# What is the graph of hedging using put options?

An investor owns 1,000 Microsoft shares. The price is \$28 per share. He is concerned about a possible share price decline in the next 2 months and wants protection. Buying ten July \$27.50 put option contracts would give him the right to sell 1,000 shares for \$27.50.

If the put premium is \$1 (\$100 cost per put)then the total cost of the hedging strategy would be 10 x 100 = \$1,000.

This strategy guarantees that the shares can be sold for \$27.50 per share during the life of the option. If the price of Microsoft falls below \$27.50, the options can be exercised so that \$27,500 is realized for the shares. When the cost of the options is taken into account, the amount realized is \$26,500. If the market price stays above \$27.50, the options will expire worthless.

Figure 1.4 shows the net value of the portfolio (after taking the cost of the options into account) as a function of Microsoft’s stock price in 2 months. The dotted line shows the value of the portfolio assuming no hedging. The graphs below are those that I have been studying and as you can see the shape of the graph of the exercise corresponds to the shape of a long call position.

Can somebody explain why the graph looks like this? I understand the explanation of the exercise and how it's helpful to hedge using a put strategy but I think that the shape of the graph (the solid Hedging line) is the shape of a long CALL option payoff. Shouldn't it be that of a long put option payoff since it is a long put? The technical answer to you question involves the Synthetic Triangle:

There are six basic synthetic positions:

1. Synthetic Long Stock = Long Call + Short Put
1. Synthetic Short Stock = Short Call + Long Put
1. Synthetic Long Call = Long Stock + Long Put
1. Synthetic Short Call = Short Stock + Short Put
1. Synthetic Short Put = Short Call + Long Stock
1. Synthetic Long Put = Long Call + Short Stock

They can be thought of as a 'synthetic triangle' of CALL, PUT, and STOCK. A combination of two elements in the synthetic triangle creates a synthetic position of the third element.

This synthetic triangular relationship is governed by the principle of Put Call Parity (requires the extrinsic values (aka time value) of call and put options to be in equilibrium so as to prevent arbitrage).

In order for the synthetic triangle relationship to work, all options used together must be of the same expiration, strike and represent the same amount of shares used in combination.

All combinations of synthetics can be created by the following formula:

S + P - C = 0

As to your question: Can somebody explain why the graph looks like this?

If you want to find out the synthetic equivalent to a put protected stock (S + P), isolate that +C position:

S + P = C

Your example involves owning 1,000 shares of MSFT and 10 July \$27.50 puts. This is equivalent to owning 10 July \$27.50 calls.

Since synthetics may be confusing at this point, to demonstrate this, just calculate the value of the stock position and the put position at \$20 and add them. Repeat at various intervals up to say \$35 and you'll get the exact same Hedging graph that you provided in Figure 1.4

• Another way to prove the equivalence -- both positions give you a choice (at a price of \$27.50) of having the stock or not Nov 18, 2019 at 18:52
• @Ben Voigt - While your explanation is true, it would be meaningless to someone who does not understand synthetics. They would say "Yes, that's true" but it would be insufficient information for them to extrapolate across the entire price spectrum. They wouldn't understand the underpinnings. Nov 18, 2019 at 19:10
• I am having some troubles understanding your explanation im sorry...i never studied synthethics. does it mean that my graph in 1.4 Synthetic Short Put = Short Call + Long Stock? Is there any way other way you can explain why figure 1.4 does not look like a long put position ?
– GGGG
Nov 18, 2019 at 19:11
• Factor the equation and you get S+P=C. That means that owning the stock and the put is syn equiv to owning the call. S+P is the synthetic. Let's simplify to get rid of the cents. You buy 100 MSFT at \$30 and a July \$30 put for \$1. I buy the Jul \$30 call for \$1. At \$25, we both are down \$1. At \$31, both are even. At \$35, both have \$4 gain. In this example, the two positions are identical. In real life it gets a little more complicated because there is a carry cost for owning the stock and the premiums shift if there is a pending dividend (same series put price expands and call price contracts). Nov 18, 2019 at 19:27
• Is there any way you can explain it without synthetics?
– GGGG
Nov 18, 2019 at 19:37