Q http://www.investopedia.com/articles/optioninvestor/03/121003.asp Dividends, Interest Rates And Their Effect On Stock Options By Jim Graham | December 29, 2003 Share While the math behind options-pricing models may seem daunting, the underlying concepts are not. The variables used to come up with a "fair value" for a stock option are the price of the underlying stock, volatility, time, dividends and interest rates. The first three deservedly get most of the attention because they have the largest effect on option prices. But it is also important to understand how dividends and interest rates affect the price of a stock option. These two variables are crucial to understanding when to exercise options early. Black Scholes Doesn't Account for Early Exercise The first option pricing model, the Black Scholes model, was designed to evaluate European-style options, which don't permit early exercise. So Black and Scholes never addressed the problem of when to exercise an option early and how much the right of early exercise is worth. Being able to exercise an option at any time should theoretically make an American-style option more valuable than a similar European-style option, although in practice there is little difference in how they are traded. Different models were developed to accurately price American-style options. Most of these are refined versions of the Black Scholes model, adjusted to take into account dividends and the possibility of early exercise. To appreciate the difference these adjustments can make, you first need to understand when an option should be exercised early. And how will you know this? In a nutshell, an option should be exercised early when the option's theoretical value is at parity and its delta is exactly 100. That may sound complicated, but as we discuss the effects interest rates and dividends have on option prices, I will also bring in a specific example to show when this occurs. First, let's look at the effects interest rates have on option prices, and how they can determine if you should exercise a put option early. The Effects of Interest Rates An increase in interest rates will drive up call premiums and cause put premiums to decrease. To understand why, you need to think about the effect of interest rates when comparing an option position to simply owning the stock. Since it is much cheaper to buy a call option than 100 shares of the stock, the call buyer is willing to pay more for the option when rates are relatively high, since he or she can invest the difference in the capital required between the two positions. When interest rates are steadily falling to a point where the Fed Funds' target is down to around 1.0% and short-term interest rates available to individuals are around 0.75% to 2.0% (like in late 2003), interest rates have a minimal effect on option prices. All the best option analysis models include interest rates in their calculations using a risk-free interest rate such as U.S. Treasury rates. Interest rates are the critical factor in determining whether to exercise a put option early. A stock put option becomes an early exercise candidate anytime the interest that could be earned on the proceeds from the sale of the stock at the strike price is large enough. Determining exactly when this happens is difficult, since each individual has different opportunity costs, but it does mean that early exercise for a stock put option can be optimal at any time provided the interest earned becomes sufficiently great. The Effects of Dividends It's easier to pinpoint how dividends affect early exercise. Cash dividends affect option prices through their effect on the underlying stock price. Because the stock price is expected to drop by the amount of the dividend on the ex-dividend date, high cash dividends imply lower call premiums and higher put premiums. While the stock price itself usually undergoes a single adjustment by the amount of the dividend, option prices anticipate dividends that will be paid in the weeks and months before they are announced. The dividends paid should be taken into account when calculating the theoretical price of an option and projecting your probable gain and loss when graphing a position. This applies to stock indices as well. The dividends paid by all stocks in that index (adjusted for each stock's weight in the index) should be taken into account when calculating the fair value of an index option. Because dividends are critical to determining when it is optimal to exercise a stock call option early, both buyers and sellers of call options should consider the impact of dividends. Whoever owns the stock as of the ex-dividend date receives the cash dividend, so owners of call options may exercise in-the-money options early to capture the cash dividend. That means early exercise makes sense for a call option only if the stock is expected to pay a dividend prior to expiration date. Traditionally, the option would be exercised optimally only on the day before the stock's ex-dividend date. But changes in the tax laws regarding dividends mean that it may be two days before now if the person exercising the call plans on holding the stock for 60 days to take advantage of the lower tax for dividends. To see why this is, let's look at an example (ignoring the tax implications since it changes the timing only). Say you own a call option with a strike price of 90 that expires in two weeks. The stock is currently trading at $100 and is expected to pay a $2 dividend tomorrow. The call option is deep in-the-money, and should have a fair value of 10 and a delta of 100. So the option has essentially the same characteristics as the stock. You have three possible courses of action: Do nothing (hold the option). Exercise the option early. Sell the option and buy 100 shares of stock. Which of these choices is best? If you hold the option, it will maintain your delta position. But tomorrow the stock will open ex-dividend at 98 after the $2 dividend is deducted from its price. Since the option is at parity, it will open at a fair value of 8, the new parity price, and you will lose two points ($200) on the position. If you exercise the option early and pay the strike price of 90 for the stock, you throw away the 10-point value of the option and effectively purchase the stock at $100. When the stock goes ex-dividend, you lose $2 per share when it opens two points lower, but also receive the $2 dividend since you now own the stock. Since the $2 loss from the stock price is offset by the $2 dividend received, you are better off exercising the option than holding it. That is not because of any additional profit, but because you avoid a two-point loss. You must exercise the option early just to ensure you break even. What about the third choice - selling the option and buying stock? This seems very similar to early exercise, since in both cases you are replacing the option with the stock. Your decision will depend on the price of the option. In this example, we said the option is trading at parity (10) so there would be no difference between exercising the option early or selling the option and buying the stock. But options rarely trade exactly at parity. Suppose your 90 call option is trading for more than parity, say $11. Now if you sell the option and purchase the stock you still receive the $2 dividend and own a stock worth $98, but you end up with an additional $1 you would not have collected had you exercised the call. Alternately, if the option is trading below parity, say $9, you want to exercise the option early, effectively getting the stock for $99 plus the $2 dividend. So the only time it makes sense to exercise a call option early is if the option is trading at or below parity, and the stock goes ex-dividend tomorrow. Conclusion Although interest rates and dividends are not the primary factors affecting an option's price, the option trader should still be aware of their effects. In fact, the primary drawback I have seen in many of the option analysis tools available is that they use a simple Black Scholes model and ignore interest rates and dividends. The impact of not adjusting for early exercise can be great, since it can cause an option to seem undervalued by as much as 15%. Remember, when you are competing in the options market against other investors and professional market makers, it makes sense to use the most accurate tools available. To read more on this subject, see Dividend Facts You May Not Know. UQ
Q http://fermatslastspreadsheet.com/2012/04/05/pricing-options-in-your-head/ Pricing an at-the-money option: well-known and easy For an at-the-money call or put (ie where K=F), the price is the same, let’s call it ATMPrice. The BS formula reduces to a simpler: ATMPrice = 1/sqrt(2*PI) * vol * sqrt(Maturity) which is very well approximated by: ATMPrice = 0.4 * vol * sqrt(Maturity). Here is an example in action: if interest rates have a normal vol of 100bps per year, then the ATM 5y5y payer would cost about: ATM 5y5y payer ~ 0.4 * 0.01 * 2.2 * 4.5 = 3.95%. Note that the ‘4.5’ accounts for the duration of the underlying swap (a 5-year swap has a duration of about 4.5 at the current level of rates). Nice so far, but it is going to get better Options which are not ATM: my new discovery The standard decomposition for an option is: Option value = Intrinsic value + Time value. Let’s compare that with the Black-Scholes normal formula from above: This is where I got my Eureka! moment. If you have a look at the term (F-K)N(d1) in a spreadsheet, you’ll see that for small levels of volatility and maturity (try, for example, sigma=0.25%, Maturity=1) it is actually quite close to Max(0,F-K) – which is the intrinsic value of the call. Consequently, the BS normal formula is almost: Call Price = Intrinsic + ATMPrice*exp(-0.5*d1*d1). Eureka! But not so fast. If you compare this formula to the correct BS formula in a spreadsheet, you’ll see that around the strike it gives too much value to the call: basically the term ATMPrice*exp(-0.5*d1*d1), is too big when d1 is non-zero and small. This is telling us that the difference between (F-K)N(d1) and Max(0,F-K) gets important near the strike. As I say, have a look in a spreadsheet. Nonetheless, this simple-but-wrong formula for the Call Price has pointed us in the right direction: it shows that the time value of the option should be written in terms of the price of the ATM option. Starting with the Black-Scholes normal flavour formula, adding and subtracting Max(0,F-K) then rearranging and using the 0.4 trick, gives (with a blatant effort to own this decomposition): The Hardy Decomposition: Option Price = Intrinsic + ATMPrice*HardyFactor where HardyFactor = exp(-0.5*d1*d1) + d1/0.4*N(d1) – Max(d1/0.4,0). Note that this is just a function of d1, which as I said above, is just a measure of how far you are from the strike in terms of the standard deviation to the option expiry date (being vol*sqrt(Maturity) ) of the underlying asset: eg “I am 2 standard devs from the strike” means d1=2, The lovely bit about the Hardy Decomposition is that the HardyFactor is well approximated by a simple expression: \text{HardyFactor} \approx e^{-1.4 \, |d_1|} . Better still, you can just remember a few values: abs(d1) HardyFactor 0 100% 0.25 70% 0.5 50% 1 25% 1.5 12% If you experiment a bit you’ll find better approximations. Here’s one I quite like and which is more accurate: \text{HardyFactor} \approx (1-0.41 |d_1|) \, e^{- |d_1|}. Let’s now look at some examples. UQ
So the title of the thread is "Why are prices of ESTX50 ATM options so different", bold added by me. Put call parity doesn't require an exact straddle and as Robert pointed out so nicely the parity price is the future not the spot. Even when not ATM, there are only a few cases where it's been shown that puts generally cost more than calls, and even then it only applies to far OTM puts (http://www.worldscientific.com/doi/abs/10.1142/S2010139214500153 is a good paper to get you started) The volatility smile is super interesting to me, it might be worth a deep dive if you have the time.
1. I think the title using the word ATM is not a proper one. 2. Practically no ATM could be found in reality for the OP's case at the time. 3. Depends on the futures, puts for equity futures (actually SPX options included) would be generally more expensive than calls. Currency futures is different.
Again, 3 is just not true. Why would fops violate put call parity? And why couldn't an ATM be found? Put call parity doesn't require an exact stradle to hold.
Odd trader, If puts and calls violated put call parity then there would be an arbitrage opportunity. ATM within even 50 points is good enough for the eurostoxx June. It's one of the most efficient vol surfaces on the planet (prob only second to the spx). The difference in premium between a put and a call at the same strike can only be explained by extrinsic value and dividends. Implied vol has to be the same in a no arbitrage world. And there will be no arbitrage in the eurostoxx.
My guess is the OP wanted to find ATM options but not available. That's why the question. However, the confusion so far is the whole thread using examples of OTM options. Perhaps sometimes options can be priced abnormally just before active market open.
Q http://www.nooptionantics.com/blog/?p=77 The reason call options are more expensive than put options ATM (at the money) Posted on September 15, 2010 by Peter Zomaya Jr. It has been a popular myth that the reason that calls trade higher than puts, when the options are at the money, is due to the fact that stocks can only drop to zero for the puts, but can go as high as infinity for the calls. Sounds great on the surface. But again this is another myth perpetuated by the uneducated in the seminar industry. The real reason that calls trade higher than puts is due to the cost of carry for the stock. Remember that a call option gives you the right to buy the stock. If you had to buy the stock, your money would tied up in the stock and not in the bank earning interest. That interest component must be added to the call price. This is the real reason calls are more expensive than puts. It is not a mystery or a made up answer about stocks being able to rise to infinity. Happy Trading! Peter Zomaya UQ