Dec

25

Options 101 #60

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Put-call parity lends many insights into option pricing and theory. But it goes far beyond theory because there is a practical application to the formula that is used by all professional traders. It is the formula that provides the foundation for synthetic options.

Synthetic Options
Synthetic options are not a type of option such as calls or puts. As the name implies, synthetic options are ways of creating positions that look, feel, and behave like one asset but are constructed from entirely different assets. By using the put-call parity equation, we can create one asset from another. The benefit to us is that one form may be more liquid or more efficient than another.

For example, let’s say we want to create a synthetic long call option. All we have to do is refer to any of our put-call parity formulas and algebraically solve for the call option. We might choose to use our original formula:

S + P – C = Pv (E)

If we are to solve for the call option, we must get C to one side of the equal sign with the other variables on the other. Using the above formula, in order to get the “C” by itself, we must add C to both sides and then subtract Pv (E) from both sides. This result is:

C = S – Pv (E) + P

This is the same formula we saw earlier in Formula 5-6. In other words, if you borrow the present value of the exercise price and buy stock plus buy a put, you have exactly created a call option – you have created a synthetic call option.

When we say that one position is the synthetic equivalent of another, we really mean that the two positions have exactly the same profit and loss profiles at expiration. Let’s see if these two positions are truly equal. Assume one investor buys a $50 call while another holds Portfolio A, which contains long stock at $50, long $50 put, and borrows the present value of the $50 exercise price. The borrowed funds mature to $50, which means the investor holding Portfolio A owes $50 at expiration. According to Formula 5-6, the profits and losses to either investor should be identical at expiration regardless of the stock’s price. Table 5-11 compares the two investors:

For example, if the stock price is $35 at expiration, the $50 call is worthless. Portfolio A has stock worth $35 and a long put worth $15 for a total of $50. However, the holder of Portfolio A must also repay $50 at expiration, which leaves him with zero. Notice that the two bolded columns have exactly the same values. No matter which stock prices we might try, both columns will always match. This shows that a call option can be synthetically created by borrowing the present value of the exercise price to buy stock and then protecting that investment with a put option. So if the two positions are equal, which should we choose? That all depends on which is more cost effective.

For instance, in Figure 5-12, Google (GOOG) was trading for $293 and the January $290 calls with 529 days to expiration (1.47 years) were asking $58. The $290 put was asking $39.40.

Our synthetic relationship tells us there is no difference between the long call and long stock + long put + borrowed funds portfolio. However, many investors get trapped into thinking that the $290 call is obviously a better deal at $58 rather than spending $293 for the stock plus $39.40 for the put. If you do not plan to purchase the stock at expiration then it may be best to spend less money and buy the $290 call. However, if you are planning to buy the stock at expiration, we need to find out which is our most efficient method. If you buy the call and the stock price is greater than $290 at expiration, you will pay $58 today for the call and $290 in 1.47 years when you exercise the call. Alternatively, you could buy the stock for $293 and the $290 put for $39.40 today and accomplish the same thing. While it may appear that the call is the better deal, we need to consider the financing rates.

If you buy the stock plus put combination it will cost you $293 + $39.40 = $332.40 today. If you buy the call, it will only cost $58 today. However, if plan to buy the stock at expiration, you will have a payment of $290 due in 1.47 years. The question we need to answer is this: Is it cheaper to pay $332.40 today or $58 today and $290 nearly a year-and-a-half in the future? In order to compare these two cash flows, we need to line them up at the same time. In this example, it’s easiest to line them up at today’s prices. We know the stock + put will cost $332.40 today and the call will cost $58, which is a difference of $274.40. Now we just need to figure out which interest rate makes a future payment of $290 equal to $274.40 today. In other words, if we deposit $274.40 into a risk-free account today, what risk-free interest rate is necessary for our money to grow to $290 at expiration?

We can solve this from the basic principle and interest relationship:

interest = principal * rate * time

In this example, the interest is the difference between the $290 future value and the $274.40 present value, or $15.60:

$15.60 = $274.40 * r * 1.47

r = $15.60 / ($274.40 * 1.47)

r = .0387, or 3.87%.

If your broker pays a risk-free rate of 3.87% on cash balances then there is no difference between buying the call or the stock plus put combination. If your broker pays more than 3.87%, you should keep your cash and buy the call. If your broker pays less, you should buy the stock and put combination. So while the cheapest combination today may seem to make the most sense, that may not be true once we consider the financing rates.

Whenever we consider a true synthetic equivalent, we must consider all four variables in the put-call parity equation. However, in the world of trading, synthetic positions usually ignore the present value of the exercise price. This means that synthetic positions are usually not calculated on total values but, instead, they are figured out on net changes between the two positions. For instance, let’s look at our previous positions but this time we’re not going to account for the borrowed funds.

In other words, we’re going to make Portfolio A the combination of long stock and a long put and forget about the borrowed funds. Table 5-13 shows that the two portfolios are not equal in terms of total value. Read the first row. If the stock price is $35 at expiration, the $50 call is worth zero while the stock plus put combination of Portfolio A is worth $50. Read the last row. If the stock is $65, the $50 call is worth $15 while the stock and put combination for Portfolio A are worth $65 so they are clearly not equal. However, look at the “total value” column for Portfolio A. If you consider the net changes for this column in relation to the “total value” column, you’ll find they match the expiration values for the $50 call and are shown in the last column:

So if our only concern is to make the net changes of two positions behave the same then we can forget about the borrowed funds. This means that any synthetic position can be calculated by only considering the stock, put, and call in the put-call parity equation.

Our first version of put-call parity was:

S + P – C = Pv (E)

We can rewrite this by subtracting Pv (E) from both sides:
Formula 5-14:
S + P – C – Pv (E) = 0

Formula 5-14 is just another variation of put-call parity. It tells us that if we buy stock, buy a put, sell a call, and borrow the present value of the exercise price, we are effectively doing nothing. This also means this portfolio should cost nothing. If we are going to ignore the borrowed funds portion, we can just drop off the Pv (E) variable to get:
Formula 5-15:
S + P – C = 0

And it is Formula 5-15 that provides our basic synthetic option formula. Notice that the combination of long stock, long put, and a short call is a conversion and are the same assets used by the market maker to create the risk-free position that we started with in this chapter. Consequently, this is sometimes referred to as a synthetic T-bill (Treasury Bill). If you buy stock, buy a put, and sell a call, you are guaranteed to receive more money than you spent – just as if you purchased a T-bill.

To be continued…….

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