Question

“Prove” that you can create a synthetic short stock by “long a put, short a call, and short a discount bond.”

Answer 1

Ans

The relationship that must exist between European put and call options with the same underlying asset, expiration and strike prices. (It doesn't apply to American-style options because they can be exercised any time up to expiration.)

Put/call parity states that the price of a call option implies a certain fair price for the corresponding put option with the same strike price and expiration (and vice versa). Support for this pricing relationship is based on the argument that arbitrage opportunities would exist whenever put and call prices diverged.

Synthetic Relationships

If you understand the put/call parity relationship, you can connect the value between a call option, put option and the underlying stock. The three are related in that a combination of any two will yield the same profit/loss profile as the remaining component. For example, to replicate the gain/loss features of a long stock position, an investor could simultaneously hold a long call and a short put (with the same strike price and expiration). Similarly, a short stock position could be replicated with a short call plus a long put, and so on. The six possibilities are:

Original Position	=	Synthetic Equivalent
Long Stock	=	Long Call	+	Short Put
Short Stock	=	Short Call	+	Long Put
Long Call	=	Long Stock	+	Long Put
Short Call	=	Short Stock	+	Short Put
Long Put	=	Short Stock	+	Long Call
Short Put	=	Long Stock	+	Short Call

S + P – C = 0

            Now that you understand synthetics, it is easier to see why the formula works. Assume you are long stock and also have another asset in your portfolio. You don’t know what this asset is but you are told that it makes your long stock position risk free. What does that tell you about the other asset? If you have no risk, then the other assets must be short stock. If you are long stock and short stock, then you are effectively flat and have no risk. Now think about our synthetic formula. If you are long stock and have other assets (shown by the box) and are also told that you have no risk then the boxed assets (long put + short call) must be equal to short stock and that’s exactly what we found out.

S   + P – C    = 0

            To find any synthetic equivalent, all you need to do is isolate the variable(s) you’re trying to solve for and the answer will be immediately visible. Let’s try another. If you want to find out the synthetic equivalent to a long put, just isolate that +P position:

S   + P   – C    = 0

            If that long put is paired with other assets so that it has no risk, then the other assets must be equal to a short put. This immediately shows that long stock plus a short call must be equal to a short put.

            It turns out that no matter which asset you pick, the other two are the synthetic opposite and fully hedge the risk. It should make sense, then, that we could also pick any two assets and know that the third will fully hedge those two.

And discounting bonds likely to be

discount bond is the opposite of a premium bond, which occurs when the market price of a bond is higher than the price for which it was originally sold. To compare the two in the current market, and to convert older bond prices to their value in the current market, you can use a calculation called yield to maturity (YTM). Yield to maturity considers the bond's current market price, par value, coupon interest rate, and time to maturity in order to calculate a bond's return.

When short a stock, you are exposed to unlimited loss and unlimited profit while losing nothing when the stock remains stagnant. A synthetic short stock completely duplicates those characteristics. The premium gained from the short call covers the premium on the long put (thus losing nothing if the stock remains stagnant), the long put option grants unlimited profits and the short call options introduced the unlimited loss.

The principal differences are the time limitation imposed by the term of the options, the absence of the large initial cash inflow that a short sale would produce, but also the absence of the practical difficulties and obligations associated with short sales.

Investor would be looking for a decline in the stock's price during the term of the options.

Since the strategy's term is limited, the longer-term outlook for the stock isn't as critical as for, say, an outright short stock position.

you are an experienced option trader, you would be able to immediately tell that the above equation is saying that Fiduciary Calls = Protective Puts! That's right! That's what this synthetic position is trying to say.

A fiduciary call is just a long call option. A protective put is a long stock and long put.

Ergo, fiduciary call = protective put.

Provides the benefits of stock ownership (potential price appreciation), with some downside protection.

Max Loss: The maximum loss is limited. The worst that can happen is for the stock price to be above the strike price at expiration, in which case the short stock position can be closed out by exercising the call option. The loss would be the selling price of the stock (where it was sold short), less the purchase price of the stock (the strike price), less the premium paid for the call option.

Max Gain: The maximum gain is limited but potentially substantial. The best that can happen is for the stock to become worthless. In that case, the investor could buy the stock for zero to close out the short stock position. The total profit, however, would be reduced by the premium paid for the call option, which expired worthless.

Are synthetics really useful?

exercise a call option, you give up the rights with the call option in exchange for the stock. exercise a call to gain the stock, you are holding all of the downside risk of the stock in exchange for collecting the dividend. Now that the understand synthetics, we can perhaps find a better way to do this. When the long the call, rather than exercising it, could, instead, sell the same strike put. This creates a synthetic long stock position (since long the call and short the put), which is effectively the same position you were going to have if you exercised the call. While the synthetic long stock position does not allow you to collect the dividend, it does let you collect the premium of the put. In many cases, this put premium will be of greater value than the dividend on the stock while either choice exposes to the same downside risk. In addition, by choosing the synthetic long call position, it

can hold onto the exercise value of the cash a little longer to earn interest. In other words, if you choose to exercise the call, must pay for the stock today.

By entering the synthetic long stock position, you will end up buying the stock at the same price but at a later date. Why? For the synthetic long stock position, you are long the call and short the put. If the stock price is above the strike at expiration, it will exercise the call and buy the stock for the exercise price.

If the stock’s price is below the strike, you will be assigned on the put and buy the stock for the strike price. No matter where the stock’s price is at expiration, it will pay the strike price and receive the stock, which is exactly what you would have done had you exercised the call only at a later date. This allows you to hold onto the cash for a longer period of time to earn interest. As long as the put premium plus interest earned exceeds the value of the dividend, it must be better off with the synthetic long stock position.

This means that it may still be advantageous to use the synthetic stock strategy even if the put premium is less than the dividend. You must compare the put premium plus interest earned to the dividend. Only when the dividend exceeds the value of the put plus interest should you exercise the call for the dividend. Please keep in mind that we are not saying that you should always capture either the dividend or put premium. In many cases, neither choice may warrant holding all of the downside risk of the stock whether synthetically or not. We just mean that if you have decided it is worthwhile, then you should consider the synthetic long stock version before exercising the call.