Editing Black–Scholes model (section)

==Black–Scholes in practice==
[[File:Crowd outside nyse.jpg|thumb|The normality assumption of the Black–Scholes model does not capture extreme movements such as [[stock market crash]]es.]]
The assumptions of the Black–Scholes model are not all empirically valid. The model is widely employed as a useful approximation to reality, but proper application requires understanding its limitations{{snd}} blindly following the model exposes the user to unexpected risk.<ref>{{cite SSRN |last=Yalincak |first=Hakan |date=2012 |title=Criticism of the Black–Scholes Model: But Why Is It Still Used? (The Answer is Simpler than the Formula |ssrn=2115141 }}</ref>{{Unreliable source?|reason=Unpublished working paper.|date=November 2020}} Among the most significant limitations are:

* the underestimation of extreme moves, yielding [[tail risk]], which can be hedged with [[out-of-the-money]] options;
* the assumption of instant, cost-less trading, yielding [[liquidity risk]], which is difficult to hedge;
* the assumption of a stationary process, yielding [[volatility risk]], which can be hedged with volatility hedging;
* the assumption of continuous time and continuous trading, yielding gap risk, which can be hedged with Gamma hedging;
* the model tends to underprice deep out-of-the-money options and overprice deep in-the-money options.<ref>{{cite journal |last1=Macbeth |first1=James D. |last2=Merville |first2=Larry J. |title=An Empirical Examination of the Black-Scholes Call Option Pricing Model |journal=The Journal of Finance |date=December 1979 |volume=34 |issue=5 |pages=1173–1186 |doi=10.2307/2327242 |jstor=2327242 |quote=With the lone exception of out of the money options with less than ninety days to expiration, the extent to which the B-S model underprices (overprices) an in the money (out of the money) option increases with the extent to which the option is in the money (out of the money), and decreases as the time to expiration decreases.}}</ref>

In short, while in the Black–Scholes model one can perfectly hedge options by simply [[Delta hedging]], in practice there are many other sources of risk.

Results using the Black–Scholes model differ from real world prices because of simplifying assumptions of the model. One significant limitation is that in reality security prices do not follow a strict stationary [[Log-normal distribution|log-normal]] process, nor is the risk-free interest actually known (and is not constant over time). The variance has been observed to be non-constant leading to models such as [[GARCH]] to model volatility changes. Pricing discrepancies between empirical and the Black–Scholes model have long been observed in options that are far [[out-of-the-money]], corresponding to extreme price changes; such events would be very rare if returns were lognormally distributed, but are observed much more often in practice.

Nevertheless, Black–Scholes pricing is widely used in practice,<ref name="bodie-kane-marcus"/>{{rp|751}}<ref name = "Wilmott Defence"/> because it is:

* easy to calculate
* a useful approximation, particularly when analyzing the direction in which prices move when crossing critical points
* a robust basis for more refined models
* reversible, as the model's original output, price, can be used as an input and one of the other variables solved for; the implied volatility calculated in this way is often used to quote option prices (that is, as a ''quoting convention'').

The first point is self-evidently useful. The others can be further discussed:

Useful approximation: although volatility is not constant, results from the model are often helpful in setting up hedges in the correct proportions to minimize risk. Even when the results are not completely accurate, they serve as a first approximation to which adjustments can be made.

Basis for more refined models: The Black–Scholes model is ''robust'' in that it can be adjusted to deal with some of its failures. Rather than considering some parameters (such as volatility or interest rates) as ''constant,'' one considers them as ''variables,'' and thus added sources of risk. This is reflected in the [[Greeks (finance)|Greeks]] (the change in option value for a change in these parameters, or equivalently the partial derivatives with respect to these variables), and hedging these Greeks mitigates the risk caused by the non-constant nature of these parameters. Other defects cannot be mitigated by modifying the model, however, notably tail risk and liquidity risk, and these are instead managed outside the model, chiefly by minimizing these risks and by [[stress testing]].

Explicit modeling: this feature means that, rather than ''assuming'' a volatility ''a priori'' and computing prices from it, one can use the model to solve for volatility, which gives the [[implied volatility]] of an option at given prices, durations and exercise prices. Solving for volatility over a given set of durations and strike prices, one can construct an [[volatility surface|implied volatility surface]]. In this application of the Black–Scholes model, a [[coordinate transformation]] from the ''price domain'' to the ''volatility domain'' is obtained. Rather than quoting option prices in terms of dollars per unit (which are hard to compare across strikes, durations and coupon frequencies), option prices can thus be quoted in terms of implied volatility, which leads to trading of volatility in option markets.

===The volatility smile===
{{Main|Volatility smile}}
One of the attractive features of the Black–Scholes model is that the parameters in the model other than the volatility (the time to maturity, the strike, the risk-free interest rate, and the current underlying price) are unequivocally observable. All other things being equal, an option's theoretical value is a [[Monotonic function|monotonic increasing function]] of implied volatility.

By computing the implied volatility for traded options with different strikes and maturities, the Black–Scholes model can be tested. If the Black–Scholes model held, then the implied volatility for a particular stock would be the same for all strikes and maturities. In practice, the [[volatility smile|volatility surface]] (the 3D graph of implied volatility against strike and maturity) is not flat.

The typical shape of the implied volatility curve for a given maturity depends on the underlying instrument. Equities tend to have skewed curves: compared to [[at-the-money]], implied volatility is substantially higher for low strikes, and slightly lower for high strikes. Currencies tend to have more symmetrical curves, with implied volatility lowest at-the-money, and higher volatilities in both wings. Commodities often have the reverse behavior to equities, with higher implied volatility for higher strikes.

Despite the existence of the volatility smile (and the violation of all the other assumptions of the Black–Scholes model), the Black–Scholes PDE and Black–Scholes formula are still used extensively in practice. A typical approach is to regard the volatility surface as a fact about the market, and use an implied volatility from it in a Black–Scholes valuation model. This has been described as using "the wrong number in the wrong formula to get the right price".<ref>{{cite book|author=Riccardo Rebonato|author-link=Riccardo Rebonato|year=1999|title=Volatility and correlation in the pricing of equity, FX and interest-rate options|publisher=Wiley|isbn=0-471-89998-4}}</ref> This approach also gives usable values for the hedge ratios (the Greeks). Even when more advanced models are used, traders prefer to think in terms of Black–Scholes implied volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on. For a discussion as to the various alternative approaches developed here, see {{slink|Financial economics|Challenges and criticism}}.

===Valuing bond options===
Black–Scholes cannot be applied directly to [[bond (finance)|bond securities]] because of [[pull to par|pull-to-par]]. As the bond reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing its volatility, and the simple Black–Scholes model does not reflect this process. A large number of extensions to Black–Scholes, beginning with the [[Black model]], have been used to deal with this phenomenon.<ref>{{cite journal|first=Andrew|last=Kalotay|author-link=Andrew Kalotay|url=http://kalotay.com/sites/default/files/private/BlackScholes.pdf|title=The Problem with Black, Scholes et al.|journal=Derivatives Strategy|date=November 1995}}</ref> See {{sectionlink|Bond option#Valuation}}.

===Interest rate curve===
In practice, interest rates are not constant—they vary by tenor (coupon frequency), giving an [[yield curve|interest rate curve]] which may be interpolated to pick an appropriate rate to use in the Black–Scholes formula. Another consideration is that interest rates vary over time. This volatility may make a significant contribution to the price, especially of long-dated options. This is simply like the interest rate and bond price relationship which is inversely related.

===Short stock rate===
Taking a [[short (finance)|short stock]] position, as inherent in the derivation, is not typically free of cost; equivalently, it is possible to lend out a long stock position [[Short (finance)#Short selling terms|for a small fee]]. In either case, this can be treated as a continuous dividend for the purposes of a Black–Scholes valuation, provided that there is no glaring asymmetry between the short stock borrowing cost and the long stock lending income.{{Citation needed|date=April 2012}}