Put-Call Parity & Option Pricing

02/01/2013 8:00 am EST


Alan Ellman

President, The Blue Collar Investor Corp.

This article has a few advanced concepts that I hope you find interesting but is not necessary to master or memorize, writes Alan Ellman of TheBlueCollarInvestor.com. Some of the material may make your head spin a bit as it did mine as I was writing the article!

Definition and Example
Put call parity is an option pricing concept that requires the time (extrinsic) values of call and put options to be in equilibrium so as to prevent arbitrage (Arbitrage is the simultaneous purchase and sale of an asset in order to profit from a difference in the price).

It is when the value of a call option, at one strike price, implies a certain fair value for the corresponding put, and vice versa. The argument, for this pricing relationship, relies on the arbitrage opportunity that results if there is difference between the value of calls and puts with the same strike price and expiration date. Arbitrageurs would step in to make profitable, risk-free trades until the departure from put-call parity is eliminated. This relationship is strictly for European-style options, but the concept also applies to American-style options, adjusting for dividends and interest rates. If the dividend increases, the puts expiring after the ex-dividend date will rise in value, while the calls will decrease by a similar amount. Changes in interest rates have the opposite effects. Rising interest rates increase call values and decrease put values.

Put-call parity graph
Click to Enlarge

The above chart depicts a put-call parity relationship. We see that a long-stock/long-put position (red line) has the same risk/return profile as a long call (blue line) with the same expiration and strike price. The only difference between the two lines is the assumed dividend that is paid during the time to expiration. The owner of the stock (red line) would receive the additional amount, while the owner of the call (blue line) would not. However, if we assume no dividend would be paid to stockholders during the holding period, then both lines would overlap. From this chart one can see that if you sold a cash-secured put instead of writing a covered call, the put-call parity concept would account for that dividend difference.

Click to Enlarge

When you buy a call, your loss is limited to the premium paid while the possible gain is unlimited. Now, consider the simultaneous purchase of a long put and 100 shares of the underlying stock. Once again, your loss is limited to the premium paid for the put, and your profit potential is unlimited if the stock price goes up.

Put Call Parity: An Example

BCI trading @ $50/share
$50 call = $2
$50 put = $2
At expiration BCI still trading @ $50

The Trades:
Buy 1 x $50 call for $200
Buy 100 x BCI and 1 x $50 put for $700

Calls expire worthless = $200 loss
Puts expire worthless, no loss or gain on shares = $200 loss

Since put call parity exists, the net losses on both positions are exactly the same so that there are no arbitrage opportunity to be exploited by being long one position and shorting the other simultaneously.

NEXT PAGE: Put-Call Parity Rules

Why Are Call Premiums Sometimes Higher Than Put Premiums?
Look at many option prices in which the stock price is close to the strike price. You are likely to see that put premiums are lower than call premiums (a dividend distribution prior to expiration will have the opposite effect). Are puts cheaper than calls? In fact, the time premiums of puts and calls at the same strike price (and the same expiration) are theoretically the same. Why then do call premiums usually appear to be higher?

The answer is that with the stock equal to the strike, the calls are considered in-the-money and the puts are out-of-the-money. This is because the real price of the underlying is the stock’s future delivery price, which is determined by the stock’s dividend and the current interest rate in addition to the stock price. If the interest rate is higher than the dividend rate (as it usually is) then the stock’s future delivery price will be higher than its current price.

As an example, if you are a market maker and you need to buy the stock (and guarantee a price) for one-year delivery (for a call we are writing), you need to borrow the funds (at the one-year interest rate) to buy the stock. This increases your actual price of the stock. However, you also collect the dividends (if any) on the stock. This will reduce your price. Thus, if the current stock price is $100, the one-year interest rate is 5% and the dividend rate is 1% annually, then the real cost of the stock for a one-year option is $104.

The equation therefore is:

Effective stock price = current stock price plus Interest minus dividend

An arbitrage opportunity: Why put and call prices are equal

If call or put time premiums get out of line with each other, option market makers can make a risk-free arbitrage profit. Here’s an example:

Let’s assume that the time premium of a particular put is greater than the time premium of the corresponding call (same stock, strike and expiration). Here the market maker can take advantage of the difference in the time premiums by selling the put and buying the call. What the market maker has done is to create a “synthetic” long stock position with a free credit of premium. The market maker can fully offset his risk by selling the stock. The net credit of time premium from this transaction will be his arbitrage profit.

Put-Call Parity Rules: Synthetic Trades (This is the head-spinning part!)
From the above paragraph, you get the first of the put/ call parity rules: If you buy a call and sell a put at the same strike price and expiration, you get the equivalent of a long stock position. These are the 6 rules:

Rule 1: (+) Buy Call & (-) Write Put = (+) Buy Stock
Rule 2: (-) Write Call & (+) Buy Put = (-) Short Stock
Rule 3: (+) Buy Stock & (+) Buy Put = (+) Buy Call
Rule 4: (-) Short Stock & (-) Write Put = (-) Write Call
Rule 5: (-) Short Stock & (+) Buy Call = (+) Buy Put
Rule 6: (+) Buy Stock & (-) Write Call = (+) Write Put = Covered Call

Notice that if you move any of these transactions to the other side of the equal sign, you change the strategy from a short to a long (or vice versa) and the “+” sign to a “-” sign (or vice versa).

NEXT PAGE: Put-Call Parity in the Real World

Practical Application of Put-Call Parity
Since the market maker has a clear advantage in trading options, it is unlikely that the average blue collar investor will capture profit from “risk-free” arbitrage opportunities. However, these rules are helpful, because knowing them can help you determine what your strategy alternatives are. Here are two examples:

  1. Let’s say you see attractive covered call opportunities, where the call is in-the-money. Here you may be reluctant to establish the position for fear of having the call exercised and perhaps be susceptible to tax liability. Here you can write a cash-secured put instead of the covered call (Rule 6).

  2. 2. Let’s assume you are interested in buying a particular stock, but you are concerned about the risk and may want to hedge it with a put. Knowing rule 3 tells you that buying the call will give you the same risk/reward as buying the stock and the put.

Put-Call Parity in the Real World
As you might have noticed above, put-call parity requires that the extrinsic (time) value of call and put options of the same strike price to be the same. However, in reality, the extrinsic value of put and call options are rarely in exact parity in option trading even though market makers have been charged with the responsibility of maintaining put-call parity. When the outlook of a stock is bullish, the time value of call options tend to be higher than put options due to higher implied volatility and when the outlook of a stock is bearish, the time value of put options tends to be higher than call options. When the put option is valued much higher than the call, look for a dividend distribution to be the reason why.

However, such option trading arbitrage opportunities are rare even under put-call disparity as the presence of bid/ask spreads and commissions tends to neutralize gains from such disparities.

A put-call parity is one of the foundations for option pricing, explaining why the price of one option can’t move very far without the price of the corresponding options changing as well. So, if the parity is violated, an opportunity for arbitrage exists. Arbitrage strategies are not a useful source of profits for the average trader.

By Alan Ellman of TheBlueCollarInvestor.com

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