This post follows up on a previous post, which is an introductory discussion on options. In this post, we focus on the two basic strategies of using options as insurance – protective put and protective call. These two strategies are for investors or traders who want to buy insurance to protect profits that come from holding either a long or short position. For every insurance buyer, there must be an insurance seller. In the next post, we discuss covered call and covered put – basic strategies for investors or traders who want to sell insurance protection.

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**Option strategies that are insurance protection**

Suppose that an investor has been holding an asset that has gained in value. The long asset position held by this investor will suffer a loss if prices fall. A long put option (a purchased put) has positive payoff when prices are less than the strike price. Buying a put option will guarantee a minimum sale price (the strike price of the put option) should the investor wishes to sell the asset at a future date. Thus the risk management strategy of buying a put option to guard against the loss of a long position is called a protective put.

A protective call deals with an opposite situation. Suppose an investor or trader is holding a short position on an asset (e.g. the investor has short sold a stock). The short asset position held by this investor will suffer a loss if prices increase. A purchased call option has positive payoff when the asset prices are greater than the strike price. When the investor buys a call option with the same underlying asset, the strike price is in effect a minimum purchased price of the asset should there be a price increase, thus keeping the loss at a minimum. Thus the risk management strategy of buying a call option to guard against the loss of a short position is called a protective call.

Protective puts and protective calls are basic insurance strategies that can be used to protect profits from either holding a long asset position or a short position. Both of these strategies will minimize the loss in the event that the prices of the asset position move in the wrong direction. Of course, the insurance protection comes at a cost in the form of an option premium, which is a cash fee paid by the buyer to the seller at the time the option contract is made. In the remainder of the post, we examine the protective put and the protective call in greater details by examining the payoff and profit diagrams.

The put and call considered here are European options, i.e., they can be exercised only at expiration.

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**Protective put**

The protective put consists of a long asset position (e.g. owning a stock) and a long put option on the same asset. Our goal is to examine the payoff and profit of this combined position. We can then make some observations based on the profit diagram. Figure 1 is the payoff of the long asset. Figure 2 is the payoff of the long put option. Figure 3 is the payoff of the combined position. Figure 4 is the profit of long asset + long put. The strike price in all the diagrams is . Instead of using a numerical example to anchor the diagrams, we believe that the following diagrams of payoff and the profit are clear. In fact, getting bogged down in a numerical example may make it harder to see the general idea. Asking questions such as – what happens when the asset is less than , etc – will make the diagrams clear. In fact, reading the diagrams is a good concept check. An even better practice is to draw the payoff and profit diagrams on paper.

Figure 1 is the payoff of the long asset. The strike price has no effect on the payoff of the long asset (Figure 1). The payoff of an asset is simply the value of the asset at a point in time. Thus the payoff is simply the asset price at a target date. The higher the asset price, the higher the payoff.

Figure 2 is the payoff of the long put option. For the long put, the payoff is when it makes sense for the put option buyer to exercise. Thus the payoff is positive to the left of . To the right of , the put option expires worthless, thus the payoff is 0. The sum of Figure 1 and Figure 2 gives Figure 3.

Figure 3 is the payoff of Long Asset + Long Put. The payoff to the right of the strike price is flat, which is a sign of the insurance at work. The positive payoff of the long put neutralizes the effect of falling prices of the long asset, minimizing the loss from holding a long asset when the prices go south. This position of long asset + long put will enjoy the upside potential in the event that prices go up. Of course, such as good insurance product is not free. The next diagram will take cost into account. First, the following formula shows the payoff of long asset + long put.

Figure 4 is the profit of long asset + long put. Recall that the profit of a position is the payoff less the cost of acquiring that position. What is the cost of acquiring a long asset and a long put? The cost of the long asset is , the future value of the price paid at time 0. The cost of the long put is , where is the put option premium paid by the buyer to the seller at time 0. Thus the cost of the long asset + long put is . As a result, the profit graph is Figure 4 is obtained by pressing down the payoff in Figure 3 by the amount of the cost. The cost of the position is likely to be more than the strike price . This is why in Figure 4 is negative.

Without the insurance (Figure 1), the long asset position will suffer substantial loss in the event that the prices are low. With insurance (Figure 4), the potential loss for the long asset position is essentially the put option premium. But the long asset position still enjoys the upside profit potential (less the option premium).

Another observation that can be made about Figure 4 is that its shape is very similar to the profit of a long call option (compare Figure 4 above with the Figure 1 in this previous post). The profit diagram of long asset + long put does not merely resemble the profit of a long call (with the same strike price); it is identical.

Based on Figure 4, the investment strategy of long asset + long put mimics the profit of the long call position (with the same strike price). Thus the position of long asset + long call is called a synthetic call option.

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**Protective call**

The protective call consists of a short asset position (e.g. shorting a stock) and a long call option on the same asset. We now examine the payoff and profit of this combined position. Figure 5 is the payoff of the short asset. Figure 6 is the payoff of the long call option. Figure 7 is the payoff of short asset + long call. Figure 8 is the profit of short asset + long call. The strike price in all the diagrams is . From Figure 8, we will see that the combined position is a synthetic put option.

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.

Figure 6 is the payoff of the long call position. When the spot price at expiration is less than the strike price , the call option expires worthless. When the spot price at expiration is greater than the strike price , the payoff is .

Figure 7 is the payoff of the combined position of a short asset and a long call. The payoff of the combined position is flat to the right of the strike price. This is a sign of the insurance at work. The upside potential of the short call is limiting the loss of the short asset position. The positive payoff of the long call is . The payoff of the short asset is when price is greater than the strike price. Then the combined payoff is when price is greater than the strike price. To further clarify, the following is the payoff of the combined position.

Figure 8 is the profit of short asset + long call. To derive the profit, we need to subtract the cost of acquiring the combined position from the payoff. The profit graph is Figure 8 is the result of lifting up the payoff graph. That suggests that in this case the profit is the payoff plus a positive amount. This is indeed correct since the cost of acquiring the position is a negative number. Thus subtracting the cost from the payoff is in effect adding a positive number.

To see the above point, the cost of acquiring the initial position is a positive number if it is a cash outflow (you pay to buy an asset) and is a negative number if it is a cash inflow (you sell an asset). In a short position, you borrow the asset and sell it to get cash, which is in this calculation. There is also the purchase of a call. Thus the total cost is , which is likely a negative amount. So subtracting this negative cost from the payoff has the effect of lifting up the payoff graph.

Without the insurance of a long call (Figure 5), the short asset position has unlimited loss. With insurance (Figure 8), the loss of the short asset position is minimized, essentially the call option premium. The short asset position still enjoys the profit potential should asset prices fall (less the option premium).

Compare the above Figure 8 with the Figure 8 in this previous post, we see that they have the same shape. This is not coincidence. Both positions have the same profit. Thus the combined position of a short asset and a long call option is called a synthetic long put option since both have the same profit diagrams.

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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 + long put) also resembles the payoff of a long call 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 long call option payoff by a uniform amount. That uniform amount can be interpreted as the payoff of a zero-coupon bond. Thus we have the following relationship.

payoff of “long asset + long put” = payoff of “long call + zero-coupon bond”

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:

profit of “long asset + long put” = profit of “long call”

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

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

payoff of “short asset + long call” = payoff of “long put – zero-coupon bond”

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:

profit of “short asset + long call” = profit of “long put”

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

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