25
Mar 18

Visualization of payoffs on calls and puts

Earlier I provided the traditional view of the call option value. In this post I give an alternative, more visual representation of the payoff on calls and puts, both long and short. When payoff is positive, it is a profit, and when payoff is negative, it is a loss. Everywhere it is assumed that the strike price $K$ is \$50, the call price is denoted $c$ and the put price is denoted $p$. The transaction cost is assumed to be zero. In practice, usually the broker charges you 4-5 dollars for the round turn (to open and close the position), which is not very much compared to the option price. You have to pay additional in case of stock assignment or unusual orders (like conditional orders), and still the transaction fees are low relative to the payoff.

Payoff on calls

Video 1. Visualization of payoff on long call

Video 1 illustrates the initial motivation for using calls: the buyer of the call wants to gain from an upside stock movement and limit his/her loss in the opposite case. The investment in this case is $c$, the call price. When to exercise the option is a difficult question. Let your profits run, on one hand, and don't be greedy, on the other. The decision to exercise depends on many considerations, of which we indicate the most certain two. Don't keep to expiration, to avoid the time value decay and last day uncertainties. Don't aim for the maximum profit. The main reason for this recommendation is volatility: the jump due to volatility may be much higher than the jump due to stock price rise, and volatility may quickly subside. If the call price leaps like crazy, take your money and run.

Video 2. Visualization of payoff on short call

Trading options is a zero-sum game: apart from transaction fees, the profit of the long call holder is exactly equal to the loss of the short call holder (also called the option writer). This is illustrated in Video 2. The potential profit is limited by the call price and the potential loss is sky-high. Besides, the call writer's role is passive: it's the call buyer who decides when to exercise the option. Then why would you sell a call? Suppose you own a stock priced, say, at \$100 and you don't believe that it will go much higher. Actually, you would be happy to sell it at a slightly higher price. Suppose, further, that you sold a call for \$5 with a strike of \$105. If the stock price exceeds the breakeven of \$110 and the call buyer exercises the option, you will have sold the stock for \$110. If the option is not exercised, you keep the \$5. The strategy described is called a covered call (it is a stock plus call).

Payoff on puts

Video 3. Visualization of payoff on long put

A put option gives its buyer the right, but not the obligation, to sell the stock at the price fixed in the option contract. All other definitions related to puts are similar to those for calls. The put option allows its buyer to gain from a fall in the stock price.

Video 4. Visualization of payoff on short put

As with calls, the payoff of the put seller is the opposite of that of the put buyer. Since the downside risk is large, many people sell puts in the following situation. Suppose there is a stock that you would like to buy and keep longer term. The current price of \$100 seems too high for you. Then you sell a put, say, at \$5 and with a strike of \$95. If the put buyer decides to exercise, you buy the stock at \$95 but the actual price will be \$90 because initially you received \$5. If the option is not exercised, you keep the \$5. I traded Tesla like this. The actual price paid was \$220, the stock continued to fall to \$180 and then rose to \$260. The total profit was \$4,000.