio pays $1 for $95 strike calls on product X
• At the expiration date, product X is trading in the market at $100

What is Fabio’s net P&L now?

We follow this process:

1. Calculate the value of the option
2. Subtract the value we paid for it.

The product is trading above the value of Fabio’s call strike. So there’s no point exercising these options. Fabio is a smart man and he wouldn’t sell product Z for less than he could get in the market.

So Fabio’s options expire “out of the money”.  They expire worthlessly.

Value of put option at expiration = $0

Now we subtract the amount Fabio paid for the call option to get her net p&l in the trade.

Net Profit = $0 - $1 = -$1

It was a loss – but because he bought options, Fabio’s loss was limited to the amount he paid for the puts.

Payoff Profile for Long Put Position

We can plot the P&L of a long options position held to expiration as a function of the price of the underlying asset.

To do this we:

• calculate the value of the option at expiry for a range of underlying asset prices
• subtract the value paid for the option (“the premium”) from each point.

min_price <- 50
max_price <- 150
strike <- 95 
premium <- 1
put_payoffs <- tibble(price = c(min_price, strike, max_price)) %>% 
  mutate(putvalue = case_when(price > strike ~ 0, TRUE ~ strike - price)) %>%
  mutate(payoff = putvalue - premium)
put_payoffs %>% 
  ggplot(aes(x = price, y = payoff)) + 
  geom_line() + 
  ggtitle(paste0('Payoff profile for long $95 put - Premium $1'))

We see that:

• Our maximum loss is the amount we paid for the put options
• Our profit is unlimited
• We break even at the price where the value of the put option at expiration is equal to the amount we paid for the puts. This is strike - premium = $95 - $1 = $94

Now we understand the basics, we will next look at a simple applied use of put options: portfolio hedging.