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蜗牛新壳

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Emanuel Derman  

2009-08-07 22:08:36|  分类: 金融 |  标签: |举报 |字号 订阅

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这些东西我不懂。留个记号。

Emanuel Derman - shu4huan4 - 蜗牛新壳 

http://www.ederman.com/new/index.html

http://www.ieor.columbia.edu/fac-bios/derman/faculty.html

http://en.wikipedia.org/wiki/Derman

http://en.wikipedia.org/wiki/Black%E2%80%93Scholes

Emanuel Derman is a South African-born academic and businessman. He is best known as a quantitative analyst, and author of the book My Life as a Quant: Reflections on Physics and Finance[1]. He is a co-author of Black-Derman-Toy model, one of the first interest-rate models, and the Derman-Kani local volatility or implied tree model, the first model consistent with the volatility smile.

He is currently a professor at Columbia University and Director of its program in financial engineering, and is also the Head of Risk and a partner at Prisma Capital Partners, a fund of funds. My Life as A Quant: Reflections on Physics and Finance was published by Wiley in September 2004, and was one of Business Week's top ten books of the year for 2004[2].

Derman studied at the University of Cape Town, and received a Ph.D. in theoretical physics from Columbia in 1973, where he wrote a thesis that proposed a test for a weak-neutral current in electron-hadron scattering. This experiment was carried out at SLAC in 1978 by a team led by Charles Prescott and Richard Taylor, and confirmed the Weinberg-Salam model. Between 1973 and 1980 he did research in theoretical particle physics at the University of Pennsylvania, the University of Oxford, Rockefeller University and the University of Colorado at Boulder. From 1980 to 1985 he worked at AT&T Bell Laboratories, where he developed computer languages for business modeling applications.

In 1985 Derman joined Goldman Sachs' fixed income division where he was one of the co-developers of the Black-Derman-Toy interest-rate model.

He left Goldman Sachs at the end of 1988 to take a position at Salomon Brothers Inc. as a Head of Adjustable Rate Mortgage Research in the Bond Portfolio Analysis group.

Rehired by Goldman Sachs, from 1990 to 2000 he led the Quantitative Strategies group in the Equities division, which pioneered the study of local volatility models and the volatility smile. He was appointed a managing director of Goldman Sachs in 1997. In 2000 he became head of the firm’s Quantitative Risk Strategies group. He retired from Goldman Sachs in 2002 and took up his current positions at Columbia University and Prisma Capital Partners.

Derman was named the IAFE/Sungard Financial Engineer of the Year 2000[3], and was elected to the Risk Hall of Fame in 2002[4]. He is the author of numerous articles on quantitative finance on the topics of volatility and the nature of financial modeling[5].

Since 1995, Derman has written many articles pointing out the essential difference between models in physics and models in finance. Good models in physics aim to predict the future accurately from the present, or to predict new previously unobserved phenomena; models in finance are used mostly to estimate the values of illiquid securities from liquid ones. Models in physics deal with objective variables; models in finance deal with subjective ones. “In physics there may one day be a Theory of Everything; in finance and the social sciences, you’re lucky if there is a useable theory of anything.”

Professor Derman is together with Paul Wilmott one of the authors of the Financial Modelers' Manifesto.[6]

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Black-Derman-Toy model

In finance, the Black-Derman-Toy model is a model of the evolution of the yield curve, sometimes referred to as a short rate model. It is a one-factor model; that is, a single stochastic factor (the short rate) determines the future evolution of all interest rates. One can calibrate the parameters in the BDT model to fit the current term structure of interest rates (yield curve) as well as volatility structure as derived from implied (from the Black-76 model) prices for interest rate caps. From here, one can value a variety of more complex interest-rate sensitive securities.

The model was introduced by Fischer Black, Emanuel Derman, and Bill Toy. It was first developed for in-house use by Goldman Sachs in the 1980s and was eventually published in the Financial Analysts Journal in 1990. A personal account of the development of the model is provided in one of the chapters in Emanuel Derman's memoir "My Life as a Quant."

Volatility smile

In finance, the volatility smile is a long-observed pattern in which at-the-money options tend to have lower implied volatilities than in- or out-of-the-money options. The pattern displays different characteristics for different markets and results from the probability of extreme moves. Equity options traded in American markets did not show a volatility smile before the Crash of 1987 but began showing one afterwards.[1]

Modelling the volatility smile is an active area of research in quantitative finance. Typically, a quantitative analyst will calculate the implied volatility from liquid vanilla options and use models of the smile to calculate the price of more exotic options.

A closely related concept is that of term structure of volatility, which refers to how implied volatility differs for related options with different maturities. An implied volatility surface is a 3-D plot that combines volatility smile and term structure of volatility into a consolidated view of all options for an underlier.

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In the Black-Scholes model, the theoretical value of a vanilla option is a monotonic increasing function of the Black-Scholes volatility. Furthermore, except in the case of American options with dividends whose early exercise could be optimal, the price is a strictly increasing function of volatility. This means it is usually possible to compute a unique implied volatility from a given market price for an option. This implied volatility is best regarded as a rescaling of option prices which makes comparisons between different strikes, expirations, and underlyings easier and more intuitive.

When implied volatility is plotted against strike price, the resulting graph is typically downward sloping for equity markets, or valley-shaped for currency markets. For markets where the graph is downward sloping, such as for equity options, the term "volatility skew" is often used. For other markets, such as FX options or equity index options, where the typical graph turns up at either end, the more familiar term "volatility smile" is used. For example, the implied volatility for upside (i.e. high strike) equity options is typically lower than for at-the-money equity options. However, the implied volatilities of options on foreign exchange contracts tend to rise in both the downside and upside directions. In equity markets, a small tilted smile is often observed near the money as a kink in the general downward sloping implicit volatility graph. Sometimes the term "smirk" is used to describe a skewed smile.

Market practitioners use the term implied-volatility to indicate the volatility parameter for ATM (at-the-money) option. Adjustments to this value is undertaken by incorporating the values of Risk Reversal and Flys (Skews) to determine the actual volatility measure that may be used for an options with a delta which is not 50.

Callx = ATMx + 0.5 RRx + Flyx

Putx = ATMx - 0.5 RRx + Flyx

Risk reversals are generally quoted X% delta risk reversal and essentially is Long X% delta call, and short X% delta put.

Butterfly, on the other hand, is Y% delta fly which mean Long Y% delta call, Long Y% delta put, and short ATM.


Image:volatility smile.svg

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< and historical volatility>

It is helpful to note that implied volatility is related to historical volatility, however the two are distinct. Historical volatility is a direct measure of the movement of the underlier’s price (realized volatility) over recent history (e.g. a trailing 21-day period). Implied volatility, in contrast, is set by the market price of the derivative contract itself, and not the underlier. Therefore, different derivative contracts on the same underlier have different implied volatilities. For instance, the IBM call option, struck at $100 and expiring in 6 months, may have an implied volatility of 18%, while the put option struck at $105 and expiring in 1 month may have an implied volatility of 21%. At the same time, the historical volatility for IBM for the previous 21 day period might be 17% (all volatilities are expressed in annualized percentage moves).

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< of structure>

For options of different maturities, we also see characteristic differences in implied volatility. However, in this case, the dominant effect is related to the market's implied impact of upcoming events. For instance, it is well-observed that realized volatility for stock prices rises significantly on the day that a company reports its earnings. Correspondingly, we see that implied volatility for options will rise during the period prior to the earnings announcement, and then fall again as soon as the stock price absorbs the new information. Options that mature earlier exhibit a larger swing in implied volatility than options with longer maturities.

Other option markets show other behavior. For instance, options on commodity futures typically show increased implied volatility just prior to the announcement of harvest forecasts. Options on US Treasury Bill futures show increased implied volatility just prior to meetings of the Federal Reserve Board (when changes in short-term interest rates are announced).

The market incorporates many other types of events into the term structure of volatility. For instance, the impact of upcoming results of a drug trial can cause implied volatility swings for pharmaceutical stocks. The anticipated resolution date of patent litigation can impact technology stocks, etc.

Volatility term structures list the relationship between implied volatilities and time to expiration. The term structures provide another method for traders to gauge cheap or expensive options.

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It is often useful to plot implied volatility as a function of both strike price and time to maturity. The result is a 3-D surface whereby the current market implied volatility (Z-axis) for all options on the underlier is plotted against strike price and time to maturity (X & Y-axes).

The implied volatility surface simultaneously shows both volatility smile and term structure of volatility. Option traders use an implied volatility plot to quickly determine the shape of the implied volatility surface, and to identify any areas where the slope of the plot (and therefore relative implied volatilities) seems out of line.

The graph shows an implied volatility surface for all the call options on a particular underlying stock price. The Z-axis represents implied volatility in percent, and X and Y axes represent the option delta, and the days to maturity. Note that to maintain put-call parity, a 20 delta put must have the same implied volatility as an 80 delta call. For this surface, we can see that the underlying symbol has both volatility skew (a tilt along the delta axis), as well as a volatility term structure indicating an anticipated event in the near future.

Image:Ivsrf.gif

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An implied volatility surface is static: it describes the implied volatilities at a given moment in time. How the surface changes over time (especially as spot changes) is called the evolution of the implied volatility surface.

Common heuristics include:

  • "sticky strike" (or "sticky-by-strike", or "stick-to-strike"): if spot changes, the implied volatility of an option with a given absolute strike does not change.
  • "sticky moneyness" (aka, "sticky delta"; see moneyness for why these are equivalent terms): if spot changes, the implied volatility of an option with a given moneyness does not change.

So if spot moves from $100 to $120, sticky strike would predict that the implied volatility of a $120 strike option would be whatever it was before the move (though it has moved from being OTM to ATM), while sticky delta would predict that the implied volatility of the $120 strike option would be whatever the $100 strike option's implied volatility was before the move (as these are both ATM at the time).

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Methods of modelling the volatility smile include stochastic volatility models and local volatility models.

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The term Black–Scholes refers to three closely related concepts:

Fischer Black and Myron Scholes first articulated the Black-Scholes formula in their 1973 paper, "The Pricing of Options and Corporate Liabilities." The foundation for their research relied on work developed by scholars such as Jack L. Treynor, Paul Samuelson, A. James Boness, Sheen T. Kassouf, and Edward O. Thorp. The fundamental insight of Black-Scholes is that the option is implicitly priced if the stock is traded.

Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model and coined the term "Black-Scholes" options pricing model.

Merton and Scholes received the 1997 The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel for this and related work. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish academy [1].

The Black-Scholes model of the market for an equity makes the following explicit assumptions:

From these ideal conditions in the market for an equity (and for an option on the equity), the authors show that "it is possible to create a hedged position, consisting of a long position in the stock and a short position in [calls on the same stock], whose value will not depend on the price of the stock."[2]

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Define

below).
risk-free interest rate, continuously compounded.
quadratic variation of the stock's log price process.
portfolio.
delta-hedging trading strategy.

standard normal cumulative distribution function, frac{1}{sqrt{2pi}}int_{-infty}^{x} e^{-frac{z^2}{2}}, dz.

x) denotes the standard normal probability density function,frac{e^{-frac{x^2}{2}}}{sqrt{2pi} } .

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Simulated Geometric Brownian Motions with Parameters from Market Data

As per the model assumptions above, we assume that the underlying (typically the stock) follows a geometric Brownian motion. That is,

 dS_t = mu S_t,dt + sigma S_t,dW_t ,

where Wt is Brownian -- the dW term here stands in for any and all sources of uncertainty in the price history of a stock.

The payoff of an option V evolves as a function of S and T. By Itō's lemma for two variables we have

 dV = left(mu S frac{partial V}{partial S} + frac{partial V}{partial t}+ frac{1}{2}sigma^2 S^2frac{partial^2 V}{partial S^2}right)dt + sigma S frac{partial V}{partial S},dW.

Now consider a trading strategy under which one holds one option and continuously trades in the stock in order to hold  - frac{partial V}{partial S} shares. At time t, the value of these holdings will be

 Pi = V - Sfrac{partial V}{partial S}.

The composition of this portfolio, called the delta-hedge portfolio, will vary from time-step to time-step. Let R denote the accumulated profit or loss from following this strategy. Then over the time period [t, t + dt], the instantaneous profit or loss is

 dR = dV - frac{partial V}{partial S},dS.

By substituting in the equations above we get

 dR = left(frac{partial V}{partial t} + frac{1}{2}sigma^2 S^2frac{partial^2 V}{partial S^2}right)dt.

This equation contains no dW term. That is, it is entirely riskless (delta neutral). Black and Scholes reason that under their ideal conditions, the rate of return on this portfolio must be equal at all times to the rate of return on any other riskless instrument; otherwise, there would be opportunities for arbitrage. Now assuming the risk-free rate of return is r we must have over the time period [t, t + dt]

 rPi,dt = dR = left(frac{partial V}{partial t} + frac{1}{2}sigma^2 S^2frac{partial^2 V}{partial S^2}right)dt.

If we now substitute in for dt we obtain the Black–Scholes PDE:

 frac{partial V}{partial t} + frac{1}{2}sigma^2 S^2frac{partial^2 V}{partial S^2} + rSfrac{partial V}{partial S} - rV = 0.

With the assumptions of the Black–Scholes model, this partial differential equation holds whenever V is twice differentiable with respect to S and once with respect to t.

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Black-Scholes European Call Option Pricing Surface

The Black Scholes formula is used for obtaining the price of European put and call options. It is obtained by solving the Black-Scholes PDE as discussed - see derivation below.

The value of a call option in terms of the Black–Scholes parameters:

 C(S,t) = SN(d_1) - Ke^{-r(T - t)}N(d_2) ,
 d_1 = frac{ln(S/K) + (r + sigma^2/2)(T - t)}{sigmasqrt{T - t}}
 d_2 = d_1 - sigmasqrt{T - t}.

The price of a put option is:

 P(S,t) = Ke^{-r(T-t)}N(-d_2) - SN(-d_1).

For both, as above:

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martingale probability measure (numéraire = stock) and the equivalent martingale probability measure (numéraire = risk free asset), respectively. The equivalent martingale probability measure is also called the risk neutral probability measure. Note that both of these are "probabilities" in a measure theoretic sense, and neither of these is the true probability of expiring in-the-money under the real probability measure.

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We now show how to get from the general Black–Scholes PDE to a specific valuation for an option. Consider as an example the Black–Scholes price of a call option, for which the PDE above has boundary conditions

 C(0,t) = 0text{ for all }t,
 C(S,t) rightarrow Stext{ as }S rightarrow infty ,
 C(S,T) = max(S - K,0). ,

The last condition gives the value of the option at the time that the option matures. The solution of the PDE gives the value of the option at any earlier time,  mathbb{E}left[max(S - K,0)right]. In order to solve the PDE we transform the equation into a diffusion equation which may be solved using standard methods. To this end we introduce the change-of-variable transformation

 tau = T - t ,
 u = Ce^{rtau} ,
 x = ln(S/K) + (r - frac{sigma^2}{2})tau . ,

Then the Black–Scholes PDE becomes a diffusion equation

 frac{partial u}{partial tau} = frac{sigma^2}{2} frac{partial^2 u}{partial x^2}.

The terminal condition S ? K,0) now becomes an initial condition

 u(x,0) = u_0(x) equiv Kmax(e^x - 1,0). ,

Using the standard method for solving a diffusion equation we have

 u(x,tau) = frac{1}{sigmasqrt{2pitau}}int_{-infty}^infty u_0(y) e^{-(x - y)^2/(2sigma^2tau)},dy.

After some algebra we obtain

 u(x,tau) = Ke^{x + sigma^2tau/2}N(d_1) - KN(d_2)

where

 d_1 = frac{x + sigma^2tau}{sigmasqrt{tau}}

and

 d_2 = frac{x}{sigmasqrt{tau}}.

Substituting for u, x, and < Black–Scholes terms obtain>

 C(S,t) = SN(d_1) - Ke^{-r(T - t)}N(d_2) ,

where

 d_1 = frac{ln(S/K) + (r + sigma^2/2)(T - t)}{sigmasqrt{T - t}}
 d_2 = d_1 - sigmasqrt{T - t}.

The price of a put option may be computed from this by put-call parity and simplifies to

 P(S,t) = Ke^{-r(T-t)}N(-d_2) - SN(-d_1). ,

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The Greeks under Black–Scholes are given in closed form, below:

What Calls Puts
delta  frac{partial C}{partial S}  N(d_1) ,  - N( - d_1) = N(d_1)-1,
gamma  frac{partial^2 C}{partial S^2}  frac{N'(d_1)}{Ssigmasqrt{T-t}} ,
vega  frac{partial C}{partial sigma}  S N'(d_1) sqrt{T-t} ,
theta  -frac{partial C}{partial t}  - frac{S N'(d_1) sigma}{2 sqrt{T-t}} - rKe^{-r(T-t)}N(d_2) ,  - frac{S N'(d_1) sigma}{2 sqrt{T-t}} + rKe^{-r(T-t)}N(-d_2) ,
rho  frac{partial C}{partial r}  K(T-t)e^{-r(T-t)}N(d_2),  -K(T-t)e^{-r(T-t)}N(-d_2),

Note that the gamma and vega formulas are the same for calls and puts. This can be seen directly from put-call parity.

In practice, some sensitivities are usually quoted in scaled-down terms, to match the scale of likely changes in the parameters. For example, rho is often reported divided by 10,000 (1bp rate change), vega by 100 (1 vol point change), and theta by 365 or 252 (1 day decay based on either calendar days or trading days per year).

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< the>

The above model can be extended to have non-constant (but deterministic) rates and volatilities. The model may also be used to value European options on instruments paying dividends. In this case, closed-form solutions are available if the dividend is a known proportion of the stock price. American options and options on stocks paying a known cash dividend (in the short term, more realistic than a proportional dividend) are more difficult to value, and a choice of solution techniques is available (for example lattices and grids).

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< yield continuous paying>

For options on indexes, it is reasonable to make the simplifying assumption that dividends are paid continuously, and that the dividend amount is proportional to the level of the index.

The dividend payment paid over the time period [t, t + dt] is then modelled as

 qS_t,dt

for some constant q (the dividend yield).

Under this formulation the arbitrage-free price implied by the Black–Scholes model can be shown to be

 C(S_0,T) = e^{-rT}(FN(d_1) - KN(d_2)) ,

where now

 F = S_0 e^{(r - q)T} ,

is the modified forward price that occurs in the terms d1 and d2:

 d_1 = frac{ln(F/K) + (sigma^2/2)T}{sigmasqrt{T}}
 d_2 = d_1 - sigmasqrt{T}.

Exactly the same formula is used to price options on foreign exchange rates, except that now q plays the role of the foreign risk-free interest rate and S is the spot exchange rate. This is the Garman–Kohlhagen model (1983).

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< paying proportional discrete>

It is also possible to extend the Black–Scholes framework to options on instruments paying discrete proportional dividends. This is useful when the option is struck on a single stock.

A typical model is to assume that a proportion t1, t2, .... The price of the stock is then modelled as

 S_t = S_0(1 - delta)^{n(t)}e^{ut + sigma W_t}

where n(t) is the number of dividends that have been paid by time t.

The price of a call option on such a stock is again

 C(S_0,T) = e^{-rT}(FN(d_1) - KN(d_2)) ,

where now

 F = S_0(1 - delta)^{n(T)}e^{rT} ,

is the forward price for the dividend paying stock.

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The normality assumption of the Black–Scholes model does not capture extreme movements such as stock market crashes.

The Black–Scholes model disagrees with reality in a number of ways, some significant. It is widely used as a useful approximation, but proper use requires understanding its limitations – blindly following the model exposes the user to unexpected risk.

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.

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 due to simplifying assumptions of the model. One significant limitation is that in reality security prices do not follow a strict stationary 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 [3], for it is easy to calculate and explicitly models the relationship of all the variables. It is a useful approximation, particularly when analyzing the directionality that prices move when crossing critical points. It is used both as a quoting convention and a basis for more refined models. Although volatility is not constant, results from the model are often useful in practice and 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.

One reason for the popularity of the Black–Scholes model is that it 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 (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.

Additionally, 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 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 and tenors), option prices can thus be quoted in terms of implied volatility, which leads to trading of volatility in option markets.

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Main article: Volatility smile

One of the attractive feature of the Black-Scholes model is that the parameters in the model (other than the volatility and the risk-free interest rate) — the time to maturity, the strike, and the current underlying price — are unequivocally observable. All other things being equal, an option's theoretical value is a 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 surface (the three-dimensional 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 behaviour 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."[4] 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 volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on.

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< bond>

Black–Scholes cannot be applied directly to bond securities because of the pull-to-par problem. 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.

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In practice, interest rates are not constant - they vary by tenor, giving an 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.

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It is not free to take a short stock position. Similarly, it may be possible to lend out a long stock position for a small fee. In either case, this can be treated as a continuous dividend for the purposes of a Black-Scholes valuation.

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Let S0 be the current price of the underlying stock and S the price when the option matures at time T. Then S0 is known, but S is a random variable. Assume that

 X equiv lnleft(frac{S}{S_0}right) ,

is a normal random variable with mean variance S is

 mathbb{E}left[ S right] = S_0 e^{qT} ,

for some constant q (independent of T). Now a simple no-arbitrage argument shows that the theoretical future value of a derivative paying one share of the stock at time T, and so with payoff S, is

 S_0 e^{rT} ,

where r is the risk-free interest rate. This suggests making the identification q = r for the purpose of pricing derivatives. Define the theoretical value of a derivative as the present value of the expected payoff in this sense. For a call option with exercise price K this discounted expectation (using risk-neutral probabilities) is

 C(S_0,T) = e^{-rT} mathbb{E}left[ max(S - K,0) right]. ,

The derivation of the formula for C is facilitated by the following lemma: Let Z be a standard normal random variable and let b be an extended real number. Define

b \\ -\infty & \mbox{otherwise}. \end{cases}" src="http://upload.wikimedia.org/math/3/c/4/3c4e58b2535e55865ad8c9ed1983a2b1.png">

If a is a positive real number, then

 mathbb{E}left[e^{aZ^+(b)}right] = e^{a^2/2}N(a - b)

where cumulative distribution function. In the special case b = ?∞, we have

 mathbb{E}left[e^{aZ}right] = e^{a^2/2}.

Now let

 Z = frac{X - uT}{sigmasqrt{T}}

and use the corollary to the lemma to verify the statement above about the mean of S. Define

K \\ 0 & \mbox{otherwise} \end{cases} " src="http://upload.wikimedia.org/math/2/b/4/2b48020fc8edb77cc3a2b6711b16264c.png">
 X^+ = ln(S^+/S_0) ,

and observe that

 frac{X^+ - uT}{sigmasqrt{T}} = Z^+(b)

for some b. Define

K \\ 0 & \mbox{otherwise} \end{cases} " src="http://upload.wikimedia.org/math/a/6/8/a683078d500978b269144258dda88645.png">

and observe that

 max(S - K,0) = S^+ - K^+. ,

The rest of the calculation is straightforward.

Although the "elementary" derivation leads to the correct result, it is incomplete as it cannot explain, why the formula refers to the risk-free interest rate while a higher rate of return is expected from risky investments. This limitation can be overcome using the risk-neutral probability measure, but the concept of risk-neutrality and the related theory is far from elementary. In elementary terms, the value of the option today is not the expectation of the value of the option at expiry, discounted with the risk-free rate. (So the basic capital asset pricing model (CAPM) results are not violated.) The value is instead computed using the expectation under another distribution of probability, of the value of the option at expiry, discounted with the risk-free rate. This other distribution of probability is called the "risk neutral" probability.

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< derivations>

Above we used the method of arbitrage-free pricing ("delta-hedging") to derive some PDE governing option prices given the Black–Scholes model. It is also possible to use a risk-neutrality argument. This latter method gives the price as the expectation of the option payoff under a particular probability measure, called the risk-neutral measure, which differs from the real world measure.

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The reader is warned of the inconsistent notation that appears in this article. Thus the letter S is used as:

(1) a constant denoting the current price of the stock
(2) a real variable denoting the price at an arbitrary time
(3) a random variable denoting the price at maturity
(4) a stochastic process denoting the price at an arbitrary time

It is also used in the meaning of (4) with a subscript denoting time, but here the subscript is merely a mnemonic.

In the partial derivatives, the letters in the numerators and denominators are, of course, real variables, and the partial derivatives themselves are, initially, real functions of real variables. But after the substitution of a stochastic process for one of the arguments they become stochastic processes.

The Black–Scholes PDE is, initially, a statement about the stochastic process S, but when S is reinterpreted as a real variable, it becomes an ordinary PDE. It is only then that we can ask about its solution.

The parameter u that appears in the discrete-dividend model and the elementary derivation is not the same as the parameter Geometric Brownian motion.

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    1. ^ Nobel prize foundation, 1997 Press release [1]
    2. ^ Black, Fischer; Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy 81 (3): 637–654.
    3. ^ http://www.wilmott.com/blogs/paul/index.cfm/2008/4/29/Science-in-Finance-IX-In-defence-of-Black-Scholes-and-Merton
    4. ^ R Rebonato: Volatility and correlation in the pricing of equity, FX and interest-rate options (1999)
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      • TE style="FONT-STYLE: normal">Black, Fischer; Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy 81 (3): 637–654. doi:10.1086/260062.TE>  [2] (Black and Scholes' original paper.)
      • TE style="FONT-STYLE: normal">Merton, Robert C. (1973). "Theory of Rational Option Pricing". Bell Journal of Economics and Management Science 4 (1): 141–183. doi:10.2307/3003143.TE>  [3]

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      < sociological>

      • TE style="FONT-STYLE: normal">Bernstein, Peter (1992). Capital Ideas: The Improbable Origins of Modern Wall Street. The Free Press. ISBN 0-02-903012-9.TE> 
      • TE style="FONT-STYLE: normal">MacKenzie, Donald (2003). "An Equation and its Worlds: Bricolage, Exemplars, Disunity and Performativity in Financial Economics". Social Studies of Science 33 (6): 831–868. doi:10.1177/0306312703336002.TE>  [4]
      • TE style="FONT-STYLE: normal">MacKenzie, Donald; Yuval Millo (2003). "Constructing a Market, Performing Theory: The Historical Sociology of a Financial Derivatives Exchange". American Journal of Sociology 109 (1): 107–145. doi:10.1086/374404.TE>  [5]
      • TE style="FONT-STYLE: normal">MacKenzie, Donald (2006). An Engine, not a Camera: How Financial Models Shape Markets. MIT Press. ISBN 0-262-13460-8.TE> 

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