-20 -15 -10 -5 0 5 10 15 20 25 60 65 70 75 80 85 90 95 Stock Price, $ Pr of it on P osit ion
Profit on Long Stock Position Payoff incl. Premium
So if the stock closes above the strike price, the put expires worthless and the investor is out the premium. The investor breaks even if the stock closes at K-Prem. If the stock closes below this point, the investor profits. The maximum profit that can be made on the put will occur if the stock price goes to zero, in which case the profit = K (less the premium).
American vs. European options
In the scenarios described above, the investor can only exercise the option at the expiry date. These types of options are called “European options.” For these options, though, the investor doesn’t have to wait until expiry to either lock in profits or get out of the position: they can execute a reversing trade, effectively closing out the position. For example if someone owned Jan 80 puts on Enron and that expired in October, but didn’t want to wait that long (perhaps because they thought that there might be a rebound and wanted to lock in profits), they could just sell their put.
There are also options that can be exercised at any time prior to expiry. These are called “American options” and are the most common. They are worth more than their equivalent European options, since the opportunity to exercise at any point is worth something. In fact, it can be shown mathematically that VAmerican ≥VEuropean. Theoretical prices for European options are calculated with the Black-Scholes equation
(shown later). For American options, numerical techniques such as Monte Carlo analysis, trees, or finite difference methods must be used. This is because the price is path-dependent.
What gives an option value?
We will see that the value of an option can be decomposed into two sources of value: intrinsic value and
time value. Intrinsic value is what we have if the option is in the money: for a call, this means that the strike price is below the asset price; for a put, this means the strike price is above the asset price. Even if the options are out of the money (call: strike price above asset price; put: strike price below asset price), they still have value because there is always the chance, given enough time, that the asset price can move sufficiently to place the option into the money (have positive value). The most important factor influencing
Looking again at the table of IBM option prices, we can easily see how much of the price is due to intrinsic value and how much is due to time (and volatility).
Stock Price, last
Expiry Strike Price Call Price
(1) Status of Call Intrinsic Value =max (S- K,0) (2) Time Value (1) – (2)
85.35 Apr 80 5.70 In the Money 5.35 0.35
85.35 Apr 85 2 Near the Money 0.35 1.65
85.35 Apr 90 0.30 Out of the Money 0 0.30
85.35 Apr 95 0.05 Out of the Money 0 0.05
85.35 May 80 7 In the Money 5.35 1.65
85.35 May 90 1.60 Out of the Money 0 1.60
85.35 Jul 80 8.60 In the Money 5.35 3.25
Looking at the series of calls with strike prices of 80,where the intrinsic value is constant, it can be seen that the time value decreases as we get closer to expiry. In fact, the time value decays exponentially. It’s also interesting that the time value for May differs between the 80 and 90 calls. One would expect this to be constant, but remember this also includes effects of volatility. It’s harder for the stock to go to 90 than to 80, where the stock is already in the money. Graphs of time value of European calls and puts follow, with the top curve being the highest time to expiry and the lowest curve the value at expiry.
80 100 120 140 S, $ 10
20 30 40
V, $ European Call Option
50 100 150 200 S, $ 2 4 6 8 10
The option price should be highly correlated with the price of the underlying. This should be the case for options on the same asset as the underlying, such as Lucent puts on Lucent stock. In such a case, when the
option is in the money, the slope of the option curve is +/- 45◦ at maturity: a $1 change in the underlying
causes a $1 change in the option. If there is a mismatch between the underlying and the hedging instrument, basis risk will occur. A cross-hedge is when you want to hedge an underlying, but perfectly correlated derivatives don’t exist, so you do the best you can. An example is using calls on the gold contract to hedge silver, or using the S&P 400 to hedge the S&P 500. The risk-free rate is also an important parameter in option pricing.
What are options used for?
Many users of options are hedgers. They wish to reduce exposure to sources of risk, or correct mismatches between their liabilities and assets. They are usually exposed to sources of risk that they don’t wish to take on, and use options to redistribute this risk to willing parties. These users include market participants such as farmers, commodity purchasers, multinational corporations, airlines, banks and other major creditors. One example of a hedger would be Ford Motor Company, which requires large supplies on precious metals such as palladium. The cost of materials can easily be 50 to 60 percent of their revenue and, because competition squeezes profit margins, Ford is very sensitive to cost. At one time Ford made large purchase of derivatives in an attempt to hedge against price increases of palladium (however, the price moved against them and they lost millions. Angus, the owner of Lucent stock in our example a the beginning of this chapter, would be a hedger if he purchased a protective put to hedge against price declines of his stock. Other users of derivatives are known as speculators. Unlike hedgers, speculators aren’t trying to hedge themselves against future price movements, but rather they’re taking a position on a stock movement. The investor Yukiko described at the beginning of this section is a speculator.
Put-call parity
There is an important and fundamental relationship between the price of the underlying asset and a put and call with the same strike price and time to expiry. If you buy a portfolio consisting of the asset at price S, a put at strike price K and expiry Texpiry, and sell a call with the same strike and expiry, then
rt
Ke c p
S+ - = -
This relationship is called put-call parity and is used for purposes such as detecting arbitrage opportunities and price mismatches, the creation of synthetic securities, and in calculating “fair value” of derivatives prices. (To remember that the equation is “S + p” and not “S–p”, just think “S and P,” as in S&P 500.)
To see this, suppose that you borrow an amount Ke-rt to finance the purchase of one share of the asset at
price S and one put at price p. You sell one call at the same strike price and expiration date as the put. The
money that you borrowed is to be paid back at expiry at the rate r. This is a risk-free position requiring
zero net cash outlay because:
At time t=0, you are long S+p. It is financed with the cash Ke-rt and the call premium c.
At time t=Texpiry, the asset is now at price St and you have to pay back the K dollars you borrowed earlier.
There are three possible states of nature:
(1) The stock price is equal to strike price K. In this situation, the call and put expire worthless and you have to pay back K. But the stock is worth K. Sell the stock and pay the loan. Net cash
Stock: St
Call: 0 Expires worthless since St <K.
Put: K- St
Loan: -K Have to pay back K.
Total: St + (K- St) – K = 0
(3) Stock price St is above strike price K.
Value of portfolio at expiry:
Stock: St
Call: -(St-K) You sold the call so payoff is max(St-K,0).
Put: 0 Expires worthless since St >K.
Loan: -K Have to pay back K.
Total: St -(St-K) – K = 0
No matter what the price of the asset at expiry, you are perfectly hedged. All we need to do to complete this is to figure out the value of the portfolio at the time of purchase and set it equal to zero,
since we can’t have a risk-free profit. We pay back K so must have borrowed Ke-rt. This is equal to
the value of the portfolio S+p-c. If it were not, we would have an arbitrage possibility. Such “no-
arbitrage” arguments are proved by squeezing from both sides as follows:
Case 1: S+p-c > Ke-rt. We would sell the portfolio and invest the proceeds (S+p-c) in the bank.
At the end of the period, our account would be worth (S+p-c)ert. We would have to make good on
our short sale of S and p and purchase of call, but we have already seen that this portfolio is perfectly
hedged and worth K. So we would have a risk-free profit (S+p-c)ert-K > 0.
Case 2: S+p-c < Ke-rt. We would buy the portfolio by borrowing S+p-c. At the end of the
period, we would pay back (S+p-c)ert from our loan, but since the portfolio is worth K, we have a risk-
free profit K - (S+p-c) ert > 0.
With put-call parity, we see how to create synthetic positions. For example, if puts aren’t traded on a certain stock, they can easily be synthesized by solving the put-call parity equation for p:
So you would buy a call and sell the stock short. If you were prohibited from selling your own company’s stock short, like Angus in the discussion at the beginning of this section, you would buy a put, short the call and lend K:
Once we know the strike price, price and volatility of the underlying, time to expiry and either the call or put price calculated from Black-Scholes, we can use put-call parity to get the price of the put or call.
Interview tip:Make sure you can draw the payoff diagrams of put-call parity and identify it when it appears on an interview question. For example, you could be asked a question including the following: “Suppose you borrow money and use it to buy a put and sell a call …” If an interviewer makes this statement in a question, they’re trying to see whether you recognized that they had synthetically created a stock. If Angus wanted to purchase a put on Lucent but it wasn’t available, or maybe a call was available for the month he wanted but a put was not, he could create one synthetically by forming a portfolio of stock, call and risk-free borrowings.
S c Ke p= -rt+ - S Ke c p- - -rt =-