## Basic insurance strategies – covered call and covered put

The use of options can be interpreted as buying or selling insurance. This post follows up on a previous post that focuses on two option strategies that can be interpreted as buying insurance – protective put and protective call. For every insurance buyer, there must be an insurance seller. In this post, we discuss two option strategies that are akin to selling insurance – covered call and covered put.

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Selling insurance against an asset position

The previous post discusses the strategies of protective put and protective call. Both of these are “buy insurance” strategies. A protective put consists of a long asset and a long put where the long put is purchased to protect against a fall in the prices of the long asset. A protective call consists of a short asset position and a long call where the long call option is purchased to protect against a rise in the prices of the asset being sold short. Both of these strategies are to buy an option to protect against the adverse price movement of the asset involved.

When an insurer sells an insurance policy, the insurer must have enough asset on hand to pay claims. Now we discuss two strategies where the investor or trader holds an asset position that can be used for paying claims on a sold option.

A covered call consists of a long asset and a short call. The insurance sold is in the form of a call option. The long asset gains in value when asset prices rise and the gains are used to cover the payments made by the call seller when the call buyer decides to exercise the call option. Therefore the covered call is to use the upside profit potential of the long asset to back up (or cover) the call option sold to the call buyer. The covered call strategy can be used by an investor or trader who believes that the long asset will appreciate further in the future but is willing to trade the long term upside potential for a short-term income (the call premium). This is especially true if the investor thinks that selling the long asset at the strike price of the call option will meet a substantial portion of his expected profit target.

A covered put consists of a short asset position and a short put. Here, the insurance sold is in the form of a put option. The short asset is used to back up (or cover) the put option sold to the put buyer. A short asset position is not something that is owned. How can a short asset position back up a put option? The short asset position gains in value when asset prices fall. A put option is exercised when the prices of the underlying asset fall. Thus a put option seller needs to pay claims exactly when the short asset position gains in value. Thus the gains in the short asset position are used to cover the payments made by the put seller when the put buyer decides the exercise the put option.

In this post, we examine covered call and covered put in greater details by examining their payoff diagrams and profit diagrams.

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Covered call

As mentioned above, a covered call is a position consisting of a long asset and a short call. Here the holder of the long asset sells a call against the long asset. Figure 1 is the payoff of the long asset. Figure 2 is the payoff of the short call. Figure 3 is the payoff of the covered call. Figure 4 is the profit of the covered call. The strike price in all the diagrams is $K$. We will see from Figure 4 that the covered call is a synthetic short put.

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Figure 1 – Long Asset Payoff

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Figure 1 is the payoff of the long asset position. When the asset prices are greater than the strike price $K$, the positive payoff is unlimited. The unlimited upside potential is used to pay claim when the seller of the call is required to pay claim to the call buyer.

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Figure 2 – Short Call Payoff

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Figure 2 is the payoff of the short call. This is the payoff of the call seller (i.e. the insurer). The call seller has negative payoff to the right of the strike price. The negative payoff occurs when the call buyer decides to exercise the call. The long asset payoff in Figure 1 is to cover this negative payoff.

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Figure 3 – Long Asset + Short Call Payoff

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Figure 3 is the payoff of the covered call, the result of combining Figure 1 and Figure 2. Unlike Figure 1, the long asset holder no longer has unlimited payoff to the right of the strike price. The payoff is now capped at the strike price $K$.

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Figure 4 – Long Asset + Short Call Profit

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Figure 4 is the profit of the covered call. The profit is the payoff less the cost of acquiring the position. At time 0, the cost is $S_0$ (the purchase price of the asset, an amount that is paid out) less $P$ (the option premium, an amount that is received). The future value of the cost of the covered call is then $S_0 e^{r T}-P e^{r T}$. The profit is then the payoff less this amount. The profit graph is in effect obtained by pressing down the payoff graph by the amount of $S_0 e^{r T}-P e^{r T}$. Because of the received option premium, $S_0 e^{r T}-P e^{r T}$ is less than the strike price $K$. As a result, the flat part of the profit graph is above the x-axis.

Without selling insurance (Figure 1), the profit potential of the long asset is unlimited. With the insurance liability (Figure 4), the profit potential is now capped at essentially at the call option premium. In effect the holder of a covered call simply sells the right for the long asset upside potential for cash received today (the option premium).

The strategy of a covered call may make sense if selling at the strike price can achieve a significant part of the profit target expected by the investor. Then the payoff from the strike price plus the call option premium may represent profit close to the expected target. Let’s look at a hypothetical example. Suppose that the stock owned by an investor was purchased at $60 a share. The investor believes that the stock has upside potential and the share price will rise to$70 in a year. The investor can then sell a call option with the strike price of $65 with an expiration of 6 months and with a call premium of$5. In exchange for a short-term income of the call option premium, the investor gives up the profit potential of $70 a share. If in 6 months, the share price is more than$65, then the investor will sell at $65 a share, producing a profit of$10 a share ($5 in share price appreciation and$5 call premium). If the share price is below the strike price is 6 months, the investor then pockets the \$5 premium.

Note the similarity between Figure 4 above and the Figure 11 in this previous post. Figure 11 in that previous post is the profit diagram of a short put. So the covered call (long asset + shot call) is also called a synthetic short put option since it has the same profit as a short put.

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Covered put

As indicated above, a covered put is to use the profit potential of a short asset position to cover the obligation of a sold put option. Figure 5 below is the profit of a short asset position. Figure 6 is the payoff of a short put option. Figure 7 is the payoff of the covered put. Figure 8 is the profit diagram of the covered put.

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Figure 5 – Short Asset Payoff

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Figure 5 is the payoff of the short asset position. Holder of a short asset position is concerned about rising prices of the asset. The holder of the short borrows the asset in a short sales and sells the asset immediately for cash, which is then accumulated at the risk-free rate. The short position will have to buy the asset back in the spot market at a future date to repay the lender. If the spot price at expiration is greater than the original sale price, then the short position will lose money. In fact the potential loss is unlimited.

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Figure 6 – Short Put Payoff

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Figure 6 is the payoff of a short put option. Recall that the short put payoff is from the perspective of the seller of the put option. When the price of the underlying asset is below the strike price, the seller has the obligation to sell at the strike price (thus experiencing a loss). When the asset price is above the strike price, the put option expires worthless.

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Figure 7 – Short Asset + Short Put Payoff

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Figure 7 is the payoff of the covered call. With the covered call, the holder of the short asset can no longer profit by paying a price lower than the strike price for the asset to repay the lender. Instead he has to pay the strike price (this is the flat part of Figure 7). To the right of the strike price, the covered call continues to have the potential for unlimited loss.

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Figure 8 – Short Asset + Short Put Profit

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Figure 8 is the profit of the covered put, which indicates the profit is essentially the option premium received by selling the put option. Without selling the insurance (Figure 5), the short asset position has good profit potential when prices fall. With selling the insurance, the profit potential to the left of the strike price is limited to the option premium. The covered put is in effect to trade the profit potential (when prices are low) with a known put option premium.

Compare Figure 8 above with Figure 5 in this previous post. Both profit diagrams are of the same shape. Figure 5 in the previous post is the profit diagram of a short call. So the combined position of short asset + short put is called a synthetic short call.

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Synthetic put and call

Just a couple of more observations to make about synthetic put and synthetic call.

Note that Figure 3 (the payoff of long asset + short call) also resembles the payoff of a short put option, except that the level part of the payoff is not at the x-axis. So Figure 3 is the lifting up of the usual short put option payoff by a uniform amount. That uniform amount can be interpreted as the payoff of a long zero-coupon bond. Thus we have the following relationship.

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payoff of “long asset + short call” = payoff of “short put + zero-coupon bond”
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Adding a bond lifts the payoff graph. However, adding a bond to a position does not change the profit. To see this, simply subtract the cost of acquiring the position from the payoff. You will see that for the bond, the same amount appears in both the cost and the payoff. Thus we have:

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profit of “long asset + short call” = profit of “short put”
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As mentioned earlier, the above relationship indicates that the combined position of long asset + short call can be viewed as a synthetic short put. We now see that the covered call is identical to a short put.

Now similar thing is going on in a covered put. Note that Figure 7 resembles the payoff of a short call except that it is the pressing down of the payoff of a usual short call. We can think of this pressing down as a borrowing. Thus we have:

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payoff of “short asset + short put” = payoff of “short call – zero-coupon bond”
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Adding a bond means lending and subtracting a bond means borrowing. As mentioned before, adding or subtracting a bond lift or depress the payoff graph but does not change the profit graph. We have:

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profit of “short asset + short put” = profit of “short call”
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The above relationship is the basis for calling “short asset + short put” as a synthetic short call.

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$\copyright \ \ 2015 \ \text{Dan Ma}$