这些东西我不懂。留个记号。
http://www.ederman.com/new/index.html
http://www.ieor.columbia.edu/facbios/derman/faculty.html
http://en.wikipedia.org/wiki/Derman
http://en.wikipedia.org/wiki/Black%E2%80%93Scholes
Emanuel Derman is a South Africanborn 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 coauthor of BlackDermanToy model, on
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 on
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 weakneutral current in electronhadron scattering. This experiment was carried out at SLAC in 1978 by a team led by Charles Prescott and Richard Taylor, and confirmed the WeinbergSalam 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 on
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 on
Professor Derman is together with Paul Wilmott on
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
BlackDermanToy model
In finance, the BlackDermanToy model is a model of the evolution of the yield curve, sometimes referred to as a short rate model. It is a on
The model was introduced by Fischer Black, Emanuel Derman, and Bill Toy. It was first developed for inhouse 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 on
In finance, the volatility smile is a longobserved pattern in which atthemoney options tend to have lower implied volatilities than in or outofthemoney 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 on
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 3D plot that combines volatility smile and term structure of volatility into a consolidated view of all options for an underlier.

In the BlackScholes model, the theoretical value of a vanilla option is a monotonic increasing function of the BlackScholes 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 valleyshaped 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 atthemoney 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 impliedvolatility to indicate the volatility parameter for ATM (atthemoney) 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.
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 21day 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).
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 wellobserved 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 beha
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.
It is often useful to plot implied volatility as a function of both strike price and time to maturity. The result is a 3D surface whereby the current market implied volatility (Zaxis) for all options on the underlier is plotted against strike price and time to maturity (X & Yaxes).
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 Zaxis represents implied volatility in percent, and X and Y axes represent the option delta, and the days to maturity. Note that to maintain putcall 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.
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:
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).
Methods of modelling the volatility smile include stochastic volatility models and local volatility models.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The term Black–Scholes refers to three closely related concepts:
Fischer Black and Myron Scholes first articulated the BlackScholes 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 BlackScholes 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 "BlackScholes" 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 BlackScholes 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]}
Define
standard normal cumulative distribution function, .
x) denotes the standard normal probability density function,.
As per the model assumptions above, we assume that the underlying (typically the stock) follows a geometric Brownian motion. That is,
where W_{t} 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
Now consider a trading strategy under which on
The composition of this portfolio, called the deltahedge portfolio, will vary from timestep to timestep. 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
By substituting in the equations above we get
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 riskfree rate of return is r we must have over the time period [t, t + dt]
If we now substitute in for dt we obtain the Black–Scholes PDE:
With the assumptions of the Black–Scholes model, this partial differential equation holds whenever V is twice differentiable with respect to S and on
The Black Scholes formula is used for obtaining the price of European put and call options. It is obtained by solving the BlackScholes PDE as discussed  see derivation below.
The value of a call option in terms of the Black–Scholes parameters:
The price of a put option is:
For both, as above:
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 inthemoney under the real probability measure.
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
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, . 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 changeofvariable transformation
Then the Black–Scholes PDE becomes a diffusion equation
The terminal condition S ? K,0) now becomes an initial condition
Using the standard method for solving a diffusion equation we have
After some algebra we obtain
where
and
Substituting for u, x, and < Black–Scholes terms obtain>
where
The price of a put option may be computed from this by putcall parity and simplifies to
The Greeks under Black–Scholes are given in closed form, below:
What  Calls  Puts  

delta  
gamma  
vega  
theta  
rho 
Note that the gamma and vega formulas are the same for calls and puts. This can be seen directly from putcall parity.
In practice, some sensitivities are usually quoted in scaleddown 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).
The above model can be extended to have nonconstant (but deterministic) rates and volatilities. The model may also be used to value European options on instruments paying dividends. In this case, closedform 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).
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
for some constant q (the dividend yield).
Under this formulation the arbitragefree price implied by the Black–Scholes model can be shown to be
where now
is the modified forward price that occurs in the terms d_{1} and d_{2}:
Exactly the same formula is used to price options on foreign exchange rates, except that now q plays the role of the foreign riskfree interest rate and S is the spot exchange rate. This is the Garman–Kohlhagen model (1983).
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 t_{1}, t_{2}, .... The price of the stock is then modelled as
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
where now
is the forward price for the dividend paying stock.
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:
In short, while in the Black–Scholes model on
Results using the Black–Scholes model differ from real world prices due to simplifying assumptions of the model. On
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.
On
Additionally, rather than assuming a volatility a priori and computing prices from it, on
On
Despite the existence of the volatility smile (and the violation of all the other assumptions of the BlackScholes model), the BlackScholes PDE and BlackScholes 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 BlackScholes 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.
Black–Scholes cannot be applied directly to bond securities because of the pulltopar 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.
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 BlackScholes formula. Another consideration is that interest rates vary over time. This volatility may make a significant contribution to the price, especially of longdated options.
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 BlackScholes valuation.
Let S_{0} be the current price of the underlying stock and S the price when the option matures at time T. Then S_{0} is known, but S is a random variable. Assume that
is a normal random variable with mean variance S is
for some constant q (independent of T). Now a simple noarbitrage argument shows that the theoretical future value of a derivative paying on
where r is the riskfree 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 riskneutral probabilities) is
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
If a is a positive real number, then
where cumulative distribution function. In the special case b = ?∞, we have
Now let
and use the corollary to the lemma to verify the statement above about the mean of S. Define
and observe that
for some b. Define
and observe that
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 riskfree interest rate while a higher rate of return is expected from risky investments. This limitation can be overcome using the riskneutral probability measure, but the concept of riskneutrality 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 riskfree 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 riskfree rate. This other distribution of probability is called the "risk neutral" probability.
Above we used the method of arbitragefree pricing ("deltahedging") to derive some PDE governing option prices given the Black–Scholes model. It is also possible to use a riskneutrality argument. This latter method gives the price as the expectation of the option payoff under a particular probability measure, called the riskneutral measure, which differs from the real world measure.
The reader is warned of the inconsistent notation that appears in this article. Thus the letter S is used as:
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 on
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 on
The parameter u that appears in the discretedividend model and the elementary derivation is not the same as the parameter Geometric Brownian motion.
external links may not follow Wikipedia's content policies or guidelines. Please ion=edit href="http://en.wikipedia.org/w/index.php?title=Black%E2%80%93Scholes&act 
评论