Put-Call Parity

Put-Call Parity     

                                                                                          

The prices of European puts and calls on the same stock with identical exercise prices and expiration dates have a special relationship. The put price, call price, stock price, exercise price, time to expiration, and risk-free rate are all related by a formula called put-call parity. Let us see how this formula is derived.

Imagine a portfolio, called portfolio A, consisting of one share of stock and one European put. This portfolio will require an investment of S₀ + P_e(S₀, T, X). Now consider a second portfolio, called portfolio B, consisting of a European call with the same exercise price and risk-free pure discount bonds with a face value off. That  portfolio will require an investment of C_e(S₀, T, X) + X(1 +r)^-T. Now let us look at what happens at expiration. Table 2.1 presents the outcomes.

Table 2.1

				Payoffs from Portfolio
Portfolio		Current Value	ST ≤ X	ST > X
A	Long put	Pe(S0, T, X)	ST – X	0
	Stock	S0	ST	ST
			X	ST
B	Long call	Ce(S0, T, X)	0	ST – X
	Long bond	X(1 +r)-T	X	X
				ST

The stock is worth S_T regardless of whether S_T is more or less than X. Likewise the risk-free bonds are worth X regardless of the outcome. If S_T exceeds X, the call expires in the money and is worth S_T – X and the put expires worthless. If S_T is less than or equal to X, the put expires in-the-money worth S_T – X and the call expires worthless.

The total values of portfolios A and B are equal regardless of the outcome of the stock       price. However, given the Law of One Price, the current value of the two portfolios must be equal. Thus we require that:

   S₀ + P_e(S₀, T, X) = C_e(S₀, T, X) + X(1 +r)^-T

This statement is referred to as put-call-parity and it is probably one of the most important results in understanding options. It says that a share of stock plus a put is equivalent to a call plus risk-free bonds. It shows the relationship between the call and put prices, the stock price, the time to expiration, and the exercise price.

Suppose the combination of the put and the stock is worth less than the combination of the call and the bonds. Then you can create an arbitrage portfolio by buying the put and stock and short selling the call and the bonds. Selling short a call just means writing the call, and selling short the bonds simply means to borrow the present value of X and promise to pay back X at the option's expiration. Since everyone would start doing this transaction, the prices would be forced back in line with the put-call parity equation.

By observing the signs in front of each term, we can easily determine which combinations replicate others. If the sign is positive, we should buy the option, stock or bond. If the sign is negative, we should sell. For example, suppose we isolate the call price:

      C_e(S₀, T, X) =  S₀ + P_e(S₀, T, X) - X(1 +r)^-T

Then owning a call is equivalent to owning a put, owning the stock, and selling the bonds (borrowing). If the stock pay dividends, once again, we simply insert

which is the stock price minus the present value of the dividends.

While put-call parity is an extremely important and useful result, it does not hold so neatly if the options are American. The put-call parity rule for American options must be stated as inequalities:

where S’₀ is, again, the stock price minus the present value of the dividends.

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The Effect of Stock Volatility

The Effect of Stock Volatility

The effect on a pat's price is the same as that for a call: Higher volatility increases the possible gains for a put holder. This is because greater volatility increases the gains on the put if the stock price increases, because the stock price can drop below the exercise price by a larger amount. On the other hand, greater volatility means that if the stock price goes up, it can also be much higher than the exercise price. To a put holder, however, this does not matter because the potential loss is limited, it is said to be truncated at the exercise price.

The Effect of Interest Rates

The Effect of Interest Rates

In contrast to call option prices, which vary directly with interest rates, put option prices vary inversely with interest rates. Purchasing a put is like deferring the sale of the stock. When you finally sell the stock by exercising the put, you receive X dollars. If interest rates increase, the X dollars will have a lower present value. Thus, a put holder forgoes higher interest while waiting to exercise the option and receive the exercise price. Higher interest rates make puts less attractive to investors.

The Early Exercise of American Puts

The Early Exercise of American Puts

Let us suppose there are no dividends. Suppose you hold an American put and the stock goes bankrupt, meaning that the stock price goes to zero. You are holding an option to sell it for X dollars. There is no reason to wait until expiration to exercise it and obtain your X dollars. You might as well exercise it now. Thus, bankruptcy is one obvious situation in which an American put would be exercised early. However, bankruptcy is not required to justify early exercise. If the stock price falls to a critical level - and thus cannot fall much further - an American option might be exercised early and the funds reinvested.

If the stock pays dividends, it might still be worthwhile to exercise it early, but because dividends drive the stock price down, they may make American puts less likely to be exercised early. In fact, if the dividends are sufficiently large, it can sometimes be shown that the put would never be exercised early, thus making it effectively a European put.

American Put Versus European Put

American Put Versus European Put

Everything that can be done with a European put can be done with an American put. In addition, an American put can be exercised at any time prior to expiration.

Therefore, the American put price must at least equal the European put price, that is

                          

   P_a(S₀, T, X) ≥   P_e(S₀, T, X)

The Lower Bound of a European Put

The Lower Bound of a European Put

We showed that the minimum value of an American put is Max(0, X-S₀). This statement does not hold for a European put, because it cannot be exercised early. The price of a European put must at least equal the greater zero or the present value of the exercise price minus the stock price:

   P_e(S₀, T, X) = Max(0, X(1+r)^-T - S₀)

In the following figure, the curved line is the European put price, which must lie above the lower bound, As time to expiration gets smaller so the lower bound moves to the right, with the put price curve following it. However, as time goes by, the put price gradually declines. At expiration, the put price and the lower bound converge to Max(0, X - S₀).   

Finally we should note that if the stock pays dividends such that the stock price minus the present value of the dividends is

Then the lower bound is:

   P_e(S₀, T, X) = Max(0, X(1+r)^-T – S’₀)

The Effect of Exercise Price

The Effect of Exercise Price     

                      

Consider two European puts that are identical in all respects except that the exercise price of one is X₁ and the other is X₂, where X₂ > X₁.

We want to know which price is greater: P_e(S₀, T, X₁) or P_e(S₀, T, X₂). To cut a long story short, the price of a European put must be at least as high as the price of a European put must be at least as high as the price of an otherwise identical European call with a lower exercise price:

   P_e(S₀, T, X₂) ≥ P_e(S₀, T, X₁).

The intuition behind why a put with a higher exercise price is more is quite simple. A put is an option to sell an asset at a fixed price. The higher the price the put holder can sell the asset, the more attractive the put.

The same relationship holds for American puts. The price of an American put must be at least as high as the price of another otherwise identical American call with a lower exercise price:

   P_a(S₀, T, X₂) ≥ P_a(S₀, T, X₁).

It follows from this result that the difference in prices of two European puts that differ only by exercise price cannot exceed the present value of the difference in their exercise prices, that is:

   (X₂-X₁)(1+r)^-T ≥ P_e(S₀, T, X₂) - P_e(S₀, T, X₁)

Also, the difference in the prices of two American puts that differ only by exercise price cannot exceed the difference in their exercise prices:

   (X₂-X₁) ≥ P_a(S₀, T, X₂) - P_a(S₀, T, X₁)

The Effect of Time to Expiration

The Effect of Time to Expiration

Consider two American puts, one with a time to expiration of T₁ and the other with time to expiration of T₂, where T₂ > T_1.

Now assume today is the expiration date of the shorter-lived put. The stock price is S_T1. The expiring put is worth Max(0, X – S_T1). The other put, which has a remaining time to expiration of T₂ - T₁, is worth at least Max(0, X – S₀). Consequently, it must be true that at time 0, we must have:

                                      P_a(S₀, T₂, X) ≥ P_a(S₀, T₁, X)

Note that the two puts could be worth the same, however, this would only occur if both puts were very deep-in or out-of-the-money. The principles that underlie the time value of a put are the same as those that underlie the time value of a call. As the following figure shows, as expiration approaches, the put price curve approaches the intrinsic value, which is due to time value decay. At expiration the put price equals the intrinsic value.

The Maximum Value of a Put

The Maximum Value of a Put

At expiration, the payoff from a European put is Max(0, X- S₀).  The best outcome that a put holder can expect is for the company to go bankrupt. In that case, the stock will be worthless and the put holder will be able to sell the shares to the put writer for X dollars. Thus the present value of the exercise price is the European pat's maximum possible value. Since an American option can be exercised at any time, its maximum value is the exercise price:

                                      P_e(S₀, T, X) ≥ X(1+r)^-T

                                      P_a(S₀, T, X) ≥ X

On the put's expiration date, no time value will remain. Expiring American puts       therefore are the same as European puts. The value of either type of put must be the intrinsic value:

                                      P(S₀, T₁, X) = Max(0, X – S₀)

The Minimum Value of a Put

The Minimum Value of a Put

A put is an option to sell a stock. A put holder is not obligated to exercise it and will not do so if exercising will decrease wealth. Thus, a put can never have negative value:

                                      P(S₀, T, X) ≥ 0

An American put can be exercised early. Therefore:

                                      P(S₀, T, X) ≥ Max(0, X- S₀)

The value, Max(0, X- S₀), is called the put's intrinsic value. An in-the-money put has a positive intrinsic value, while an out-of-the-money put has an intrinsic value of zero. The difference between the put price and the intrinsic value is the time value speculative value. The time value is defined as P(S₀, T, X) - Max(0, X- S₀). As with calls, the time value reflects what an investor is willing to pay for the uncertainty of the final outcome.

Principles of Put Option Pricing

Principles of Put Option Pricing   

Many of the rules that apply to call option pricing apply in a straightforward manner to put options. However, there are some significant differences.

The Effect of Stock Volatility

The Effect of Stock Volatility

One of the basic principles of investor behavior is that individuals prefer less risk to more. For holders of stocks, higher risk means lower value. But higher risk in a stock translates into greater value for a call option on it. This is because greater volatility increases the gains on the call if the stock price increases, because the stock price can exceed the exercise price by a larger amount. On the other hand, greater volatility means that if the stock price goes down, it can be much lower than the exercise price. To a call holder, however, this does not matter because the potential loss is limited, it is said to be truncated at the exercise price. In fact, die option holder will not care how low the stock price can go. If the possibility of a lower stock price is accompanied by the possibility of higher stock prices, the option holder will benefit, and the option will be priced higher when the volatility is higher.

The Effect of Interest Rates

The Effect of Interest Rates

Interest rates affect a call option's price. Although the effect of interest rates is somewhat complex, an easy way to think of it is to think of a call as a way to purchase stocky paying an amount of money less than the face value of the stock. By paying the call premium, you save the difference between the stock price and the exercise price, the price you are willing to pay for the stock. The higher the interest rate, the more interest you can earn on the amount saved by buying a call. Thus, when interest rates are higher, calls are more attractive than buying stock.

American Call versus European Call

American Call versus European Call

Many of the results so far only apply to European calls. In many cases, however, American calls behave exactly like European calls. In fact, an American call can be viewed as a European call with the added feature that it can be exercised early. Since exercising an option is never mandatory an American call will be at least as valuable as a European call with the same terms:

                                                  C_a(S₀, T, X) ≥ C_e(S₀, T, X)

We already proved that the minimum value of an American call is Max(0, S₀ - X) while the lower bound of a European call is Max(0, S₀ - X(1+r)^-T). Because S₀ - X(1+r)^-T is greater than S_O - X, the lower bound value of the American call must also be Max(0, S₀ - X(1+r)^-T).

With the lower bound of an American call established, we can now examine whether an American call should ever be exercised early. If the stock price is S_o, exercising the call produces a cash flow of S_O – X. Since the cash flow from exercising, S_O - X, can never exceed the call's lower bound, S₀ - X(1+r)^-T, it will always be better to sell the call in the market.

When the transaction cost of exercising is compared to the transaction cost selling a call, the argument that a call should not be exercised early is strengthened.