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The Put-Call Parity Formula
We have shown that the market maker’s three-sided position (conversion) is guaranteed to be worth the present value of the exercise price. No matter what happens to the stock’s price, the market maker is guaranteed to receive the $50 exercise price at expiration. Because he’s guaranteed the $50 exercise price, the long stock + long put + short call position must be worth the present value of the exercise price ($50 in this example).
 
If we use S for stock price, P for put price, C for call price, and Pv (E) for the present value of the exercise price, then we can write the relationship of long stock + long put + short call equals the present value of the exercise price more concisely:
 
Formula 5-1:
S + P – C = Pv (E)
 
Notice the use of plus and minus signs. The long positions are denoted by a plus sign while any short positions are indicated by a minus sign. (If no sign precedes a letter such as with the “S” then it is assumed to be a plus sign.) This formula is stating that values of long stock + long put + short call are equal to the present value of the exercise price. Using our previous example, Formula 5-1 states that $50,000 + $3,000 – $3,980 = $49,020. Formula 5-1 is one variation of the put-call parity. If Formula 5-1 or any of the upcoming algebraic variations do not hold in the real world then arbitrage is possible.
 
Let’s see if the formula works by using the above example. If we have a portfolio consisting of stock purchased at $50, a long $50 put, and a short $50 call, what will happen at expiration to the value of the portfolio at different stock prices?
 
Table 5-2
Portfolio
Stock price
Stock
$50 put
$50 Call
Total Value
35
35
15
0
50
40
40
10
0
50
45
45
5
0
50
50
50
0
0
50
55
55
0
5
50
60
60
0
10
50
65
65
0
15
50
 
The last column in Table 5-2 shows the total value of the portfolio at expiration, which is found by adding the “stock” column to the “$50 put” column and then subtracting the “$50 call” column, which is what Formula 5-1 tells us to do. Reading the top row, for example, if the stock price is $35 at expiration (column one) then your stock position must also be worth $35 (column two). The $50 put would be worth $15 (column three) and the $50 call would be worthless (column four). This means that your total portfolio would be worth $35 + $15 + $0 = $50 if the stock closes at $35.
 
The “total value” column shows us that this portfolio always sums to $50 no matter what happens to the stock’s price at expiration, and that is exactly what we expected to happen. Of course, these are not the only stock prices that could occur at expiration. But you can check for yourself that any stock price results in a final portfolio value of exactly $50.
 
If the value of this package of long stock, long put, and short call is guaranteed to be worth $50 at expiration then today, that package must be the present value of $50, which is what Formula 5-1 tells us.
 
As stated before, Formula 5-1 is just one variation of the put-call parity formula. We can rearrange it algebraically and come up with other forms that will provide different views of the pricing connections between calls and puts. We will present some of the more interesting forms but, as you read through these, don’t worry so much about the formulas as much as the insights they provide. Formula 5-3 shows an algebraic rearrangement of Formula 5-1 that was created by adding the call option (C) to both sides of the equation:
 
Formula 5-3:
S + P = C + Pv (E)
This formula tells a very interesting story. The left-hand side of the equation represents an investor who owns stock plus a protective put. The right-hand side represents an investor who owns a call option plus a deposit of cash, Pv (E), which will grow to the exercise price at expiration by earning the risk-free rate. You can think of the deposit of cash as an investment into a Treasury bill or CD or other guaranteed security. Formula 5-3 tells us that an investor who buys stock and a put is financially doing the same thing as someone who buys a call and deposits sufficient cash to grow to the exercise price at expiration by earning the risk-free rate. In other words, owning stock with the right to sell it is financially the same thing as someone who has enough cash at expiration with the right to buy stock.
Let’s see if that’s true by comparing two investors; one owns the stock + put (left side of equation), while the other owns a call plus the present value of the exercise price in cash (right side). If the stock is above the exercise price at expiration the investor on the left side of the equation will let the put expire worthless and be left with the stock. The investor on the right side will exercise the call and pay for the stock with the cash that has grown to the exercise price. In other words, both investors will be holding the stock. However, if the stock price is below the exercise price at expiration, the investor on the left-hand side of the equation will exercise the put and receive the strike price in cash. The investor on the right side will let the call expire worthless and be left holding an amount of cash equal to the exercise price. No matter what happens to the stock’s price, both investors are equal at expiration.
What’s most interesting about Formula 5-3 is that it refutes one of the most persistent myths in options trading. That is, most brokerage firms view investors who buy stock and a put as insurance (left hand side of equation) as being responsible, conservative investors while those who buy calls (right hand side of equation) as being risk takers and reckless. Formula 5-3 shows that as long as the call buyer has enough cash to buy the stock at expiration, both investors are doing exactly the same thing. Both investors must be conservative or both must be reckless but not one of each. Yet most brokerage firms maintain a split view on each even though they are identical strategies.
Let’s take a look at another variation:
Formula 5-4:
C – P = S – Pv (E)
 
Formula 5-4 is saying that the difference between the same-strike call and put prices (left side of equation) must be separated by the same difference as the stock and present value of the exercise price (right side). In Chapter Two, Pricing Principle #4 showed us that the right hand side, S – Pv (E), is the minimum value for a call option. This variation tells us that the call and put prices must be separated by an amount equal to the minimum value for a call option.
 
For example, assume the stock is $50 and interest rates are 5% with one year to expiration. The cost of carry on the stock is $50 * .05 = $2.50. In other words, if you buy the stock, you will miss out on $2.50 worth of interest in one year. In turn, the value of this missed opportunity is $2.50/1.05 = $2.38 today.
 
Formula 5-4 tells us that the difference between the call and put prices must be at least $2.38. If you were looking at a $50 call and a $50 put with one year to expiration and 5% risk-free interest rates, you would find that the call price is $2.38 higher than the put price. The volatility of the underlying stock changes the total prices but their difference will be exactly $2.38. For instance, if the put were priced at $2 then the call must be $2 + $2.38 = $4.38. If the put were worth $10 then the call must be worth $10 + $2.38 = $12.38.
 
What happens if the stock and strike prices are not equal? Assume the stock is $55, the strike is $50 ($5 intrinsic value), and interest rates are 5% with one year to expiration. Formula 5-4 tells us that the difference between the call and put prices must be $55 – ($50/1.05) = $7.38. This price arises from the fact that we must immediately pay for the $5 intrinsic value plus the cost of carry, or $5 + $2.38 = $7.38. Hopefully, you’re starting to see that call and put prices are tied together.
 
Formula 5-4 shows that all call options must be priced higher than the puts by the cost of carry on the exercise price. New traders often confuse this relationship and think that call options are priced higher by the cost of carry on the stock price. The reason it is the cost of carry on the exercise price and not the stock price is because it is the exercise price that is effectively being borrowed when you buy a call option. If you buy a call and exercise it to get the stock then you get to defer the payment of the exercise price until expiration. It is the exercise price that determines the cost of carry.
 
To be continued……

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