Tag Archives: Covered Put

Put-Call Parity, Part 1

Put–call parity is a relationship between the price of a European call option and European put option with the same strike price and time to expiration. It is one of the most important relationships in option pricing. It provides a tool for constructing equivalent positions. This post is a general discussion of put-call parity. In the next post, we discuss put-call parity in greater details for various underlying assets – e.g. stocks, treasuries and currencies.

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Synthetic forward – buying a call and selling a put

Suppose you follow the strategy of buying a call and selling a put (at time 0) where both options have the same underlying asset, the same strike price K and the same time T to expiration. At time T, it is certain that you will buy the underlying asset by paying the strike price K. Too see this, if at expiration of the options, the asset price is more than K, then you, as a call buyer will want to exercise the call option and pay K to buy the asset. If the asset price at expiration is less than K, then you as a call buyer will not want to exercise but the put buyer that bought from you will want to exercise the put option. As a result, you will also buy the asset by paying the strike price K. Thus by entering into a long call and a short put (on the same underlying asset, with the same strike and same time to expiration), you will end up buying the underlying asset at time T at the strike price K. What is being described sounds very much like a forward contract – a contract in which you can lock in a price today to pay for an asset a time T in the future. For this reason, the strategy of buying a call and selling a put is called a synthetic forward contract.

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Put-call parity

The above discussion on synthetic forward suggests that there are two ways to buy an underlying asset (e.g. a stock) at time T in the future. They are:

  1. Enter into a forward contract to buy the underlying asset by paying the forward price F_{0,T} at time T.
  2. Buy a call and sell a put today (on the same underlying asset, with the same strike price K and the same time T to expiration).

The two different strategies generate the same payoff. Hence they must have the same cost. Otherwise there would be arbitrage opportunities. By the “no-arbitrage pricing” principle, the net cost of the two strategies must equal. The cost at time 0 of the “buy call sell put” strategy is C(K,T)-P(K,T), plus the present value of the strike price K, where C(K,T) and P(K,T) represent the call option premium and put option premium, respectively. The cost at time T of the forward contract strategy is the forward price F_{0,T}. Thus cost at time 0 of the forward contract strategy is the present value of F_{0,T}. We can now equate the costs of the two strategies.

    \text{ }
    Put-Call Parity
    \displaystyle PV(F_{0,T})=C(K,T)-P(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)
    \text{ }

The notation PV(\cdot) denotes the time 0 value of an amount at the time T. Equation (0) is one form of the put-call parity, which is a statement that buying a call and selling a put is equivalent to a synthetic forward contract. It also tells us that buying a call and selling a put plus lending the present value of the strike price is equivalent to buying the underlying asset.

Other versions can be derived by algebraically rearranging equation (0), some of which have interesting interpretations. The following is one of them.

    \text{ }
    Put-Call Parity
    \displaystyle C(K,T)-P(K,T)=PV(F_{0,T}-K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)
    \text{ }

The left hand side of (1) is the net option premium – the premium paid for the call less the premium received for the put. When this amount is not zero, it is in effect the premium of the synthetic forward contract (this amount is the initial cash outlay for the synthetic forward contract). This is one difference between a synthetic forward and an actual forward. Note that an actual forward contract has zero premium (the initial cash outlay is zero). Another difference is that the “forward price” of the synthetic forward is the strike price K of the options and while the forward price of the actual forward is F_{0,T}.

Suppose that the strike price K is chosen to be less than the actual forward price F_{0,T}. Then the holder of the synthetic forward contract can buy the asset at a price lower than the forward price. This is certainly a benefit. In order to get this benefit, the holder of the synthetic forward contract has to pay the net option premium, which is the result of the call being more expensive than the put. In this scenario, the net payment is a little higher at time 0. As a result, the payment at time T is a little less.

Suppose that the strike price K is chosen to be more than the actual forward price F_{0,T}. Then the holder of the synthetic forward position is obliged to pay for the underlying asset at a price higher than the forward. It then makes sense for the holder of the synthetic forward position to be compensated by receiving a payment initially. This would occur if the put is more expensive than the call. In this scenario, the net payment is a little less at time 0, leading to a larger payment at time T.

If the strike price is chosen to be the same as the forward price F_{0,T}, then equation (1) suggests that the synthetic forward mimic exactly the actual forward (both have zero premium). For this to happen, premiums for the put and the call must be equal.

The right hand side of (1) is the value of the discount resulted from paying the strike price instead of the forward price. This version of the put-call parity says that the discount is identical to the net option premium.

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Protective put and covered call

The next two versions can be interpreted in terms of a protective put and a covered call. A protective put consists of a long asset position and a long put. It is the strategy of buying a put option to protect against the risk of falling prices of a long asset position. A covered call consists of a long asset position and a short call. The covered call uses the upside profit potential of the long asset to back up (or cover) the call option sold to the call buyer. First, the protective call version:

    \text{ }
    Put-Call Parity
    \displaystyle PV(F_{0,T})+P(K,T)=C(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)
    \text{ }

The left hand side of (2) is the time 0 cash outlay of buying the underlying asset and buying a put. The right hand side of (2) is time 0 cash outlay of buying a call option (with the same strike and time to expiration as the put) and buying a zero-coupon bond costing PV(K). Thus equation (2) tells us that buying the underlying asset and buying a put on that asset (i.e. a protective put) have the same cost and generate the same payoff as the buying a call option and buying a zero-coupon bond. Adding a bond lifts the payoff graph but does not change the profit graph. Thus buying the asset and buying a put has the same profit as buying a call. Because of Equation (2), buying the underlying asset and buying a put is called a synthetic long call option. This point is also discussed in this previous post. Here’s the version of the put-call parity involving covered call.

    \text{ }
    Put-Call Parity
    \displaystyle PV(F_{0,T})-C(K,T)=PV(K)-P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)
    \text{ }

The left hand side of (3) is the time 0 cash outlay of buying the underlying asset and selling a call on that asset (i.e. a covered call). The right hand side of (3) is the time 0 cash outlay of buying a zero-coupon bond costing PV(K) and selling a put. Thus a covered call has the same cost and same payoff as buying a bond and selling a put. Once again, adding a bond does not change the profit. Thus a covered call has the same profit as selling a put. For this reason, a buying the underlying asset and selling a call is called a synthetic short put option. This point is also discussed in this previous post.

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Summary

As a summary, we gather the various versions of the put-call parity in one place along with their interpretations.

    \text{ }
    Versions of Put-Call Parity
    \text{ }
    \displaystyle PV(F_{0,T})=C(K,T)-P(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)
    Interpretation: Time 0 cost of a long asset = Time 0 cost of (Long Call + Short Put + Long Bond).

    \text{ }

    \displaystyle C(K,T)-P(K,T)=PV(F_{0,T}-K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)
    Interpretation: Net option premium (call option premium that is paid out less put option premium received) = the value of the discount as a result of paying the strike price instead of the forward price.
    \text{ }

    \displaystyle PV(F_{0,T})+P(K,T)=C(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)
    Interpretation: Time 0 cost of (Long Asset + Long Put) = Time 0 cost of (Long Call + Long Bond).
    The portfolio on the left (Long Asset + Long Put) is called a protective put.
    Because of (2), a protective put is considered a synthetic long call option.
    \text{ }

    \displaystyle PV(F_{0,T})-C(K,T)=PV(K)-P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)
    Interpretation: Time 0 cost of (Long Asset + Short Call) = Time 0 cost of (Long Bond + Short Put).
    The portfolio on the left (Long Asset + Short Call) is called a covered call.
    Because of (3), a covered call is considered a synthetic short put option.
    \text{ }

In each of the above versions of parity, the portfolio of investments on the left side is equivalent to the portfolio of investment on the right side. More specifically, each version equates the costs of obtaining the portfolios at time 0. The bond indicated in the interpretations is a zero-coupon bond. A long position on a bond means lending.

One comment about the four parity relations discussed here. We derive the first one, which is version (0) by comparing the cash flows of two equivalent investments. The other three versions are then derived by algebraically rearranging the first version. As a learning device, it is a good idea to think through the cash flows and payoff of versions (2) through (3) independently of version (0). Doing so is a great practice and will help solidify the understanding of put-call parity. Drawing payoff diagrams can make the comparison easier. It is also possible to just think through the cash flows of both sides of the equation. For example,

    let’s look at version (2). On the right side, you lend PV(K) and buy a call at time 0. Then at time T, you get K back. If the price of the underlying asset at that time is more than K, then you exercise the call – using the K that you receive to buy the asset. So on the right hand, side, the payoff is S_T-K if asset price is more than K and the payoff is K if asset price is less than K (you would not exercise the call in this case). On the left hand side, you lend PV(F_{0,T}) and buy a put at time 0. At time T, you get F_{0,T} back and you use it to pay for the asset. So you own the asset at time T. If the asset price at time T is less than K, you exercise by selling the asset you own and receive K. Thus the payoff on the left hand side is S_T-K if asset price is more than K (in this case you don’t exercise the put and instead you profit from holding the asset). The payoff is K if the asset price at time T is less than K (this is the case where you exercise the put option). The comparison shows that both sides of (2) have the same payoff at time T. Then it must be the case that they also have the same cost at time 0. Otherwise, there would be an arbitrage opportunity by buying the side that is low and sell the other side.

The basic put-call parity relations discussed in this post can be used in a “cookbook” fashion to create synthetic assets. For example, version (0) indicates that buying a call, selling a put and lending the present value of the strike price K has the same cost and payoff as buying a non-dividend paying stock. Thus version (0) is a basis for constructing a synthetic stock. In the next post, we discuss the put-call parity for different underlying assets.

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

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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.

    \text{ }

    Figure 1 – Long Asset Payoff
    long asset position payoff

    \text{ }

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.

    \text{ }

    Figure 2 – Short Call Payoff
    Short call position payoff

    \text{ }

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.

    \text{ }
    Figure 3 – Long Asset + Short Call Payoff
    long asset position short call payoff
    \text{ }

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.

    \text{ }
    Figure 4 – Long Asset + Short Call Profit
    long asset position short call profit
    \text{ }

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.

    \text{ }

    Figure 5 – Short Asset Payoff
    short asset payoff
    \text{ }

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.

    \text{ }

    Figure 6 – Short Put Payoff
    Short put payoff
    \text{ }

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.

    \text{ }

    Figure 7 – Short Asset + Short Put Payoff
    short asset short put payoff
    \text{ }

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.

    \text{ }

    Figure 8 – Short Asset + Short Put Profit
    short asset short put profit
    \text{ }

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.

    \text{ }
    payoff of “long asset + short call” = payoff of “short put + zero-coupon bond”
    \text{ }

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:

    \text{ }
    profit of “long asset + short call” = profit of “short put”
    \text{ }

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:

    \text{ }
    payoff of “short asset + short put” = payoff of “short call – zero-coupon bond”
    \text{ }

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:

    \text{ }
    profit of “short asset + short put” = profit of “short call”
    \text{ }

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}