The Black–Scholes /ˌblæk ˈʃoʊlz/[1] or Black–Scholes–Merton
model is a mathematical model of a financial market containing
derivative investment instruments. From the partial differential
equation in the model, known as the Black–Scholes equation, one can
deduce the Black–Scholes formula, which gives a theoretical estimate
of the price of European-style options and shows that the option has a
unique price regardless of the risk of the security and its expected
return (instead replacing the security's expected return with the
risk-neutral rate). The formula led to a boom in options trading and
provided mathematical legitimacy to the activities of the Chicago
Board Options Exchange and other options markets around the world.[2]
It is widely used, although often with adjustments and corrections, by
options market participants.[3]:751 Many empirical tests have shown
that the Black–Scholes price is "fairly close" to the observed
prices, although there are well-known discrepancies such as the
"option smile".[3]:770–771
Based on works previously developed by market researchers and
practitioners, such as Louis Bachelier,
Contents 1 The Black–Scholes world 2 Notation 3 Black–Scholes equation 4 Black–Scholes formula 4.1 Alternative formulation 4.2 Interpretation 4.2.1 Derivations 5 The Greeks 6 Extensions of the model 6.1 Instruments paying continuous yield dividends 6.2 Instruments paying discrete proportional dividends 6.3 American options 6.4 Binary options 6.4.1 Cash-or-nothing call 6.4.2 Cash-or-nothing put 6.4.3 Asset-or-nothing call 6.4.4 Asset-or-nothing put 6.4.5 Foreign exchange 6.4.6 Skew 6.4.7 Relationship to vanilla options' Greeks 7 Black–Scholes in practice 7.1 The volatility smile 7.2 Valuing bond options 7.3 Interest-rate curve 7.4 Short stock rate 8 Criticism and comments 9 See also 10 Notes 11 References 11.1 Primary references 11.2 Historical and sociological aspects 11.3 Further reading 12 External links 12.1 Discussion of the model 12.2 Derivation and solution 12.3 Computer implementations 12.4 Historical The Black–Scholes world[edit]
The
(riskless rate) The rate of return on the riskless asset is constant and thus called the risk-free interest rate. (random walk) The instantaneous log return of stock price is an infinitesimal random walk with drift; more precisely, it is a geometric Brownian motion, and we will assume its drift and volatility is constant (if they are time-varying, we can deduce a suitably modified Black–Scholes formula quite simply, as long as the volatility is not random). The stock does not pay a dividend.[Notes 1] Assumptions on the market: There is no arbitrage opportunity (i.e., there is no way to make a riskless profit). It is possible to borrow and lend any amount, even fractional, of cash at the riskless rate. It is possible to buy and sell any amount, even fractional, of the stock (this includes short selling). The above transactions do not incur any fees or costs (i.e., frictionless market). With these assumptions holding, suppose there is a derivative security also trading in this market. We specify that this security will have a certain payoff at a specified date in the future, depending on the value(s) taken by the stock up to that date. It is a surprising fact that the derivative's price is completely determined at the current time, even though we do not know what path the stock price will take in the future. For the special case of a European call or put option, Black and Scholes showed that "it is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock".[12] Their dynamic hedging strategy led to a partial differential equation which governed the price of the option. Its solution is given by the Black–Scholes formula. Several of these assumptions of the original model have been removed in subsequent extensions of the model. Modern versions account for dynamic interest rates (Merton, 1976),[citation needed] transaction costs and taxes (Ingersoll, 1976),[citation needed] and dividend payout.[13] Notation[edit] Let S displaystyle S , be the price of the stock, which will sometimes be a random variable and other times a constant (context should make this clear); V ( S , t ) displaystyle V(S,t) , the price of a derivative as a function of time and stock price; C ( S , t ) displaystyle C(S,t) the price of a European call option and P ( S , t ) displaystyle P(S,t) the price of a European put option; K displaystyle K , the strike price of the option; r displaystyle r , the annualized risk-free interest rate, continuously compounded (the force of interest); μ displaystyle mu , the drift rate of S displaystyle S , annualized; σ displaystyle sigma , the standard deviation of the stock's returns; this is the square root of the quadratic variation of the stock's log price process; t displaystyle t , a time in years; we generally use: now = 0 displaystyle =0 , expiry = T displaystyle =T ; Π displaystyle Pi , the value of a portfolio. Finally, we will use N ( x ) displaystyle N(x) to denote the standard normal cumulative distribution function, N ( x ) = 1 2 π ∫ − ∞ x e − z 2 2 d z displaystyle N(x)= frac 1 sqrt 2pi int _ -infty ^ x e^ - frac z^ 2 2 ,dz . N ′ ( x ) displaystyle N'(x) will denote the standard normal probability density function, N ′ ( x ) = 1 2 π e − x 2 2 displaystyle N'(x)= frac 1 sqrt 2pi e^ - frac x^ 2 2 Black–Scholes equation[edit] Main article: Black–Scholes equation Simulated geometric Brownian motions with parameters from market data As above, the
∂ V ∂ t + 1 2 σ 2 S 2 ∂ 2 V ∂ S 2 + r S ∂ V ∂ S − r V = 0 displaystyle frac partial V partial t + frac 1 2 sigma ^ 2 S^ 2 frac partial ^ 2 V partial S^ 2 +rS frac partial V partial S -rV=0 The key financial insight behind the equation is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk".[citation needed] This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula (see the next section). Black–Scholes formula[edit] A European call valued using the Black–Scholes pricing equation for varying asset price S and time-to-expiry T. In this particular example, the strike price is set to unity. The Black–Scholes formula calculates the price of European put and call options. This price is consistent with the Black–Scholes equation as above; this follows since the formula can be obtained by solving the equation for the corresponding terminal and boundary conditions. The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is: C ( S t , t ) = N ( d 1 ) S t − N ( d 2 ) K e − r ( T − t ) d 1 = 1 σ T − t [ ln ( S t K ) + ( r + σ 2 2 ) ( T − t ) ] d 2 = d 1 − σ T − t displaystyle begin aligned C(S_ t ,t)&=N(d_ 1 )S_ t -N(d_ 2 )Ke^ -r(T-t) \d_ 1 &= frac 1 sigma sqrt T-t left[ln left( frac S_ t K right)+left(r+ frac sigma ^ 2 2 right)(T-t)right]\d_ 2 &=d_ 1 -sigma sqrt T-t \end aligned The price of a corresponding put option based on put–call parity is: P ( S t , t ) = K e − r ( T − t ) − S t + C ( S t , t ) = N ( − d 2 ) K e − r ( T − t ) − N ( − d 1 ) S t displaystyle begin aligned P(S_ t ,t)&=Ke^ -r(T-t) -S_ t +C(S_ t ,t)\&=N(-d_ 2 )Ke^ -r(T-t) -N(-d_ 1 )S_ t end aligned , For both, as above: N ( ⋅ ) displaystyle N(cdot ) is the cumulative distribution function of the standard normal distribution T − t displaystyle T-t is the time to maturity (expressed in years) S t displaystyle S_ t is the spot price of the underlying asset K displaystyle K is the strike price r displaystyle r is the risk free rate (annual rate, expressed in terms of continuous compounding) σ displaystyle sigma is the volatility of returns of the underlying asset Alternative formulation[edit] Introducing some auxiliary variables allows the formula to be simplified and reformulated in a form that is often more convenient (this is a special case of the Black '76 formula): C ( F , τ ) = D ( N ( d + ) F − N ( d − ) K ) d ± = 1 σ τ [ ln ( F K ) ± 1 2 σ 2 τ ] d ± = d ∓ ± σ τ displaystyle begin aligned C(F,tau )&=Dleft(N(d_ + )F-N(d_ - )Kright)\d_ pm &= frac 1 sigma sqrt tau left[ln left( frac F K right)pm frac 1 2 sigma ^ 2 tau right]\d_ pm &=d_ mp pm sigma sqrt tau end aligned The auxiliary variables are: τ = T − t displaystyle tau =T-t is the time to expiry (remaining time, backwards time) D = e − r τ displaystyle D=e^ -rtau is the discount factor F = e r τ S = S D displaystyle F=e^ rtau S= frac S D is the forward price of the underlying asset, and S = D F displaystyle S=DF with d+ = d1 and d− = d2 to clarify notation. Given put–call parity, which is expressed in these terms as: C − P = D ( F − K ) = S − D K displaystyle C-P=D(F-K)=S-DK the price of a put option is: P ( F , τ ) = D [ N ( − d − ) K − N ( − d + ) F ] displaystyle P(F,tau )=Dleft[N(-d_ - )K-N(-d_ + )Fright] Interpretation[edit] The Black–Scholes formula can be interpreted fairly handily, with the main subtlety the interpretation of the N ( d ± ) displaystyle N(d_ pm ) (and a fortiori d ± displaystyle d_ pm ) terms, particularly d + displaystyle d_ + and why there are two different terms.[14] The formula can be interpreted by first decomposing a call option into the difference of two binary options: an asset-or-nothing call minus a cash-or-nothing call (long an asset-or-nothing call, short a cash-or-nothing call). A call option exchanges cash for an asset at expiry, while an asset-or-nothing call just yields the asset (with no cash in exchange) and a cash-or-nothing call just yields cash (with no asset in exchange). The Black–Scholes formula is a difference of two terms, and these two terms equal the value of the binary call options. These binary options are much less frequently traded than vanilla call options, but are easier to analyze. Thus the formula: C = D [ N ( d + ) F − N ( d − ) K ] displaystyle C=Dleft[N(d_ + )F-N(d_ - )Kright] breaks up as: C = D N ( d + ) F − D N ( d − ) K displaystyle C=DN(d_ + )F-DN(d_ - )K , where D N ( d + ) F displaystyle DN(d_ + )F is the present value of an asset-or-nothing call and D N ( d − ) K displaystyle DN(d_ - )K is the present value of a cash-or-nothing call. The D factor is for discounting, because the expiration date is in future, and removing it changes present value to future value (value at expiry). Thus N ( d + ) F displaystyle N(d_ + )~F is the future value of an asset-or-nothing call and N ( d − ) K displaystyle N(d_ - )~K is the future value of a cash-or-nothing call. In risk-neutral terms, these are the expected value of the asset and the expected value of the cash in the risk-neutral measure. The naive, and not quite correct, interpretation of these terms is that N ( d + ) F displaystyle N(d_ + )F is the probability of the option expiring in the money N ( d + ) displaystyle N(d_ + ) , times the value of the underlying at expiry F, while N ( d − ) K displaystyle N(d_ - )K is the probability of the option expiring in the money N ( d − ) , displaystyle N(d_ - ), times the value of the cash at expiry K. This is obviously incorrect, as either both binaries expire in the money or both expire out of the money (either cash is exchanged for asset or it is not), but the probabilities N ( d + ) displaystyle N(d_ + ) and N ( d − ) displaystyle N(d_ - ) are not equal. In fact, d ± displaystyle d_ pm can be interpreted as measures of moneyness (in standard deviations) and N ( d ± ) displaystyle N(d_ pm ) as probabilities of expiring ITM (percent moneyness), in the respective numéraire, as discussed below. Simply put, the interpretation of the cash option, N ( d − ) K displaystyle N(d_ - )K , is correct, as the value of the cash is independent of movements of the underlying, and thus can be interpreted as a simple product of "probability times value", while the N ( d + ) F displaystyle N(d_ + )F is more complicated, as the probability of expiring in the money and the value of the asset at expiry are not independent.[14] More precisely, the value of the asset at expiry is variable in terms of cash, but is constant in terms of the asset itself (a fixed quantity of the asset), and thus these quantities are independent if one changes numéraire to the asset rather than cash. If one uses spot S instead of forward F, in d ± displaystyle d_ pm instead of the 1 2 σ 2 displaystyle frac 1 2 sigma ^ 2 term there is ( r ± 1 2 σ 2 ) τ , displaystyle left(rpm frac 1 2 sigma ^ 2 right)tau , which can be interpreted as a drift factor (in the risk-neutral measure for appropriate numéraire). The use of d− for moneyness rather than the standardized moneyness m = 1 σ τ ln ( F K ) displaystyle m= frac 1 sigma sqrt tau ln left( frac F K right) – in other words, the reason for the 1 2 σ 2 displaystyle frac 1 2 sigma ^ 2 factor – is due to the difference between the median and mean of the log-normal distribution; it is the same factor as in Itō's lemma applied to geometric Brownian motion. In addition, another way to see that the naive interpretation is incorrect is that replacing N(d+) by N(d−) in the formula yields a negative value for out-of-the-money call options.[14]:6 In detail, the terms N ( d 1 ) , N ( d 2 ) displaystyle N(d_ 1 ),N(d_ 2 ) are the probabilities of the option expiring in-the-money under the equivalent exponential martingale probability measure (numéraire=stock) and the equivalent martingale probability measure (numéraire=risk free asset), respectively.[14] The risk neutral probability density for the stock price S T ∈ ( 0 , ∞ ) displaystyle S_ T in (0,infty ) is p ( S , T ) = N ′ [ d 2 ( S T ) ] S T σ T displaystyle p(S,T)= frac N^ prime [d_ 2 (S_ T )] S_ T sigma sqrt T where d 2 = d 2 ( K ) displaystyle d_ 2 =d_ 2 (K) is defined as above. Specifically, N ( d 2 ) displaystyle N(d_ 2 ) is the probability that the call will be exercised provided one assumes that the asset drift is the risk-free rate. N ( d 1 ) displaystyle N(d_ 1 ) , however, does not lend itself to a simple probability interpretation. S N ( d 1 ) displaystyle SN(d_ 1 ) is correctly interpreted as the present value, using the risk-free
interest rate, of the expected asset price at expiration, given that
the asset price at expiration is above the exercise price.[15] For
related discussion – and graphical representation –
see section "Interpretation" under Datar–Mathews method for real
option valuation.
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. To calculate the probability under the real
("physical") probability measure, additional information is
required—the drift term in the physical measure, or equivalently,
the market price of risk.
Derivations[edit]
See also: Martingale pricing
A standard derivation for solving the Black–Scholes PDE is given in
the article Black–Scholes equation.
The
Calls Puts Delta ∂ C ∂ S displaystyle frac partial C partial S N ( d 1 ) displaystyle N(d_ 1 ), − N ( − d 1 ) = N ( d 1 ) − 1 displaystyle -N(-d_ 1 )=N(d_ 1 )-1, Gamma ∂ 2 C ∂ S 2 displaystyle frac partial ^ 2 C partial S^ 2 N ′ ( d 1 ) S σ T − t displaystyle frac N'(d_ 1 ) Ssigma sqrt T-t , Vega ∂ C ∂ σ displaystyle frac partial C partial sigma S N ′ ( d 1 ) T − t displaystyle SN'(d_ 1 ) sqrt T-t , Theta ∂ C ∂ t displaystyle frac partial C partial t − S N ′ ( d 1 ) σ 2 T − t − r K e − r ( T − t ) N ( d 2 ) displaystyle - frac SN'(d_ 1 )sigma 2 sqrt T-t -rKe^ -r(T-t) N(d_ 2 ), − S N ′ ( d 1 ) σ 2 T − t + r K e − r ( T − t ) N ( − d 2 ) displaystyle - frac SN'(d_ 1 )sigma 2 sqrt T-t +rKe^ -r(T-t) N(-d_ 2 ), Rho ∂ C ∂ r displaystyle frac partial C partial r K ( T − t ) e − r ( T − t ) N ( d 2 ) displaystyle K(T-t)e^ -r(T-t) N(d_ 2 ), − K ( T − t ) e − r ( T − t ) N ( − d 2 ) displaystyle -K(T-t)e^ -r(T-t) N(-d_ 2 ), Note that from the formulae, it is clear that the gamma is the same value for calls and puts and so too is the vega the same value for calls and put options. This can be seen directly from put–call parity, since the difference of a put and a call is a forward, which is linear in S and independent of σ (so a forward has zero gamma and zero vega). N' is the standard normal probability density function. 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 (1 basis point 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). (Vega is not a letter in the Greek alphabet; the name arises from reading the Greek letter ν (nu) as a V.) Extensions of the model[edit] The above model can be extended for variable (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). Instruments paying continuous yield dividends[edit] For options on indices, 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 + d t ] displaystyle [t,t+dt] is then modelled as q S t d t displaystyle qS_ t ,dt for some constant q displaystyle q (the dividend yield).
Under this formulation the arbitrage-free price implied by the
C ( S t , t ) = e − r ( T − t ) [ F N ( d 1 ) − K N ( d 2 ) ] displaystyle C(S_ t ,t)=e^ -r(T-t) [FN(d_ 1 )-KN(d_ 2 )], and P ( S t , t ) = e − r ( T − t ) [ K N ( − d 2 ) − F N ( − d 1 ) ] displaystyle P(S_ t ,t)=e^ -r(T-t) [KN(-d_ 2 )-FN(-d_ 1 )], where now F = S t e ( r − q ) ( T − t ) displaystyle F=S_ t e^ (r-q)(T-t) , is the modified forward price that occurs in the terms d 1 , d 2 displaystyle d_ 1 ,d_ 2 : d 1 = 1 σ T − t [ ln ( S t K ) + ( r − q + 1 2 σ 2 ) ( T − t ) ] displaystyle d_ 1 = frac 1 sigma sqrt T-t left[ln left( frac S_ t K right)+(r-q+ frac 1 2 sigma ^ 2 )(T-t)right] and d 2 = d 1 − σ T − t = 1 σ T − t [ ln ( S t K ) + ( r − q − 1 2 σ 2 ) ( T − t ) ] displaystyle d_ 2 =d_ 1 -sigma sqrt T-t = frac 1 sigma sqrt T-t left[ln left( frac S_ t K right)+(r-q- frac 1 2 sigma ^ 2 )(T-t)right] .[18] Instruments paying discrete proportional dividends[edit] 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 δ displaystyle delta of the stock price is paid out at pre-determined times t 1 , t 2 , … displaystyle t_ 1 ,t_ 2 ,ldots . The price of the stock is then modelled as S t = S 0 ( 1 − δ ) n ( t ) e u t + σ W t displaystyle S_ t =S_ 0 (1-delta )^ n(t) e^ ut+sigma W_ t where n ( t ) displaystyle n(t) is the number of dividends that have been paid by time t displaystyle t . The price of a call option on such a stock is again C ( S 0 , T ) = e − r T [ F N ( d 1 ) − K N ( d 2 ) ] displaystyle C(S_ 0 ,T)=e^ -rT [FN(d_ 1 )-KN(d_ 2 )], where now F = S 0 ( 1 − δ ) n ( T ) e r T displaystyle F=S_ 0 (1-delta )^ n(T) e^ rT , is the forward price for the dividend paying stock.
American options[edit]
The problem of finding the price of an
∂ V ∂ t + 1 2 σ 2 S 2 ∂ 2 V ∂ S 2 + r S ∂ V ∂ S − r V ≤ 0 displaystyle frac partial V partial t + frac 1 2 sigma ^ 2 S^ 2 frac partial ^ 2 V partial S^ 2 +rS frac partial V partial S -rVleq 0 [19] with the terminal and (free) boundary conditions: V ( S , T ) = H ( S ) displaystyle V(S,T)=H(S) and V ( S , t ) ≥ H ( S ) displaystyle V(S,t)geq H(S) where H ( S ) displaystyle H(S) denotes the payoff at stock price S displaystyle S . In general this inequality does not have a closed form solution, though an American call with no dividends is equal to a European call and the Roll-Geske-Whaley method provides a solution for an American call with one dividend;[20][21] see also Black's approximation. Barone-Adesi and Whaley[22] is a further approximation formula. Here, the stochastic differential equation (which is valid for the value of any derivative) is split into two components: the European option value and the early exercise premium. With some assumptions, a quadratic equation that approximates the solution for the latter is then obtained. This solution involves finding the critical value, s ∗ displaystyle s* , such that one is indifferent between early exercise and holding to maturity.[23][24] Bjerksund and Stensland[25] provide an approximation based on an exercise strategy corresponding to a trigger price. Here, if the underlying asset price is greater than or equal to the trigger price it is optimal to exercise, and the value must equal S − X displaystyle S-X , otherwise the option "boils down to: (i) a European up-and-out call option… and (ii) a rebate that is received at the knock-out date if the option is knocked out prior to the maturity date". The formula is readily modified for the valuation of a put option, using put–call parity. This approximation is computationally inexpensive and the method is fast, with evidence indicating that the approximation may be more accurate in pricing long dated options than Barone-Adesi and Whaley.[26] Binary options[edit] By solving the Black–Scholes differential equation, with for boundary condition the Heaviside function, we end up with the pricing of options that pay one unit above some predefined strike price and nothing below.[27] In fact, the Black–Scholes formula for the price of a vanilla call option (or put option) can be interpreted by decomposing a call option into an asset-or-nothing call option minus a cash-or-nothing call option, and similarly for a put – the binary options are easier to analyze, and correspond to the two terms in the Black–Scholes formula. Cash-or-nothing call[edit] This pays out one unit of cash if the spot is above the strike at maturity. Its value is given by C = e − r ( T − t ) N ( d 2 ) . displaystyle C=e^ -r(T-t) N(d_ 2 )., Cash-or-nothing put[edit] This pays out one unit of cash if the spot is below the strike at maturity. Its value is given by P = e − r ( T − t ) N ( − d 2 ) . displaystyle P=e^ -r(T-t) N(-d_ 2 )., Asset-or-nothing call[edit] This pays out one unit of asset if the spot is above the strike at maturity. Its value is given by C = S e − q ( T − t ) N ( d 1 ) . displaystyle C=Se^ -q(T-t) N(d_ 1 )., Asset-or-nothing put[edit] This pays out one unit of asset if the spot is below the strike at maturity. Its value is given by P = S e − q ( T − t ) N ( − d 1 ) . displaystyle P=Se^ -q(T-t) N(-d_ 1 )., Foreign exchange[edit] Further information: Foreign exchange derivative If we denote by S the FOR/DOM exchange rate (i.e., 1 unit of foreign currency is worth S units of domestic currency) we can observe that paying out 1 unit of the domestic currency if the spot at maturity is above or below the strike is exactly like a cash-or nothing call and put respectively. Similarly, paying out 1 unit of the foreign currency if the spot at maturity is above or below the strike is exactly like an asset-or nothing call and put respectively. Hence if we now take r F O R displaystyle r_ FOR , the foreign interest rate, r D O M displaystyle r_ DOM , the domestic interest rate, and the rest as above, we get the following results. In case of a digital call (this is a call FOR/put DOM) paying out one unit of the domestic currency we get as present value, C = e − r D O M T N ( d 2 ) displaystyle C=e^ -r_ DOM T N(d_ 2 ), In case of a digital put (this is a put FOR/call DOM) paying out one unit of the domestic currency we get as present value, P = e − r D O M T N ( − d 2 ) displaystyle P=e^ -r_ DOM T N(-d_ 2 ), While in case of a digital call (this is a call FOR/put DOM) paying out one unit of the foreign currency we get as present value, C = S e − r F O R T N ( d 1 ) displaystyle C=Se^ -r_ FOR T N(d_ 1 ), and in case of a digital put (this is a put FOR/call DOM) paying out one unit of the foreign currency we get as present value, P = S e − r F O R T N ( − d 1 ) displaystyle P=Se^ -r_ FOR T N(-d_ 1 ), Skew[edit]
In the standard Black–Scholes model, one can interpret the premium
of the binary option in the risk-neutral world as the expected value =
probability of being in-the-money * unit, discounted to the present
value. The
σ displaystyle sigma across all strikes, incorporating a variable one σ ( K ) displaystyle sigma (K) where volatility depends on strike price, thus incorporating the volatility skew into account. The skew matters because it affects the binary considerably more than the regular options. A binary call option is, at long expirations, similar to a tight call spread using two vanilla options. One can model the value of a binary cash-or-nothing option, C, at strike K, as an infinitessimally tight spread, where C v displaystyle C_ v is a vanilla European call:[28][29] C = lim ϵ → 0 C v ( K − ϵ ) − C v ( K ) ϵ displaystyle C=lim _ epsilon to 0 frac C_ v (K-epsilon )-C_ v (K) epsilon Thus, the value of a binary call is the negative of the derivative of the price of a vanilla call with respect to strike price: C = − d C v d K displaystyle C=- frac dC_ v dK When one takes volatility skew into account, σ displaystyle sigma is a function of K displaystyle K : C = − d C v ( K , σ ( K ) ) d K = − ∂ C v ∂ K − ∂ C v ∂ σ ∂ σ ∂ K displaystyle C=- frac dC_ v (K,sigma (K)) dK =- frac partial C_ v partial K - frac partial C_ v partial sigma frac partial sigma partial K The first term is equal to the premium of the binary option ignoring skew: − ∂ C v ∂ K = − ∂ ( S N ( d 1 ) − K e − r ( T − t ) N ( d 2 ) ) ∂ K = e − r ( T − t ) N ( d 2 ) = C no skew displaystyle - frac partial C_ v partial K =- frac partial (SN(d_ 1 )-Ke^ -r(T-t) N(d_ 2 )) partial K =e^ -r(T-t) N(d_ 2 )=C_ text no skew ∂ C v ∂ σ displaystyle frac partial C_ v partial sigma is the Vega of the vanilla call; ∂ σ ∂ K displaystyle frac partial sigma partial K is sometimes called the "skew slope" or just "skew". If the skew is typically negative, the value of a binary call will be higher when taking skew into account. C = C no skew − Vega v ⋅ Skew displaystyle C=C_ text no skew - text Vega _ v cdot text Skew Relationship to vanilla options' Greeks[edit] Since a binary call is a mathematical derivative of a vanilla call with respect to strike, the price of a binary call has the same shape as the delta of a vanilla call, and the delta of a binary call has the same shape as the gamma of a vanilla call. Black–Scholes in practice[edit] The normality assumption of the
The assumptions of the
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
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
Binomial options model, a discrete numerical method for calculating
option prices
Black model, a variant of the Black–Scholes option pricing model
Black Shoals, a financial art piece
Brownian model of financial markets
Notes[edit] ^ Although the original model assumed no dividends, trivial extensions to the model can accommodate a continuous dividend yield factor. References[edit] ^ "Scholes on merriam-webster.com". Retrieved March 26, 2012.
^ MacKenzie, Donald (2006). An Engine, Not a Camera: How Financial
Models Shape Markets. Cambridge, MA: MIT Press.
ISBN 0-262-13460-8.
^ a b c Bodie, Zvi; Alex Kane; Alan J. Marcus (2008). Investments (7th
ed.). New York: McGraw-Hill/Irwin. ISBN 978-0-07-326967-2.
^ Taleb, 1997. pp. 91 and 110–111.
^ Mandelbrot & Hudson, 2006. pp. 9–10.
^ Mandelbrot & Hudson, 2006. p. 74
^ Mandelbrot & Hudson, 2006. pp. 72–75.
^ Derman, 2004. pp. 143–147.
^ Thorp, 2017. pp. 183–189.
^
https://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1997/press.html
^ "Nobel Prize Foundation, 1997" (Press release). October 14, 1997.
Retrieved March 26, 2012.
^ Black, Fischer; Scholes, Myron. "The Pricing of Options and
Corporate Liabilities". Journal of Political Economy. 81 (3):
637–654. doi:10.1086/260062.
^ Merton, Robert. "Theory of Rational Option Pricing". Bell Journal of
Economics and Management Science. 4 (1): 141–183.
doi:10.2307/3003143.
^ a b c d e Nielsen, Lars Tyge (1993). "Understanding N(d1) and N(d2):
Risk-Adjusted Probabilities in the Black–Scholes Model" (PDF). Revue
Finance (Journal of the French Finance Association). 14 (1): 95–106.
Retrieved Dec 8, 2012, earlier circulated as
Primary references[edit] Black, Fischer;
Historical and sociological aspects[edit] Bernstein, Peter (1992). Capital Ideas: The Improbable Origins of Modern Wall Street. The Free Press. ISBN 0-02-903012-9. Derman, Emanuel. "My Live as a Quant" John Wiley & Sons, Inc. 2004. ISBN 0471394203 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. [4] 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. [5] MacKenzie, Donald (2006). An Engine, not a Camera: How Financial Models Shape Markets. MIT Press. ISBN 0-262-13460-8. Mandelbrot & Hudson, "The (Mis)Behavior of Markets" Basic Books, 2006. ISBN 9780465043552 Szpiro, George G., Pricing the Future: Finance, Physics, and the 300-Year Journey to the Black–Scholes Equation; A Story of Genius and Discovery (New York: Basic, 2011) 298 pp. Taleb, Nassim. "Dynamic Hedging" John Wiley & Sons, Inc. 1997. ISBN 0471152803 Thorp, Ed. "A Man for all Markets" Random House, 2017. ISBN 9781400067961 Further reading[edit] Haug, E. G (2007). "Option Pricing and Hedging from Theory to Practice". Derivatives: Models on Models. Wiley. ISBN 978-0-470-01322-9. The book gives a series of historical references supporting the theory that option traders use much more robust hedging and pricing principles than the Black, Scholes and Merton model. Triana, Pablo (2009). Lecturing Birds on Flying: Can Mathematical Theories Destroy the Financial Markets?. Wiley. ISBN 978-0-470-40675-5. The book takes a critical look at the Black, Scholes and Merton model. External links[edit] Discussion of the model[edit] Ajay Shah. Black, Merton and Scholes: Their work and its consequences. Economic and Political Weekly, XXXII(52):3337–3342, December 1997 The mathematical equation that caused the banks to crash by Ian Stewart in The Observer, February 12, 2012 When You Cannot Hedge Continuously: The Corrections to Black–Scholes, Emanuel Derman The Skinny On Options TastyTrade Show (archives) Derivation and solution[edit] Derivation of the Black–Scholes Equation for Option Value, Prof. Thayer Watkins Solution of the Black–Scholes Equation Using the Green's Function, Prof. Dennis Silverman Solution via risk neutral pricing or via the PDE approach using Fourier transforms (includes discussion of other option types), Simon Leger Step-by-step solution of the Black–Scholes PDE, planetmath.org. The Black–Scholes Equation Expository article by mathematician Terence Tao. Computer implementations[edit] Black–Scholes in Multiple Languages Black–Scholes in Java -moving to link below- Black–Scholes in Java Chicago Option Pricing Model (Graphing Version) Black–Scholes–Merton Implied Volatility Surface Model (Java) Online Black–Scholes Calculator On-line financial calculator with Black–Scholes Historical[edit] Trillion Dollar Bet—Companion Web site to a Nova episode originally
broadcast on February 8, 2000. "The film tells the fascinating story
of the invention of the Black–Scholes Formula, a mathematical Holy
Grail that forever altered the world of finance and earned its
creators the 1997 Nobel Prize in Economics."
BBC Horizon A TV-programme on the so-called
v t e Derivatives market
Options Terms Credit spread Debit spread Exercise Expiration Moneyness Open interest Pin risk Risk-free interest rate Strike price the Greeks Volatility Vanilla options Bond option Call Employee stock option Fixed income FX Option styles Put Warrants Exotic options Asian Barrier Basket Binary Chooser Cliquet Commodore Compound Forward start Interest rate Lookback Mountain range Rainbow Swaption Combinations Collar Covered call Fence Iron butterfly Iron condor Straddle Strangle Protective put Risk reversal Spreads Back Bear Box Bull Butterfly Calendar Diagonal Intermarket Ratio Vertical Valuation Binomial Black Black–Scholes model Finite difference Garman-Kohlhagen Margrabe's formula Put–call parity Simulation Real options valuation Trinomial Vanna–Volga pricing Swaps Amortising Asset Basis Conditional variance Constant maturity Correlation Credit default Currency Dividend Equity Forex Forward Rate Agreement Inflation Interest rate Overnight indexed Total return Variance Volatility Year-on-Year Inflation-Indexed Zero Coupon Inflation-Indexed Zero Coupon Swap Forwards Futures Contango
Currency future
Exotic derivatives Energy derivative Freight derivative Inflation derivative Property derivative Weather derivative Other derivatives
Market issues Consumer debt Corporate debt Government debt Great Recession Municipal debt Tax policy v t e Hedge funds Investment strategy
Capital structure arbitrage
Convertible arbitrage
Equity market neutral
Event-driven Activist shareholder
Distressed securities
Risk arbitrage
Directional Convergence trade Commodity trading advisors / Managed futures account Dedicated short Global macro Long/short equity Trend following Other Fund of hedge funds / Multi-manager Trading Algorithmic trading Day trading High-frequency trading Prime brokerage Program trading Proprietary trading Related terms Markets Commodities Derivatives Equity Fixed income Foreign exchange Money markets Structured securities Misc
Investors Vulture funds Family offices Financial endowments Fund of hedge funds High-net-worth individual Institutional investors Insurance companies Investment banks Merchant banks Pension funds Sovereign wealth funds Regulatory Fund governance Hedge Fund Standards Board Alternative investment management companies
Hedge funds
v t e Stochastic processes Discrete time Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy Continuous time Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage Both Branching process
Galves–Löcherbach model
Gaussian process
Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise Fields and other Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Financial models Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie Actuarial models Bühlmann Cramér–Lundberg Risk process Sparre–Anderson Queueing models Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c Properties
Limit theorems Central limit theorem
Donsker's theorem
Doob's martingale convergence theorems
Ergodic theorem
Fisher–Tippett–Gnedenko theorem
Large deviation principle
Inequalities Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe Tools Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract Disciplines Actuarial mathematics
Econometrics
Ergodic theory
List |