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2OptionsWHAT IS AN OPTION?A call (put) option is the contract right to buy (sell) a specified amount of some real or financial asset at a fixed price on or before a given date.If the option purchaser acts upon this right to buy, he or she is exercising this right; and the fixed price of the transaction is known an the strike price. The seller of the option, known as the writer, must be prepared to sell the specified asset when the option purchaser exercises these rights. When the option buyer exercises, the seller is assigned. The maturity of the contract is known as the expiration date, and exchange option trading takes place in any one of a number of set contract months, or cycles. An American option allows the holder to exercise the right any time before the expiration, and a European option restricts the right only to expiration and not before.A call is in the money at expiration when its asset price is above the exercise price, and a put is in the money when the asset price 1h below the exercise price. A call is out of the money at expiration when its asset price is below the exercise price, and a put is out of the money when the asset price is above the exercise price. Options In the money have real value, and those out of the money have no remaining value at expiration. An option is at the money when the asset price expires or trades right at the strike.The underlying asset may be either the cash market (spot) or futures contracts on the underlying spot instrument (stocks, bonds, currency, or spot commodities). The spot option market holds a right to exercise over the cash asset itself. The futures option in on the futures contract only, which is then a right to delivery of the spot instrument. There may be both a cash and a futures option market trading in the same asset market, in which case cash/futures option arbitration will become important, whether in the stock, bond, and currency futures or over-the-counter markets.The gross profit/loss (P/L), or expiration, payoff of a $100 strike call and put is illustrated graphically in Figures 2.1 and 2.2, where the purchase or trade price of both the put and call is $1.70. The 100 call allows the long holder to exercise the right to buy the underlying asset at $100 on (or sometimes before) expiration, and the short call writer must fulfill this demand. The 100 put allows the long holder to sell the asset at $100 to the short put writer on or before expiration.Profit/loss is based on the expiration value of the option minus its trade price. If the asset settles at expiration at $102, for example, the long 100 call will expire in-the-money valued at $2.00 for a profit of 30 cents ($2.00 - 1.70) and a 30-cent loss for the short call. At an asset price of $110 at expiration, the call will be worth $10.00, for a net profit of $8.30 for the long call and a loss of $8.30 for the short call. The breakeven point for both the long and short call is an asset price of $101.70 at expiration, based on the trade price of the option at $1.70. Below this asset value the long call will show a maximum loss of $1.70, and the short call will show a maximum profit of $1.70.100 Strike CallAsset price (dollars)Figure 2.1 Payoff for long/short 100 strike options at expirations.100 Strike PutAsset price (dollars)Figure 2.2 Payoff for long/short 100 strike options at expirations.The breakeven point for the long or short 100 put will be an asset price of $98.30 at expiration, above which the long put will have a maximum loss of $1.70 and the short call a maximum profit of $1.70. Below the breakeven point, the long put will show increas¬ing profit and the short call increasing losses.While determination of the value of an option at expiration is relatively straightforward, its value before expiration is unknown. How much should an option be worth, or what is its fair value, before expiration? Clearly, it must equal at least the immediate exercise in-or out-of-the-money value or else an arbitrage profit generally becomes available, forcing price and value back in line. But how much more should be added above the in-the-money value to determine the fair value?An options price depends on both the intrinsic and the extrinsic value of the option. The intrinsic value of an option is simply the value of the option if it were exercised immediately for cash value. The intrinsic value is always known and is a simple function of the relationship between asset price and the option strike. If the asset price is below the strike price of a call, the call will have no intrinsic value; and if the asset price is above the strike of the call, then the intrinsic value is the positive difference between the asset price and the strike. Puts have intrinsic value if the asset price is below the strike, but no intrinsic value if the price is above the strike. Options that have positive intrinsic value are in the money, and those without intrinsic value are out of the money.For example, if a May future is at $110, then the 100 May call will be in the money and will be worth exactly $10 in intrinsic value, because a trader can exercise his or her long call rights to buy futures at $100 and immediately sell them for $110 on the open market. If futures are selling below $100, however, the 100 call will have no intrinsic value and will expire with no worth if this relationship holds until expiration. At expiration an option will be worth exactly its intrinsic value.Whether a position has a net profit at expiration will, of course, depend on what the trader paid for the call. Only if the trader bought the call for less than $10 will there be a net profit if the futures price is at or above 110 at expiration. If the futures price closes below 100, however, the trader will lose money no matter what he or she paid for the 100 call.In theory, the intrinsic value of an option will always set the minimum price that an option must have, since if it fell below that value, an arbitrage profit for market makers would quickly close the gap. Although option prices are limited on the downside by their intrinsic value, in practice, they usually trade above this value as long as there is time remaining to expiration.Remaining time to expiration will add some additional value to an option’s intrinsic value, which is known as the extrinsic value or time premium. Generally, the longer the time remaining, the higher the time premium. This valuation makes intuitive sense since a longer time remains for the option to become intrinsic- value profitable and, therefore, the option must be more valuable than at expiration.Although the intrinsic value of an option is a direct function of futures or asset prices, it is the addition of extrinsic value or time premium that sets the total value of an option. Option valuation is essentially time premium pricing. How does the market set the price of the extrinsic value of options? How much should the time premium of an option be worth? Answers to these questions will he developed in I he next section.OPTION FAIR VALUEAn option may be considered fairly valued when it is priced so that trading at that price will produce neither gain nor loss over the long run, whether trading from the long or the short side. It is fairly valued because each side of the trade has equal economic advantage. When options are fairly valued in the market, they are best able to shift risk away from the underlying asset market. Acting essentially as an insurance factor to the underlying asset market, market makers in options must evaluate the risks of price changes carefully in order to charge an appropriate price for the insurance. How should options be fairly priced?Modern fair-value option models, based on statistical models of probability, attempt to provide the answer to this question. The worth of an option is related to how likely it is to earn a profit or loss. To know fair value, however, requires a trader to know the probability of gain and loss for any specific option and, therefore, the probable future underlying asset price as well.Before modern fair-value options models were developed, no theoretical model of option pricing could successfully generate hypothetical fair values of option time premiums. The price risk exposure of an option contract was evaluated subjectively, with only past empirical price and time relations somewhat imperfectly known. One can still value options in a relative way without a fair- value model if one knows the fixed arbitrageable synthetic price relations between puts, calls, and futures or asset prices (see Chapter 6). But synthetic option price relations do not establish a model of fixed absolute levels, that is, fair value itself.Modern stock option pricing theory dates from the early stalintical models of option premiums, especially the work of Fischer Black and Myron Scholes (1973), which was modified by Robert Merton (1973) into the BSM model. This model was adapted to commodity futures options by Fischer Black. The BSM model quickly came to be the most widely recognized and used stock and futures option value model; and it provides the statistical basis of this study, although alternative models will also be discussed for bond markets.To understand how options are fairly valued statistically, let us review the probability of any kind of payoff-game outcome. As an example, consider a game in which, after a trial of only one period, there are only two possible outcomes, say heads or tails, with one outcome counting as a win and the other as a loss. Suppose a win results in a reward of $5, and a loss results in no reward, or $0. If the probability of either a win or a loss on one trial is 50 percent, what bet will neither gain to nor lose money if this game is played again and again? The answer to what a fair-valued bet should be is the probability of outcomes times the payoff, summed, or .50 X $0 + .50 x $5 = $2.50. Out of every two trials, there will be a total gain of just $5 on average (one win and one loss, or $5 + $0). Since it takes two trials on average to win $5, each bet must be $2.50 in order neither to win nor to lose money over a two-trial run over many plays.The general formula for the fair value of a bet over a one-period trial, then, is the probability payoff schedule:Where V = value of fair-value betP = probability of outcome$ = outcome payoffNow consider a game in which there is only a one-period trial and for which the outcome values are either 105 or 95. Assume that an outcome of 105 will bring a profit of $5 and an outcome of 95, $0. Assume also that the probability of 105 or 95 occurring is exactly 50 percent on each trial. What is the value of a bet that will produce neither a profit nor a loss over the long run?It may be seen that the answer again is exactly $2.50. Both games just described represent a one-event, binomial distribution with identical outcome payoffs and event probabilities. Calling a win heads, or 105, and a loss tails, or 95, does not change the essential similarity of the two games in any way. In each case, the bet that will produce neither a win nor a loss is calculated as the sum of the individual outcome probabilities times the outcome payoff, that is, .50 X $0 + .50 X $5 = $2.50.Observe that the bet in the second game is equivalent to an investment or purchase of a hypothetical 100 call that can take only an exercise value of either 95 or 105 over a single period. By using elementary probability theory, one may calculate the fair value of a call option under those restricted specifications.Now consider the possibility that outcomes are distributed within a range of 95 to 105 in one-point increments (that is, 95, 96,..., 105) and that the gain on each outcome is $0 at or below 100, and the numerical outcome less $100 when above 100; for example, 95 = $0, 96 = $0,..., 103 = $3, 104 = $4, 105 = $5. This result is similar to the payoff schedule of a hypothetical 100 call taken over a one-trial event.If the probabilities of the specific outcomes are known, then, as in the previous examples, it is possible to calculate that fair-value investment or bet that will produce neither profit nor loss in the long run. For sake of illustration, assume that the outcomes from 95 to 105 have the following probabilities of occurring over a one period trial:Possible Outcomes Probability of Occurring (percent)95 296 497 898 1299 14100 20101 14102 12103 8104 4105 2These probabilities are graphed in Figure 2.3, where the mean strike is 100.OutcomeFigure 2.3 Outcome probability.Since the specific payoffs as well as the probability for each outcome are known, the fair value of a bet may be determined exactly as in the previous games. That is, one multiplies the probability of an outcome by its expected payoff and sums the results to find the exact fair-value bet or investment. Thus,Outcome 95 $0 X .02 = $0Outcome 96 $0 X .04 = 0Outcome 97 $0 X .08 = 0Outcome 98 $0 X .12 = 0Outcome 99 $0 X .14 = 0Outcome 100 $0 X .20 = 0Outcome 101 $1 X.14 = 0.14Outcome 102 $2 x .12 = 0.24Outcome 103 $3 x .08 = 0.24Outcome 104 $4 x .04 = 0.16Outcome 105 $5 X .02 = 0.10Sum = $0.88Summing the expected payoffs gives a result of 88 cents, which represents the fair-value bet in this example, that is, that invest¬ment that neither wins nor loses repeatedly on one-period trials over the long run.In the above example, one readily sees that the payoff schedule of each outcome (before the probability is taken into account) is just the payoff schedule of a hypothetical 100 call at expiration, or its intrinsic value from 95 to 105. Therefore, the only unknown in calculating the fair-value of such an option is the specific probability of each outcome. In other words, once the probability of outcome for an option’s intrinsic value over some trial period is known, that option’s fair-value price can be determined exactly.In the preceding examples, we have assumed that the probability of outcomes is known exactly. But in the real world, the exact probability of occurrence for any specific intrinsic value of an option at expiration in not known directly. How, then, is it possible to derive such outcome probabilities for option analysis? How can we know what the probabilities are for the payoff of a hypothetical 100 call from 95 to 105, or from 50 to 150? The answers to these questions will be discussed in the next section.OPTION PRICING MODELSThe key element of any option fair-value model is the probability assumptions about changes in underlying asset prices. If the probability distributions of asset price changes are known or can be successfully estimated, then these may be used to derive the probability densities of the expected payoff schedule of options at expiration, from which fair value may be derived.More generally, if it is theoretically known that an asset price has an x percent chance of increasing by y points or more over the next z days, then an option price will be related to the outcome of this price change probability. A fair-value option price only reflects the intrinsic value at expiration, which is linked to the probability of underlying asset price change. If the distribution of futures prices is accurately estimated, so too will be the fair value of the option.The probability of asset price change is usually referred to as volatility by option traders, and it is usually unknown. As a consequence, volatility must be estimated. Doing so gives rise to a number of different option pricing models.Two other statistical assumptions or unknowns must also be incorporated in order to derive fair value for any model. (1) The risk-free interest rate must be known or estimated over the life of the option; and, (2) if the option is on a yield bearing asset, the dividend or yield must be known or estimated over the life of the option. For bond options in particular, additional estimates of the term structure of interest rates may also be necessary. This chapter will review the Black-Scholes-Merton (BSM) option pricing model, but we shall also discuss some recent work in lattice-based and advanced bond option models.The work of Black and Scholes in 1973, followed quickly by that of Morton the same year, proposed to link the probability of stock price changes to stock options using a log-normal distribution as the probability estimator. The BSM model was based on earlier statistical work that had shown that stock price changes could be modeled on a normal or Gaussian curve (Cootner, 1964). That is, futures (and stock) price changes resemble a random sample drawn from a universe that can be described by a log-normally distributed curve.The finding that stock and asset price changes resemble known probability distributions makes it possible to develop theoretical models of price changes. The earliest fair value model of Black and Scholes used the normal-curve model of futures prices to derive option prices. The BSM model deduces option prices on the basis of statistical theory and thus supercedes subjective or graphical option price valuation. For example, the Fischer Black (1976) formula for European futures option fair value is:Where N(d) = cumulative normal integral r = risk-free interest rateS = standard deviation of log percentage change in annualized pricesX - futures priceK = strike pricet = time to expiration, annualizedC = call premiumP = put premiume = exponent (2.7183)In =natural logarithmThe BSM model requires information on five independent variables in order to estimate fair value for a non-income-earning future option:1 The current futures price2 Option strike price3 Days to expiration4 Risk-free interest rate5 Standard deviation of futures price change or volatilityThe current futures price is an empirical approximation of the mean of the normal curve; it is necessary in order to center the probability distribution. The option strike price represents our interest in a specific futures price outcome along the normal curve. Days to expiration, or the number of trial periods, is the empirical case frequency (or number of trial events) of the distribution. Calendar days are used in the BSM model rather than trading days. The risk-free interest rate is the opportunity cost of capital; it is taken to be the Treasury bill or note for the appropriate term of the option. Finally, volatility is the estimated standard deviation of futures price change over the number of trial periods, expressed as a logarithm. Volatility represents the probability of outcome of futures prices. To convert equation (1) for use with income-earning assets (for example, stocks, bonds, or currencies), one would need to subtract an expected future yield from the expected futures price. Colburn (1990:157-158), and Labuszewski and Sinquefield (1985:117-119), have presented empirical examples of the BSM formula.Unfortunately, there is some evidence that the assumption of log-normality for financial and commodity asset price change may be inaccurate. Records in the stock and commodity markets over many years appear to show that financial asset price changes have a higher cental tendency with longer tails than the theory of normality would suggest (Cootner, 1964; Brealey, 1969; Turner and Weigel, 1990; Sterge, 1989; Nelson, 1988; Peters, 1991). This tendency for long-term financial assets to deviate from the normal curve is illustrated in Figure 2.4. While there are very few observations in a normal distribution above or below two or three standard deviations from the mean, actual long-term financial returns may include observations as much as six standard deviations away, as found by Turner and Weigel and by Peters in stock returns from 1928 to 1988. On the other hand, long-term empirical stock returns show more data points in the center of the distribution than the assumption of normality predicts; these indicate greater peakedness.Standard deviationFigure 2.4 Normal distribution and financial asset returns. (Normal distribution from Hastings, 1975.)Nelson (1988) demonstrated the wide variations possible in year-to-year wheat futures price change from 1967 to 1987. These data also show the empirical distribution has a greater central tendency and longer tails than the standard normal curve (Figure 2.5).Research on stock and asset price change has consistently found that the standard normal distribution does not fit actual events smoothly and Cootner (1964) suggested stock prices may be only approximate standard normal. The normal curve is specifically defined by different measures, or moments. The first and second moments are the mean and standard deviation. The third moment of the normal distribution is skewness or tilt; in these cases the mode is not the same as the mean, and the mode may be either skewed to the left, or right. Skewness or tilt is evident in long-term financial asset price change.The fourth moment of the normal distribution is kurtosis, or the degree of thickness in the tails and peakedness of the center; a thick-tailed distribution shows platykurtosis while a more peaked distribution shows leptokurtosis. Most studies of long-term stock and asset returns show that price changes have both leptokurtic (peakedness) and platykurtic (thick-tailed) features but that neither theoretical curve alone fits the empirical data.Figure 2.5 Logarithmic September wheat futures. (Selected and all years, 1967-1987, from Nelson, 1988.)When differences between the actual returns and a standard normal curve are calculated, an error curve is derived. Peters (1991) computed this error curve for actual stock returns from 1928 to 1988 as shown in Figure 2.6. The normal curve was com¬pared with the frequencies of occurrence for Standard & Poor’s 500 five-day returns.Another reason why the assumption of log normality for asset price change has been questioned is that these models cannot easily handle American options, which have the possibility of early exercise. Additionally, log-normal models become increasingly in-accurate as the term of the option lengthens, especially longer than a year (Wong, 1991).All fair-value models accept the theoretical link between asset price change and probability theory, but some have abandoned the normality assumption and proposed alternative price change models based on different probability assumptions. Alternative explanations of price change seek, in effect, to eliminate the error curve formed as the difference between the normal and the asset price change.Figure 2.6 Error curve derived as the difference in frequency between the Standard & Poor's 500 five-day returns (1928-1988) and a normal curve. (Copyright©1991 Edgar Peters, Reprinted by permission of john Wiley & Sons, Inc.)Mandlebrot (1964) suggested that stock returns may resemble a class of Paretian distributions also known as fractals, which have unstable variance. Peters (1991) shows, however, that a wide range of recent financial asset price changes fit fractal assumptions and models.The Cauchy distribution (Figure 2.7) has a density function with longer tails than the normal distribution and also tends to¬ward peakedness. Indeed, the Cauchy seems to provide a better fit of long-term wheat futures change than a normal distribution. Unfortunately, the Cauchy has an unstable mean and infinite vari¬ance, which complicate its statistical use. Nevertheless, the Cauchy has been used effectively in a number of scientific fields (see Olkin, Gleser, and Derman, 1980).Recently, fair-value models based on the probability of binomial walks have been proposed, following the work of William Sharpe (1978) who derived the same result as Black and Scholes using only elementary mathematics. These binomial models, also known «h lattice-based models, are an exciting area of option research at present. Cox and Rubenstein (1985), who have developed binomial models, also suggest that asset changes may follow some sort of diffusion-jump process. Gastineau (1988) has proposed models based on normal distributions of change in option and asset pricing. For European options, binomial models will converge with the BSM model at the limit, but binomial models appear better able to incorporate American option features (Wong, 1991).Standard deviationFigure 2.7 Cauchy distribution and wheat price change (1967-1987), (Source of wheat data: R. D. Nelson, 1988.)Both the log-normal and the lattice option models depend upon what one assumes short-term interest rates will be over a long time for back-month options. For this reason, these models are somewhat insensitive to the true costs of carry of the underlying asset, which are very important for option prices. An accurate option pricing model, therefore, seeks to take into account both short- and long-term rates of interest, or the term structure. Recently, a number of innovative term-structure binomial models have been put forward for bond options and bond option, traders may wish to study these models more closely (Wong, 1991).All option traders should be familiar with the statistical as¬sumptions of the model they are using and feel comfortable with them. Most option traders and market makers, however, are traders, not statisticians. To aid the understanding of basic concepts, this book will continue to use the BSM model for illus¬tration, but some modifications of this model will be introduced in later chapters so as to fit prevailing market conditions better.Fortunately, the BSM model may often be used by option market makers for practical trading to produce reliable results since making option markets is more a question of relative pricing than absolute pricing. Market makers are not so much interested in what real or true fair value may be as in which options are mispriced relative to each other under the same model.Moreover, one can introduce sophisticated modifications to the BSM model, if needed, by incorporating a strike skew function, which will be discussed elsewhere. For these reasons, the BSM model is used as the statistical basis of this study, and the model will be considered robust to the violation of the assumptions of normality. A partial list of software that may be used for calcu¬lating option fair values from the BSM model is included in the Appendix.VOLATILITYThe standard normal fair-value models, such as the BSM model, depend upon five independent variables or unknowns for nonincome-earning asset options: interest rates, asset price, strike, expiration date, and standard deviation. Volatility in the BSM model is just the standard deviation of asset prices around the mean over a one-year period. If an underlying asset has a 10% volatility and current price of 100, then there would be a 68% chance (assuming a normal distribution of price changes) that the asset price at the end of the year would be within the range of 90 to 110.The general formula for the standard deviation is:Where SD = standard deviationP = priceP = mean priceN = number of trialsHere is an example:(P)rice (P – P)290 100100110 100Mean = 100 ∑= 200The standard deviation, then, would be In this example, one standard deviation is 8.16 above or below the mean of 100. The probability of an event being within one standard deviation on one side of the mean is about 34 percent, or within one standard deviation above or below the mean is about 68 percent. In this example, the final price over one year will be within 91.84 and 108.16 about 68 percent of the time.To avoid the possibility of negative asset prices, standard deviation is customarily calculated using natural logs of rates of change in lieu of absolute difference. In this case, the formula for the standard deviation is expressed as a rate of change.WhereComputation of the SDV gives a percentage change of the standard deviation and is what is meant by volatility. This volatility is always historical, since it depends for its data upon actual past market fluctuations. SDV may also be called historical volatility (HV). Historical volatility may be used to derive the standard de¬viation in absolute points (SDP) through the general formula:SDP = Px HVFor example, if historical volatility is 0.10 and the current price is 100, then SDP will be 10 points over one year. Knowing how to compute the standard deviation in points is useful to know, as will be evident in later chapters.To adjust for shorter time intervals than one year, volatility must be divided by a time factor. This factor is the square root of the number of periods in a year ( ). For example, the weekly time factor is composed of 52 periods, and = 7.21. Common time factor adjustments (PY) are:DailyWeeklyMonthlyBimonthlyQuarterlyThe BSM model calls for the use of calendar days (365), not trading days (250) to expiration, as a time adjustment, but market practitioners often use the number of trading days as the adjustment factor. Is it reasonable to assume that all calendar days are equal for calculating price change, or is the number of trading days a better adjustment divisor?The answer to this question is a straightforward empirical matter. If the futures price change between Friday’s close and Monday’s close is equal to or greater than the price change within an average consecutive three-day trading period, then calendar days carry implicit trading volatility risk that should be taken into account. If the weekend price change is equal to or less than an intraweek daily price change, than there would appear to be no weekend volatility effect, and thus the time between Friday’s close and Monday's open could be treated as a one-day change, justifying the use of a 250-day yearly period. In either case, the answer is easily settled empirically for any specific market.In summary the formula to find the standard deviation in points (SDPt) over period PY is:For example, if V is 15 percent and P is 100, then the standard deviation over one day is 0.95 points, using 250 calendar days.If one knows the point standard deviation change over a year and wishes to calculate volatility, the above formula is solved in this manner:Thus, a 1.25-point SD over a year for a commodity priced at 100 Indicates that volatility is 0.0125, or 1.25 percent.To find the volatility (Vt) over any period PY:This relationship is sometimes also known as the T1/2 Rule; variance is equal to the standard deviation times the square root of time. For example, the historical volatility over one day is 0.382 (or 38.2%) if SDP daily standard deviation is 2.00 and P is 100:Given the importance of volatility in option pricing models, alternative measures of expressing volatility have been variously proposed. Instead of using only price change from daily close to close, it is possible to use a volatility measure based on daily high and low prices. Conceptually this measure makes sense since differences in day-to-day price closes may be flat even though large price volatility occur intraday. If the underlying asset is regularly traded in a liquid market, the high/low estimate of volatility may be more accurate than the close/close volatility estimate for identical time periods (Bookstaber, 1987).In particular, a price change measure of volatility may not work well with bond options, since bond prices converge to par at maturity. For debt markets, yield change has been found to be a better measure of standard deviation in bond option models (Wong, 1991). Finally, one must remember that these volatility and standard deviation measures will only work as long as actual price change can be described by a normal curve.Standard deviation, or volatility, is perhaps the most important variable in the BSM model. While four of the five variables in the BSM model all are easily known at any one time (futures price, strike, expiration, and interest rates), volatility is not. Thus, even if the distribution of futures price changes tends to resemble a log- normal curve over time, it is still necessary to know the width of the curve (that is, its standard deviation) before one can calculate an option’s fair value. But here a problem arises: What measure of volatility is to be used?When discussing price volatility one must distinguish among three different aspects of volatility: future, historical, and implied. Future volatility is the standard deviation that would be calculated from the present to some period in the future. In a sense, this is the real volatility. For the BSM formula to work as theoretically intended, the measure of standard deviation that one must use is the future (or real) volatility. Obviously, what is of interest is not what has happened, but what will happen.However, since the future is unknowable, so is future volatility. Since the fair-value model cannot work without knowledge of future volatility some estimate must be made in order to use the BSM formula. There are two ways of estimating what future volatility may be: use either historical or implied levels of volatility.Historical volatility is the empirical standard deviation of futures prices from some time in the past up to the present. Since it has already happened, historical volatility is always known with certainty. The only judgment to be exercised concerning historical volatility is the period to be selected. Half-year, 90-day, 20-day, and even 10-day periods have been used. However, each of these different periods is likely to provide a different estimate of volatility level, and indeed there may be no single host estimate of historical volatility. Generally, long-period estimates are the most stable and shorter periods the most unstable and erratic because the latter are subject to immediate market conditions.The use of historical volatility in the BSM model (for whatever period chosen) would not present a problem for option fair-value determinations if future volatility levels happened to coincide with past volatility levels. If what has happened will always happen, then future volatility can be estimated directly from historical levels confidently.Unfortunately, not only is there no a priori reason for past and future volatility to be the same, but also the record shows they usually are not, whatever period is used. Future volatility often changes over time, rendering historical volatility a poor and unreliable estimator of future volatility, whatever the period of historical volatility used.In statistical terms the difference between past and future volatility is known as heteroscedasticity, or the variance of the variance over time. This difference represents a problem to the uncritical use of estimators based on the assumption of statistical normality in any time series model. The existence of severe heteroscedasticity over commodity cycles is intuitively well known; commodity prices are sometimes quiet for extended periods, and then break out in extreme price runs in one direction or another, before returning to quiescence again. Historical volatility is a poor predictor of future volatility.Implied option volatility may be used as an alternative to historical volatility in estimating future volatility. Whenever an option is traded at market price, it is possible to work backward in the BSM formula and solve for volatility. The resultant implied volatility is the market’s current estimate of future volatility, and is an alternative to using historical volatility as a forecaster of future volatility.There is little evidence, however, that current implied levels of volatility are reliable or accurate estimates of future volatility To see his intuitively, suppose that “real” future volatility will be indented in the 20-day historical volatility lagged backward 20 days, Current 20-day historical volatility, for example, is only the future volatility 20 days ago. When historical volatility is lagged and then compared with the level of implied volatility at that time, there is little indication that implied levels will necessarily equal real volatility. Sometimes implied levels fall below real future volatility and sometime they are above, as illustrated for October cotton futures and options in Figure 2.8.Figure 2.8 Real and implied volatility for October cotton futures and option prices. (Source: Mr. Tom Bertolini, New York Cotton Exchange.)Generally, there is no consistent short-term pattern between real future volatility and implied levels in most option markets. Since real or future volatility may never be known until after the fact, one must estimate future volatility from either unlagged historical volatility or current implied volatilities in the market in order to derive accurate fair-value option prices. If neither is a reliable empirical predictor of future volatility, then the robustness of the BSM option pricing model is weakened.For the most part, this uncertainty is not a serious impediment for market makers to using the BSM model. Prudently profitable market making does not depend, for the most part, on knowledge of true future volatility and, therefore, true fair-value prices, but rather on relative pricing. In practice, market makers almost always initially set bid/asked prices around recent implied volatility levels, and only use historical levels for reference.FINANCIAL AND FUTURES OPTIONSThe discussion of options thus far has been generally restricted to options on non-income-earning spot instruments, or to all options in general. Nevertheless, options will display different characteristics depending upon the differences of the underlying asset: stocks, sock indexes, bonds, currency, commodities, or futures on these.One important distinction among option markets is the cost of carry for the underlying asset, whether positive or negative.For most financial assets (stocks, bonds, and currency) there is an initial positive cost of carry, with income from dividends, yield, or interest. In the case of stock dividends for example, where the dividend will reduce the price of the stock by that amount, the current price of the stock must be discounted by this dividend amount when considering the value of the in-the-money or intrinsic option at expiration. For example, if a stock priced at 100 is paying a $1 dividend quarterly, then the 90 day option with strike of 100 will be on a stock worth only $99 after dividend payout, assuming no further price changes by expiration. What is of importance for the option is not necessarily the current stock price but the present estimate of expected future value. In this example this estimate would be $99 and, in effect, the forward price.For some assets, such as commodities like gold nr copper, there is a negative cost of carry due to costs of storage, insurance, shipment, and so on for physical assets. This negative cost of carry will tend to raise the forward price. In the bond market in particular, where bonds are often financed by other bonds, both positive and negative cost-of-carry positions are possible. For example, if a holding in long-term U.S. Treasury bonds is being financed by a sale of short-term Treasury bills, there will be a positive cost of carry if bill rates are below long rates, but a negative carry if short rates exceed long rates. In summary, differences between income- or yield-producing assets (stocks, bonds, currency) and non-income- or negative-earning assets require that option models be adjusted for asset price cost of carry.Options on real underlying assets and options on futures also need to be distinguished. For example, an option on stocks or stock indexes is a call or put right that corresponds to an ongoing similar underlying asset. In this sense, a January call is a right on the same stock asset as a May call. However, futures options are rights to futures contracts and not to the underlying asset itself. A January futures call holds a right against a different underlying asset (the January futures contract) than a May call (the May futures contract). Since futures options introduce new complexities into option market making, let us briefly review futures markets here.A futures contract is the right to take delivery of a specified quantity and quality of a commodity (spot) at the monthly expiration specified in the contract. Thus, if a trader is long (bought) a December sugar futures, at a specific day in September, the trader can take delivery of 112,000 pounds of a specified grade of sugar at a certified warehouse. If a trader is short (sold) a similar futures contract, then he or she must stand ready to deliver spot at expiration. A futures contract of itself does not specify any price at which delivery is to take place. Rather, the purchase (or sale) cost of the contract, when first purchased or sold on the market, becomes the basis cost to buyer or seller. In modern futures markets, contract expiration and delivery dates are sequenced in cycles every quarter or several months apart.In economic theory, futures markets are useful because they allow for efficient price discovery and for investment risk shifting, or price insurance, Futures markets provide an efficient and competitive price-setting mechanism (“discovery”) by allowing prices to respond immediately to shifting supply-and-demand conditions through an ongoing open outcry auction market. Futures markets also provide a risk-shifting function to producers, merchants, and industrial consumers, allowing industry to shift future price- change risk to other commercial interests, dealers, or speculators. This risk shifting allows business to achieve neutralization of risk and thereby improve trade efficiency.For example, a farmer will typically sell forward to a merchant all or part of his or her crop that has not yet been harvested. This sale protects, or hedges, the price of the crop while it is still being grown. This action helps avoid the price pressures associated with seasonal harvesting and temporary oversupply. Mining or petroleum producers also use futures contracts in this way, to hedge against oversupplies.Likewise, a merchant will sell forward if shipping to the destination market requires a long time, exposing the commodity to unexpected price shifts before sale. In the early days of forward trading, a merchant would buy cotton in Savannah and then ship it north to New York, before reshipping it to New England or Liverpool textile mills. By selling forward, the merchant protected his investment in inventory during the long lag between purchase and final delivery. Hedging crude oil during its long transport to market is a more recent example.Also, manufacturers who use commodity raw products will typically hedge their cost of inventory during production by either selling or buying forward. Cotton mills today, much like the nineteenth century, still have routine recourse to forward contracts in controlling inventory costs.Futures markets in the nineteenth century and on into the twentieth century were dominated by trading in agricultural products and precious or industrial metals; these remain a significant proportion of all modern futures trading. In the last several decades, however, currency, bond, and stock index futures have been traded, and they now account for the largest proportion of all futures trading. Worldwide, the growth of futures trading has been explosive as the financial industry learned that futures trading may be used to manage and hedge portfolios of bonds, stocks, or currencies.Futures need not and usually do not trade at the same price as the underlying spot commodity; the difference in price is referred to as the basis. If the underlying commodity is a non-income- earning asset that must be stored somewhere, the futures price is usually higher than the spot commodity price in order to account for the cost of storage or negative carry. For non-income stored negative carry markets, futures prices will trade over spot (positive basis), and the more distant futures will trade over near futures prices, to account for the requisite costs of storage and insurance. When the distant futures price exceeds the near futures price, the market is said to be in contango.If the near futures price is higher than the distant price, however, the market is said to be backward (Figure 2.9). Backwardation in the futures time spread for non-income-earning durable commodities may come about for several reasons. In agricultural commodities there is often a normal backwardation between crop year futures cycles, which comes about because the storage costs of the new crop are unrelated to storage costs of the old or current crop. If the new crop has not yet been harvested, no storage costs have been incurred; and the far distant futures contract may trade, therefore, lower than the less distant or near contracts. Also, in both seasonal and nonseasonal commodity markets, backwardation may occur as a result of severe supply shortages or sharp increases in unfilled demand. For example, although gold and silver futures normally trade in contango, during the 1980-81 attempted silver corner by Bunker Hunt, precious metals futures went backward.Figure 2.9 Future contango and backwardation.For income-earning assets—stocks, bonds, or currency—the futures time spread may be somewhat different from that of non- income-earning assets. For financial assets, the cost of carry is not negative, and may even be positive. Stocks and bonds earn dividends or interest, and do not entail large costs of storage. For assets with a positive cost of carry the futures time spread is normally backward. In effect, the future price is discounted by the positive yield on the asset over time, and thus distant futures prices may trade lower than near-term prices, all else equal. Consider, for example, the situation in which one-year Treasury notes, par 100, yield 5 percent over one year. If one year forward futures contracts are priced at 95, they would approximately discount the positive cost of carry and trade at normal backwardation. Positive carry futures markets, however, also may display contango time spreads at times, for different reasons. Further discussion on time and basis spreads of futures may be found in works listed in the bibliography (Williams, 1986).Futures options have rights over a futures contract, not the un¬derlying asset. This distinction introduces important new risks to option traders, who make markets in futures options time spreads. The risk does not exist for stock option traders, however. It is also possible to have option markets on both the underlying asset and the futures on the underlying asset. For example, there are options markets on stock indexes at the same time that options on the futures of the stock index are being traded in a different market. This situation opens up the possibility of intermarket option arbitrage. This study will emphasize futures options as the general case, with discussion of differences arising with stock or bond options where appropriate.世聯(lián)翻譯-讓世界自由溝通！專業(yè)的全球語言翻譯供應(yīng)商，上海翻譯公司專業(yè)品牌。絲路沿線56種語言一站式翻譯與技術(shù)解決方案，專業(yè)英語翻譯、日語翻譯等文檔翻譯、同傳口譯、視頻翻譯、出國外派服務(wù)，加速您的全球交付。 世聯(lián)翻譯公司在北京、上海、深圳等國際交往城市設(shè)有翻譯基地，業(yè)務(wù)覆蓋全國城市。每天有近百萬字節(jié)的信息和貿(mào)易通過世聯(lián)走向全球！積累了大量政商用戶數(shù)據(jù)，翻譯人才庫數(shù)據(jù)，多語種語料庫大數(shù)據(jù)。世聯(lián)品牌和服務(wù)品質(zhì)已得到政務(wù)防務(wù)和國際組織、跨國公司和大中型企業(yè)等近萬用戶的認(rèn)可。
