335:
2889:
1686:, as used in Delta, is subtler, and can be interpreted most elegantly as change of numéraire. In more elementary terms, the probability that the option expires in the money and the value of the underlying at exercise are not independent – the higher the price of the underlying, the more likely it is to expire in the money
594:
When quantifying moneyness, it is computed as a single number with respect to spot (or forward) and strike, without specifying a reference option. There are thus two conventions, depending on direction: call moneyness, where moneyness increases if spot increases relative to strike, and put moneyness,
304:
With an "in the money" call stock option, the current share price is greater than the strike price so exercising the option will give the owner of that option a profit. That will be equal to the market price of the share, minus the option strike price, times the number of shares granted by the option
1640:
The other quantities – (percent) standardized moneyness and Delta – are not identical to the actual percent moneyness, but in many practical cases these are quite close (unless volatility is high or time to expiry is long), and Delta is commonly used by traders as a measure of (percent) moneyness.
395:
Assets can have a forward price (a price for delivery in future) as well as a spot price. One can also talk about moneyness with respect to the forward price: thus one talks about ATMF, "ATM Forward", and so forth. For instance, if the spot price for USD/JPY is 120, and the forward price one year
204:
The above is a traditional way of defining ITM, OTM and ATM, but some new authors find the comparison of strike price with current market price meaningless and recommend the use of
Forward Reference Rate instead of Current Market Price. For example, a put option will be in the money if the strike
236:
of an option is the total value of the option, less the intrinsic value. It partly arises from the uncertainty of future price movements of the underlying. A component of the time value also arises from the unwinding of the discount rate between now and the expiry date. In the case of a
European
1641:
Delta is more than moneyness, with the (percent) standardized moneyness in between. Thus a 25 Delta call option has less than 25% moneyness, usually slightly less, and a 50 Delta "ATM" call option has less than 50% moneyness; these discrepancies can be observed in prices of binary options and
632:
This section outlines moneyness measures from simple but less useful to more complex but more useful. Simpler measures of moneyness can be computed immediately from observable market data without any theoretical assumptions, while more complex measures use the implied volatility, and thus the
611:
While moneyness is a function of both spot and strike, usually one of these is fixed, and the other varies. Given a specific option, the strike is fixed, and different spots yield the moneyness of that option at different market prices; this is useful in option pricing and understanding the
115:
to express the moneyness as a number, measuring how far the asset is in the money or out of the money with respect to the strike – or, conversely, how far a strike is in or out of the money with respect to the spot (or forward) price of the asset. This quantified notion of moneyness is most
1601:
the correct moneyness. The percent moneyness is the implied probability that the derivative will expire in the money, in the risk-neutral measure. Thus a moneyness of 0 yields a 50% probability of expiring ITM, while a moneyness of 1 yields an approximately 84% probability of expiring ITM.
793:
In the log simple moneyness, ATM corresponds to 0, while ITM is positive and OTM is negative, and corresponding levels of ITM/OTM corresponding to switching sign. Note that once logs are taken, moneyness in terms of forward or spot differ by an additive factor (log of discount factor), as
237:
option, the option cannot be exercised before the expiry date, so it is possible for the time value to be negative; for an
American option if the time value is ever negative, you exercise it (ignoring special circumstances such as the security going ex dividend): this yields a
951:
of the underlying asset. Unlike previous inputs, volatility is not directly observable from market data, but must instead be computed in some model, primarily using ATM implied volatility in the Black–Scholes model. Dispersion is proportional to volatility, so
603:
expires ITM, as these are complementary events). Switching spot and strike also switches these conventions, and spot and strike are often complementary in formulas for moneyness, but need not be. Which convention is used depends on the purpose. The sequel uses
878:
The above measures are independent of time, but for a given simple moneyness, options near expiry and far from expiry behave differently, as options far from expiry have more time for the underlying to change. Accordingly, one may incorporate time to maturity
742:
In (call) simple moneyness, ATM corresponds to moneyness of 1, while ITM corresponds to greater than 1, and OTM corresponds to less than 1, with equivalent levels of ITM/OTM corresponding to reciprocals. This is linearized by taking the log, yielding the
404:
Buying an ITM option is effectively lending money in the amount of the intrinsic value. Further, an ITM call can be replicated by entering a forward and buying an OTM put (and conversely). Consequently, ATM and OTM options are the main traded ones.
1197:
1318:
1052:. Standardized moneyness is measured in standard deviations from this point, with a positive value meaning an in-the-money call option and a negative value meaning an out-of-the-money call option (with signs reversed for a put option).
269:
For example, with an "at the money" call stock option, the current share price and strike price are the same. Exercising the option will not earn the seller a profit, but any move upward in stock price will give the option value.
317:(OTM) option has no intrinsic value. A call option is out of the money when the strike price is above the spot price of the underlying security. A put option is out of the money when the strike price is below the spot price.
616:. Conversely, given market data at a given point in time, the spot is fixed at the current market price, while different options have different strikes, and hence different moneyness; this is useful in constructing an
301:(ITM) option has positive intrinsic value as well as time value. A call option is in the money when the strike price is below the spot price. A put option is in the money when the strike price is above the spot price.
1028:
131:, which is the ratio of spot (or forward) to strike, or the reciprocal, depending on convention. A particularly important measure of moneyness is the likelihood that the derivative will expire in the money, in the
320:
With an "out of the money" call stock option, the current share price is less than the strike price so there is no reason to exercise the option. The owner can sell the option, or wait and hope the price changes.
438:. The condition of being a change of variables is that this function is monotone (either increasing for all inputs, or decreasing for all inputs), and the function can depend on the other parameters of the
942:
201:
with a strike at $ 80 is out-of-the-money (80 − 100 = −20 < 0). Conversely, a call option with a $ 120 strike is out-of-the-money and a put option with a $ 120 strike is in-the-money.
1574:
108:
There are two slightly different definitions, according to whether one uses the current price (spot) or future price (forward), specified as "at the money spot" or "at the money forward", etc.
874:
595:
where moneyness increases if spot decreases relative to strike. These can be switched by changing sign, possibly with a shift or scale factor (e.g., the probability that a put with strike
1411:
586:
This definition is abstract and notationally heavy; in practice relatively simple and concrete moneyness functions are used, and arguments to the function are suppressed for clarity.
503:
791:
1373:
1094:
1208:
273:
Since an option will rarely be exactly at the money, except for when it is written (when one may buy or sell an ATM option), one may speak informally of an option being
169:. (Standard deviations refer to the price fluctuations of the underlying instrument, not of the option itself.) Another measure closely related to moneyness is the
883:
into moneyness. Since dispersion of
Brownian motion is proportional to the square root of time, one may divide the log simple moneyness by this factor, yielding:
2726:
664:, and correspond to not changing coordinates, instead using the raw prices as measures of moneyness; the corresponding volatility surface, with coordinates
2638:
696:, with analogous forward simple moneyness. Conventionally the fixed quantity is in the denominator, while the variable quantity is in the numerator, so
712:
for different options at a given spot, such as when constructing a volatility surface. A volatility surface using coordinates a non-trivial moneyness
1645:. Note that for puts, Delta is negative, and thus negative Delta is used – more uniformly, absolute value of Delta is used for call/put moneyness.
422:
for valuing options (either in currency (dollar) value or in implied volatility), and changing from spot (or forward, or strike) to moneyness is a
962:
944:
This effectively normalizes for time to expiry – with this measure of moneyness, volatility smiles are largely independent of time to expiry.
547:
In order for this function to reflect moneyness – i.e., for moneyness to increase as spot and strike move relative to each other – it must be
544:
All of these are observables except for the implied volatility, which can computed from the observable price using the Black–Scholes formula.
285:; "near the money" may narrowly refer specifically to the nearest the money strike. Conversely, one may speak informally of an option being
2432:
2007:
2437:
1044:
the current forward price is above the strike price. Thus the moneyness is zero when the forward price of the underlying equals the
886:
266:
is the same as the current spot price of the underlying security. An at-the-money option has no intrinsic value, only time value.
2772:
1413:
in each case. This is often small, so the quantities are often confused or conflated, though they have distinct interpretations.
583:(put moneyness). Somewhat different formalizations are possible. Further axioms may also be added to define a "valid" moneyness.
213:
The intrinsic value (or "monetary value") of an option is its value assuming it were exercised immediately. Thus if the current (
2462:
739:
will be used in the sequel. In practice, for low interest rates and short tenors, spot versus forward makes little difference.
1953:
1889:
1738:
162:, measuring how far above or below the strike price the current price is, in terms of volatility; this quantity is given by
1972:
1931:
1497:
1733:(First ed.). New Delhi: Dorling Kindersly (India) Pvt Ltd, licensees of Pearson Education in South Asia. p. 60.
2594:
1763:
1060:
The standardized moneyness is closely related to the auxiliary variables in the Black–Scholes formula, namely the terms
797:
2330:
1908:
382:
1416:
As these are all in units of standard deviations, it makes sense to convert these to percentages, by evaluating the
364:
2832:
2000:
1420:
281:. Similarly, given standardized options (at a fixed set of strikes, say every $ 1), one can speak of which one is
608:
moneyness – as spot increases, moneyness increases – and is the same direction as using call Delta as moneyness.
2919:
2767:
2412:
731:
While the spot is often used by traders, the forward is preferred in theory, as it has better properties, thus
360:
1676:
1381:
452:
1456:, or the risk-neutral likelihood that the option will expire ITM, with numéraire cash (the risk-free asset);
85:
If the derivative would be worthless if expiring with the underlying at its current price, it is said to be
749:
1329:
2807:
2345:
2199:
1993:
1192:{\displaystyle d_{\pm }={\frac {\ln \left(F/K\right)\pm (\sigma ^{2}/2)\tau }{\sigma {\sqrt {\tau }}}}.}
2914:
2663:
2604:
2426:
617:
356:
2708:
2519:
1884:. Lecture Notes in Economics and Mathematical Systems (Paperback ed.). Berlin: Springer-Verlag.
1728:
1668:
1606:
1313:{\displaystyle m={\frac {\ln(F/K)}{\sigma {\sqrt {\tau }}}}={\tfrac {1}{2}}\left(d_{-}+d_{+}\right),}
613:
173:
of a call or put option. There are other proxies for moneyness, with convention depending on market.
155:
67:
1427:
2827:
2822:
567:), with at least one of these strictly monotone, and have opposite direction: either increasing in
345:
1590:) is the (risk-neutral) "likelihood of expiring in the money", and thus the theoretically correct
2777:
2477:
2447:
2422:
2305:
2146:
2078:
1672:
439:
349:
434:
with input the spot price (or forward, or strike) and output a real number, which is called the
2574:
2559:
2524:
2467:
919:
1714:
2787:
2554:
2452:
2131:
1828:
1804:
96:
And if the current underlying price and strike price are equal, the derivative is said to be
2741:
2698:
2688:
2678:
2673:
2399:
2340:
2275:
2229:
2224:
2098:
2058:
2025:
1431:
132:
124:
71:
51:
17:
8:
2746:
2534:
2457:
2280:
1780:
1041:
423:
159:
2797:
2782:
2751:
2736:
2703:
2569:
2360:
2325:
2088:
2053:
2016:
1483:, or the risk-neutral likelihood that the option will expire ITM, with numéraire asset.
1453:
548:
443:
238:
140:
127:
in terms of moneyness, rather than absolute price. The most basic of these measures is
120:
2802:
2792:
2731:
2718:
2693:
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2365:
2161:
1968:
1949:
1927:
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1904:
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233:
2683:
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2599:
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2214:
2093:
2033:
621:
255:
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2812:
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2589:
2499:
2407:
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2320:
2315:
2285:
2234:
2219:
2136:
2116:
1767:
1667:
this corresponds to the difference between the median and mean (respectively) of
1642:
1480:
1417:
170:
39:
1941:
23:
Difference in the price of an underlying asset and its derivative's strike price
2868:
2853:
2653:
2564:
2514:
2491:
2472:
2300:
2242:
2209:
2204:
2184:
2108:
1617:
the implied volatility. Drift is the mean, with the corresponding median (50th
525:
1760:
2908:
2848:
2817:
2658:
2584:
2544:
2539:
2375:
2247:
2194:
2189:
2171:
2068:
2048:
953:
1967:. Macmillan Business: Finance and Capital Markets (2nd ed.). Palgrave.
1690:
the higher the value at exercise, hence why Delta is higher than moneyness.
217:) price of the underlying security (or commodity etc.) is above the agreed (
2668:
2442:
2370:
2350:
2310:
2179:
2151:
2141:
2083:
1922:
Option
Volatility & Pricing: Advanced Trading Strategies and Techniques
1045:
263:
218:
47:
1435:
418:
Intuitively speaking, moneyness and time to expiry form a two-dimensional
2549:
2417:
2388:
2384:
2335:
2126:
2121:
1629:/2, which is the reason for the correction factor. Note that this is the
1023:{\displaystyle m={\frac {\ln \left(F/K\right)}{\sigma {\sqrt {\tau }}}}.}
222:
194:
186:
136:
55:
2873:
2509:
2504:
2270:
2156:
1618:
226:
214:
198:
190:
59:
1430:
of these quantities is somewhat subtle, and consists of changing to a
197:
with a strike of $ 80 is in-the-money (100 − 80 = 20 > 0). A
1985:
225:
has positive intrinsic value (and is called "in the money"), while a
334:
2633:
2355:
2252:
2073:
676:. The simplest non-trivial moneyness is the ratio of these, either
599:
expires ITM is one minus the probability that a call with strike
27:
205:
price of the option is greater than the
Forward Reference Rate.
2888:
1903:(4th ed.). New York : New York Institute of Finance.
1037:(forward), and measures moneyness in standard deviation units.
446:(concretely the ATM implied volatility), yielding a function:
396:
hence is 110, then a call struck at 110 is ATMF but not ATM.
43:
35:
1466:) is the percentage corresponding to standardized moneyness;
1055:
937:{\displaystyle \ln \left(F/K\right){\Big /}{\sqrt {\tau }}.}
1438:. In brief, these are interpreted (for a call option) as:
139:
of expiring in the money, which is the forward value of a
182:
111:
This rough classification can be quantified by various
16:"In the money" redirects here. For the poker term, see
1263:
1040:
In words, the standardized moneyness is the number of
229:
has zero intrinsic value (and is "out of the money").
1569:{\displaystyle N(d_{-})<N(m)<N(d_{+})=\Delta .}
1500:
1384:
1332:
1211:
1097:
965:
889:
800:
752:
455:
143:
with the given strike, and is equal to the auxiliary
62:. Moneyness is firstly a three-fold classification:
1730:
1202:
The standardized moneyness is the average of these:
1919:
1827:, Section 5.3.1, Choice of Moneyness Measure, pp.
1568:
1405:
1367:
1312:
1191:
1022:
936:
868:
785:
497:
1882:Stochastic Implied Votality: A Factor-Based Model
947:This measure does not account for the volatility
869:{\displaystyle \ln \left(F/K\right)=\ln(S/K)+rT.}
208:
2906:
442:, notably time to expiry, interest rates, and
2001:
532:is the implied volatility. The forward price
1754:
704:for a single option and varying spots, and
363:. Unsourced material may be challenged and
2008:
1994:
1819:
1817:
1815:
1813:
1795:
1793:
1791:
1789:
193:with a strike of $ 100 is at-the-money. A
1917:
1854:
1841:
1677:Itō's lemma for geometric Brownian motion
1056:Black–Scholes formula auxiliary variables
383:Learn how and when to remove this message
1962:
1898:
1867:
1705:
1703:
1675:), and is the same correction factor in
1605:This corresponds to the asset following
1406:{\displaystyle \sigma {\sqrt {\tau }}/2}
34:is the relative position of the current
2833:Power reverse dual-currency note (PRDC)
2773:Constant proportion portfolio insurance
1810:
1786:
498:{\displaystyle M(S,K,\tau ,r,\sigma ),}
2907:
2015:
1940:
1879:
1824:
1800:
1710:
324:
66:If the derivative would have positive
1989:
1870:), who uses spot rather than forward.
1773:
1726:
1700:
786:{\displaystyle \ln \left(F/K\right).}
512:is the spot price of the underlying,
413:
2768:Collateralized debt obligation (CDO)
1717:, 11.2 How Can We Define Moneyness?)
1368:{\displaystyle d_{-}<m<d_{+},}
536:can be computed from the spot price
361:adding citations to reliable sources
328:
1946:Principles of Financial Engineering
1452:) is the (Future Value) price of a
648:and the simplest call moneyness is
308:
181:Suppose the current stock price of
135:. It can be measured in percentage
38:(or future price) of an underlying
13:
1613:the risk-free rate, and diffusion
1560:
1491:is monotonic (since it is a CDF):
627:
575:(call moneyness) or decreasing in
244:
14:
2931:
1901:Options as a Strategic Investment
1487:These have the same ordering, as
116:importantly used in defining the
2887:
1948:(2nd ed.). Academic Press.
1421:cumulative distribution function
636:The simplest (put) moneyness is
333:
724:(with respect to the moneyness
292:
249:
158:. This can also be measured in
112:
2595:Year-on-year inflation-indexed
1899:McMillan, Lawrence G. (2002).
1860:
1847:
1834:
1720:
1648:The meaning of the factor of (
1554:
1541:
1532:
1526:
1517:
1504:
1241:
1227:
1165:
1144:
851:
837:
589:
489:
459:
209:Intrinsic value and time value
1:
2605:Zero-coupon inflation-indexed
1693:
692:which is known as the (spot)
408:
1637:the real-world probability.
1378:differing only by a step of
620:, or more simply plotting a
7:
2808:Foreign exchange derivative
2200:Callable bull/bear contract
1918:Natenberg, Sheldon (1994).
722:relative volatility surface
674:absolute volatility surface
10:
2936:
1656:is relatively subtle. For
618:implied volatility surface
176:
15:
2882:
2841:
2760:
2717:
2709:Stock market index future
2613:
2490:
2398:
2261:
2170:
2107:
2041:
2032:
2023:
1963:Tompkins, Robert (1994).
1880:Häfner, Reinhold (2004).
1669:geometric Brownian motion
1607:geometric Brownian motion
1323:and they are ordered as:
2828:Mortgage-backed security
2823:Interest rate derivative
2798:Equity-linked note (ELN)
2783:Credit-linked note (CLN)
1679:. The interpretation of
1434:with specific choice of
1088:, which are defined as:
660:These are also known as
305:(minus any commission).
74:today, it is said to be
2778:Contract for difference
2079:Risk-free interest rate
1761:At the Money Definition
1673:log-normal distribution
540:and the risk-free rate
520:is the time to expiry,
2560:Forward Rate Agreement
1803:, Definition 3.12, p.
1570:
1426:for these values. The
1407:
1369:
1314:
1193:
1035:standardized moneyness
1024:
956:by volatility yields:
938:
870:
787:
638:fixed-strike moneyness
559:(equivalently forward
499:
399:
46:) with respect to the
2920:Derivatives (finance)
2788:Credit default option
2132:Employee stock option
1571:
1408:
1370:
1315:
1194:
1048:, when the option is
1033:This is known as the
1025:
939:
871:
788:
633:Black–Scholes model.
614:Black–Scholes formula
563:which is monotone in
516:is the strike price,
500:
156:Black–Scholes formula
2742:Inflation derivative
2727:Commodity derivative
2699:Single-stock futures
2689:Normal backwardation
2679:Interest rate future
2520:Conditional variance
2026:Derivative (finance)
1727:Chugh, Aman (2013).
1498:
1432:risk-neutral measure
1382:
1330:
1209:
1095:
1050:at-the-money-forward
963:
887:
798:
750:
745:log simple moneyness
650:fixed-spot moneyness
453:
357:improve this section
133:risk-neutral measure
18:In the money (poker)
2894:Business portal
2747:Property derivative
1042:standard deviations
716:and time to expiry
440:Black–Scholes model
424:change of variables
325:Spot versus forward
160:standard deviations
2752:Weather derivative
2737:Freight derivative
2719:Exotic derivatives
2639:Commodities future
2326:Intermarket spread
2089:Synthetic position
2017:Derivatives market
1766:2012-06-16 at the
1566:
1454:binary call option
1403:
1365:
1310:
1272:
1189:
1020:
934:
866:
783:
684:or its reciprocal
662:absolute moneyness
579:and increasing in
571:and decreasing in
495:
444:implied volatility
428:moneyness function
414:Moneyness function
287:far from the money
279:close to the money
239:boundary condition
141:binary call option
121:volatility surface
54:, most commonly a
2915:Options (finance)
2902:
2901:
2803:Equity derivative
2793:Credit derivative
2761:Other derivatives
2732:Energy derivative
2694:Perpetual futures
2575:Overnight indexed
2525:Constant maturity
2486:
2485:
2433:Finite difference
2366:Protective option
1965:Options Explained
1955:978-0-12-373574-4
1891:978-3-540-22183-8
1770:, Cash Bauer 2012
1740:978-81-317-7433-5
1592:percent moneyness
1393:
1271:
1257:
1254:
1184:
1181:
1015:
1012:
929:
420:coordinate system
393:
392:
385:
283:nearest the money
2927:
2892:
2891:
2664:Forwards pricing
2438:Garman–Kohlhagen
2039:
2038:
2010:
2003:
1996:
1987:
1986:
1978:
1974:978-0-33362807-2
1959:
1942:Neftçi, Salih N.
1937:
1933:978-1-55738486-7
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694:simple moneyness
622:volatility smile
504:
502:
501:
496:
388:
381:
377:
374:
368:
337:
329:
315:out of the money
309:Out of the money
129:simple moneyness
87:out of the money
2935:
2934:
2930:
2929:
2928:
2926:
2925:
2924:
2905:
2904:
2903:
2898:
2886:
2878:
2864:Great Recession
2859:Government debt
2837:
2813:Fund derivative
2756:
2713:
2674:Futures pricing
2649:Dividend future
2644:Currency future
2627:
2609:
2482:
2458:Put–call parity
2394:
2381:Vertical spread
2316:Diagonal spread
2286:Calendar spread
2257:
2166:
2103:
2028:
2019:
2014:
1983:
1981:
1975:
1956:
1934:
1926:. McGraw-Hill.
1911:
1892:
1875:
1874:
1865:
1861:
1852:
1848:
1839:
1835:
1822:
1811:
1798:
1787:
1783:", Investopedia
1778:
1774:
1768:Wayback Machine
1759:
1755:
1745:
1743:
1741:
1725:
1721:
1708:
1701:
1696:
1685:
1662:
1600:
1589:
1548:
1544:
1511:
1507:
1499:
1496:
1495:
1478:
1451:
1418:standard normal
1395:
1388:
1383:
1380:
1379:
1356:
1352:
1337:
1333:
1331:
1328:
1327:
1296:
1292:
1283:
1279:
1278:
1274:
1262:
1249:
1245:
1233:
1220:
1218:
1210:
1207:
1206:
1176:
1172:
1157:
1151:
1147:
1128:
1124:
1120:
1113:
1111:
1102:
1098:
1096:
1093:
1092:
1087:
1080:
1073:
1066:
1058:
1007:
1003:
989:
985:
981:
974:
972:
964:
961:
960:
924:
918:
917:
904:
900:
896:
888:
885:
884:
843:
815:
811:
807:
799:
796:
795:
767:
763:
759:
751:
748:
747:
672:(tenor) is the
630:
628:Simple examples
592:
454:
451:
450:
416:
411:
402:
389:
378:
372:
369:
354:
338:
327:
311:
295:
252:
247:
245:Moneyness terms
211:
179:
168:
153:
68:intrinsic value
24:
21:
12:
11:
5:
2933:
2923:
2922:
2917:
2900:
2899:
2897:
2896:
2883:
2880:
2879:
2877:
2876:
2871:
2869:Municipal debt
2866:
2861:
2856:
2854:Corporate debt
2851:
2845:
2843:
2839:
2838:
2836:
2835:
2830:
2825:
2820:
2815:
2810:
2805:
2800:
2795:
2790:
2785:
2780:
2775:
2770:
2764:
2762:
2758:
2757:
2755:
2754:
2749:
2744:
2739:
2734:
2729:
2723:
2721:
2715:
2714:
2712:
2711:
2706:
2701:
2696:
2691:
2686:
2681:
2676:
2671:
2666:
2661:
2656:
2654:Forward market
2651:
2646:
2641:
2636:
2630:
2628:
2626:
2625:
2620:
2614:
2611:
2610:
2608:
2607:
2602:
2597:
2592:
2587:
2582:
2577:
2572:
2567:
2562:
2557:
2552:
2547:
2542:
2537:
2535:Credit default
2532:
2527:
2522:
2517:
2512:
2507:
2502:
2496:
2494:
2488:
2487:
2484:
2483:
2481:
2480:
2475:
2470:
2465:
2460:
2455:
2450:
2445:
2440:
2435:
2430:
2420:
2415:
2410:
2404:
2402:
2396:
2395:
2393:
2392:
2378:
2373:
2368:
2363:
2358:
2353:
2348:
2343:
2338:
2333:
2331:Iron butterfly
2328:
2323:
2318:
2313:
2308:
2303:
2301:Covered option
2298:
2293:
2288:
2283:
2278:
2273:
2267:
2265:
2259:
2258:
2256:
2255:
2250:
2245:
2240:
2239:Mountain range
2237:
2232:
2227:
2222:
2217:
2212:
2207:
2202:
2197:
2192:
2187:
2182:
2176:
2174:
2168:
2167:
2165:
2164:
2159:
2154:
2149:
2144:
2139:
2134:
2129:
2124:
2119:
2113:
2111:
2105:
2104:
2102:
2101:
2096:
2091:
2086:
2081:
2076:
2071:
2066:
2061:
2056:
2051:
2045:
2043:
2036:
2030:
2029:
2024:
2021:
2020:
2013:
2012:
2005:
1998:
1990:
1980:
1979:
1973:
1960:
1954:
1938:
1932:
1915:
1909:
1896:
1890:
1876:
1873:
1872:
1859:
1855:Natenberg 1994
1846:
1844:, pp. 106–110)
1842:Natenberg 1994
1833:
1809:
1785:
1781:Near The Money
1772:
1753:
1739:
1719:
1698:
1697:
1695:
1692:
1683:
1660:
1598:
1587:
1577:
1576:
1565:
1562:
1559:
1556:
1551:
1547:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1514:
1510:
1506:
1503:
1485:
1484:
1476:
1467:
1457:
1449:
1428:interpretation
1402:
1398:
1392:
1387:
1376:
1375:
1364:
1359:
1355:
1351:
1348:
1345:
1340:
1336:
1321:
1320:
1309:
1305:
1299:
1295:
1291:
1286:
1282:
1277:
1270:
1267:
1261:
1253:
1248:
1243:
1240:
1236:
1232:
1229:
1226:
1223:
1217:
1214:
1200:
1199:
1188:
1180:
1175:
1170:
1167:
1164:
1160:
1154:
1150:
1146:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1116:
1110:
1105:
1101:
1085:
1078:
1071:
1064:
1057:
1054:
1031:
1030:
1019:
1011:
1006:
1000:
996:
992:
988:
984:
980:
977:
971:
968:
933:
928:
921:
915:
911:
907:
903:
899:
895:
892:
865:
862:
859:
856:
853:
850:
846:
842:
839:
836:
833:
830:
826:
822:
818:
814:
810:
806:
803:
782:
778:
774:
770:
766:
762:
758:
755:
720:is called the
629:
626:
591:
588:
555:and in strike
526:risk-free rate
506:
505:
494:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
461:
458:
430:is a function
415:
412:
410:
407:
401:
398:
391:
390:
341:
339:
332:
326:
323:
310:
307:
294:
291:
275:near the money
251:
248:
246:
243:
210:
207:
178:
175:
166:
154:) term in the
151:
123:: the implied
106:
105:
94:
83:
70:if it were to
22:
9:
6:
4:
3:
2:
2932:
2921:
2918:
2916:
2913:
2912:
2910:
2895:
2890:
2885:
2884:
2881:
2875:
2872:
2870:
2867:
2865:
2862:
2860:
2857:
2855:
2852:
2850:
2849:Consumer debt
2847:
2846:
2844:
2842:Market issues
2840:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2818:Fund of funds
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2796:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2765:
2763:
2759:
2753:
2750:
2748:
2745:
2743:
2740:
2738:
2735:
2733:
2730:
2728:
2725:
2724:
2722:
2720:
2716:
2710:
2707:
2705:
2702:
2700:
2697:
2695:
2692:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2665:
2662:
2660:
2659:Forward price
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2631:
2629:
2624:
2621:
2619:
2616:
2615:
2612:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2570:Interest rate
2568:
2566:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2531:
2528:
2526:
2523:
2521:
2518:
2516:
2513:
2511:
2508:
2506:
2503:
2501:
2498:
2497:
2495:
2493:
2489:
2479:
2476:
2474:
2471:
2469:
2466:
2464:
2463:MC Simulation
2461:
2459:
2456:
2454:
2451:
2449:
2446:
2444:
2441:
2439:
2436:
2434:
2431:
2428:
2424:
2423:Black–Scholes
2421:
2419:
2416:
2414:
2411:
2409:
2406:
2405:
2403:
2401:
2397:
2390:
2386:
2382:
2379:
2377:
2376:Risk reversal
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2327:
2324:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2306:Credit spread
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2279:
2277:
2274:
2272:
2269:
2268:
2266:
2264:
2260:
2254:
2251:
2249:
2246:
2244:
2241:
2238:
2236:
2233:
2231:
2230:Interest rate
2228:
2226:
2225:Forward start
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2203:
2201:
2198:
2196:
2193:
2191:
2188:
2186:
2183:
2181:
2178:
2177:
2175:
2173:
2169:
2163:
2160:
2158:
2155:
2153:
2152:Option styles
2150:
2148:
2145:
2143:
2140:
2138:
2135:
2133:
2130:
2128:
2125:
2123:
2120:
2118:
2115:
2114:
2112:
2110:
2106:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2069:Open interest
2067:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2049:Delta neutral
2047:
2046:
2044:
2040:
2037:
2035:
2031:
2027:
2022:
2018:
2011:
2006:
2004:
1999:
1997:
1992:
1991:
1988:
1984:
1976:
1970:
1966:
1961:
1957:
1951:
1947:
1943:
1939:
1935:
1929:
1924:
1923:
1916:
1912:
1910:0-7352-0197-8
1906:
1902:
1897:
1893:
1887:
1883:
1878:
1877:
1869:
1868:Tompkins 1994
1863:
1856:
1850:
1843:
1837:
1830:
1826:
1820:
1818:
1816:
1814:
1806:
1802:
1796:
1794:
1792:
1790:
1782:
1776:
1769:
1765:
1762:
1757:
1742:
1736:
1732:
1731:
1723:
1716:
1712:
1706:
1704:
1699:
1691:
1689:
1682:
1678:
1674:
1670:
1666:
1659:
1655:
1651:
1646:
1644:
1638:
1636:
1633:probability,
1632:
1628:
1624:
1620:
1616:
1612:
1608:
1603:
1597:
1593:
1586:
1582:
1563:
1557:
1549:
1545:
1538:
1535:
1529:
1523:
1520:
1512:
1508:
1501:
1494:
1493:
1492:
1490:
1482:
1475:
1471:
1468:
1465:
1461:
1458:
1455:
1448:
1444:
1441:
1440:
1439:
1437:
1433:
1429:
1425:
1422:
1419:
1414:
1400:
1396:
1390:
1385:
1362:
1357:
1353:
1349:
1346:
1343:
1338:
1334:
1326:
1325:
1324:
1307:
1303:
1297:
1293:
1289:
1284:
1280:
1275:
1268:
1265:
1259:
1251:
1246:
1238:
1234:
1230:
1224:
1221:
1215:
1212:
1205:
1204:
1203:
1186:
1178:
1173:
1168:
1162:
1158:
1152:
1148:
1141:
1137:
1133:
1129:
1125:
1121:
1117:
1114:
1108:
1103:
1099:
1091:
1090:
1089:
1084:
1077:
1070:
1063:
1053:
1051:
1047:
1043:
1038:
1036:
1017:
1009:
1004:
998:
994:
990:
986:
982:
978:
975:
969:
966:
959:
958:
957:
955:
954:standardizing
950:
945:
931:
926:
913:
909:
905:
901:
897:
893:
890:
882:
876:
863:
860:
857:
854:
848:
844:
840:
834:
831:
828:
824:
820:
816:
812:
808:
804:
801:
780:
776:
772:
768:
764:
760:
756:
753:
746:
740:
738:
734:
729:
727:
723:
719:
715:
711:
707:
703:
699:
695:
691:
687:
683:
679:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
634:
625:
623:
619:
615:
609:
607:
602:
598:
587:
584:
582:
578:
574:
570:
566:
562:
558:
554:
551:in both spot
550:
545:
543:
539:
535:
531:
527:
523:
519:
515:
511:
492:
486:
483:
480:
477:
474:
471:
468:
465:
462:
456:
449:
448:
447:
445:
441:
437:
433:
429:
425:
421:
406:
397:
387:
384:
376:
366:
362:
358:
352:
351:
347:
342:This section
340:
336:
331:
330:
322:
318:
316:
306:
302:
300:
290:
288:
284:
280:
276:
271:
267:
265:
262:(ATM) if the
261:
257:
242:
240:
235:
230:
228:
224:
220:
216:
206:
202:
200:
196:
192:
188:
184:
174:
172:
165:
161:
157:
150:
146:
142:
138:
134:
130:
126:
122:
119:
114:
109:
103:
99:
95:
92:
88:
84:
81:
77:
73:
69:
65:
64:
63:
61:
57:
53:
49:
45:
41:
37:
33:
29:
19:
2669:Forward rate
2580:Total return
2468:Real options
2371:Ratio spread
2351:Naked option
2311:Debit spread
2142:Fixed income
2084:Strike price
2063:
1982:
1964:
1945:
1921:
1900:
1881:
1862:
1849:
1836:
1775:
1756:
1744:. Retrieved
1729:
1722:
1687:
1680:
1664:
1657:
1653:
1649:
1647:
1639:
1634:
1630:
1626:
1622:
1614:
1610:
1604:
1595:
1591:
1584:
1580:
1578:
1488:
1486:
1473:
1469:
1463:
1459:
1446:
1442:
1423:
1415:
1377:
1322:
1201:
1082:
1075:
1068:
1061:
1059:
1049:
1046:strike price
1039:
1034:
1032:
948:
946:
880:
877:
744:
741:
736:
732:
730:
725:
721:
717:
713:
709:
705:
701:
697:
693:
689:
685:
681:
677:
673:
669:
665:
661:
657:
653:
649:
645:
641:
637:
635:
631:
610:
605:
600:
596:
593:
585:
580:
576:
572:
568:
564:
560:
556:
552:
546:
541:
537:
533:
529:
521:
517:
513:
509:
507:
435:
431:
427:
417:
403:
394:
379:
370:
355:Please help
343:
319:
314:
312:
303:
299:in the money
298:
296:
293:In the money
286:
282:
278:
274:
272:
268:
264:strike price
260:at the money
259:
253:
250:At the money
231:
212:
203:
185:is $ 100. A
180:
163:
148:
144:
128:
117:
110:
107:
101:
98:at the money
97:
90:
86:
79:
76:in the money
75:
48:strike price
31:
25:
2600:Zero Coupon
2530:Correlation
2478:Vanna–Volga
2336:Iron condor
2122:Bond option
1825:Häfner 2004
1801:Häfner 2004
1713:, pp.
1711:Neftçi 2008
1609:with drift
590:Conventions
221:) price, a
137:probability
113:definitions
56:call option
2909:Categories
2874:Tax policy
2590:Volatility
2500:Amortising
2341:Jelly roll
2276:Box spread
2271:Backspread
2263:Strategies
2099:Volatility
2094:the Greeks
2059:Expiration
1694:References
1619:percentile
1579:Of these,
409:Definition
234:time value
199:put option
191:put option
125:volatility
60:put option
52:derivative
2565:Inflation
2515:Commodity
2473:Trinomial
2408:Bachelier
2400:Valuation
2281:Butterfly
2215:Commodore
2064:Moneyness
1746:18 August
1561:Δ
1513:−
1479:) is the
1436:numéraire
1391:τ
1386:σ
1339:−
1285:−
1252:τ
1247:σ
1225:
1179:τ
1174:σ
1169:τ
1149:σ
1142:±
1118:
1104:±
1010:τ
1005:σ
979:
927:τ
894:
835:
805:
757:
487:σ
475:τ
436:moneyness
426:. Thus a
373:June 2008
344:does not
42:(e.g., a
32:moneyness
2704:Slippage
2634:Contango
2618:Forwards
2585:Variance
2545:Dividend
2540:Currency
2453:Margrabe
2448:Lattices
2427:equation
2413:Binomial
2361:Strangle
2356:Straddle
2253:Swaption
2235:Lookback
2220:Compound
2162:Warrants
2137:European
2117:American
2109:Vanillas
2074:Pin risk
2054:Exercise
1944:(2008).
1764:Archived
1621:) being
652:, where
640:, where
549:monotone
118:relative
2623:Futures
2243:Rainbow
2210:Cliquet
2205:Chooser
2185:Barrier
2172:Exotics
2034:Options
1715:458–460
1661:−
1631:implied
1625:−
1599:−
1594:, with
1588:−
1450:−
1079:−
524:is the
365:removed
350:sources
177:Example
28:finance
2684:Margin
2550:Equity
2443:Heston
2346:Ladder
2296:Condor
2291:Collar
2248:Spread
2195:Binary
2190:Basket
1971:
1952:
1930:
1907:
1888:
1737:
528:, and
508:where
256:option
219:strike
72:expire
2555:Forex
2510:Basis
2505:Asset
2492:Swaps
2418:Black
2321:Fence
2180:Asian
2042:Terms
1829:85–87
1671:(the
1481:Delta
171:Delta
58:or a
50:of a
44:stock
40:asset
36:price
2389:Bull
2385:Bear
2127:Call
1969:ISBN
1950:ISBN
1928:ISBN
1905:ISBN
1886:ISBN
1748:2014
1735:ISBN
1663:and
1536:<
1521:<
1350:<
1344:<
1074:and
668:and
606:call
348:any
346:cite
232:The
223:call
215:spot
195:call
187:call
2157:Put
1688:and
1652:/2)
1635:not
728:).
400:Use
359:by
313:An
297:An
277:or
258:is
254:An
227:put
189:or
183:IBM
102:ATM
91:OTM
80:ITM
26:In
2911::
2387:,
2147:FX
1812:^
1805:42
1788:^
1702:^
1615:σ,
1611:r,
1222:ln
1115:ln
1081:=
1067:=
976:ln
891:ln
832:ln
802:ln
754:ln
690:S,
658:S.
646:K,
624:.
561:F,
542:r.
289:.
241:.
104:).
93:);
82:);
30:,
2429:)
2425:(
2391:)
2383:(
2009:e
2002:t
1995:v
1977:.
1958:.
1936:.
1913:.
1894:.
1866:(
1857:)
1853:(
1840:(
1831:)
1823:(
1807:)
1799:(
1779:"
1750:.
1709:(
1684:+
1681:d
1665:m
1658:d
1654:τ
1650:σ
1627:σ
1623:r
1596:d
1585:d
1583:(
1581:N
1564:.
1558:=
1555:)
1550:+
1546:d
1542:(
1539:N
1533:)
1530:m
1527:(
1524:N
1518:)
1509:d
1505:(
1502:N
1489:N
1477:+
1474:d
1472:(
1470:N
1464:m
1462:(
1460:N
1447:d
1445:(
1443:N
1424:N
1401:2
1397:/
1363:,
1358:+
1354:d
1347:m
1335:d
1308:,
1304:)
1298:+
1294:d
1290:+
1281:d
1276:(
1269:2
1266:1
1260:=
1242:)
1239:K
1235:/
1231:F
1228:(
1216:=
1213:m
1187:.
1166:)
1163:2
1159:/
1153:2
1145:(
1138:)
1134:K
1130:/
1126:F
1122:(
1109:=
1100:d
1086:2
1083:d
1076:d
1072:1
1069:d
1065:+
1062:d
1018:.
999:)
995:K
991:/
987:F
983:(
970:=
967:m
949:σ
932:.
920:/
914:)
910:K
906:/
902:F
898:(
881:τ
864:.
861:T
858:r
855:+
852:)
849:K
845:/
841:S
838:(
829:=
825:)
821:K
817:/
813:F
809:(
781:.
777:)
773:K
769:/
765:F
761:(
737:K
735:/
733:F
726:M
718:τ
714:M
710:S
708:/
706:K
702:K
700:/
698:S
688:/
686:K
682:K
680:/
678:S
670:T
666:K
656:=
654:M
644:=
642:M
601:K
597:K
581:K
577:S
573:K
569:S
565:S
557:K
553:S
538:S
534:F
530:σ
522:r
518:τ
514:K
510:S
493:,
490:)
484:,
481:r
478:,
472:,
469:K
466:,
463:S
460:(
457:M
432:M
386:)
380:(
375:)
371:(
367:.
353:.
167:2
164:d
152:2
149:d
147:(
145:N
100:(
89:(
78:(
20:.
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