THE BLACK SCHOLES FORMULA

THE BLACK SCHOLES FORMULA MARK H.A. DAVIS If option are correctly priced in the market, it hould not be poible to make ure profit by creating portfolio of long and hort poition in option and their underlying tock. Uing thi principle, a theoretical valuation formula for option i derived. Thee are the firt two entence of the abtract of the great paper [2] by Ficher Black and Myron Schole on option pricing, and encapulate the baic idea, which i that with the aet price model they employ initing on abence of arbitrage i enough to obtain a unique value for a call option on that aet. The reulting formula, 1.4) below, i the mot famou formula in financial economic, and in fact that whole ubject plit deciively into the pre-black-schole and pot-black-schole era. Thi article aim to give a elf-contained derivation of the formula, ome dicuion of the hedge parameter, and ome extenion of the formula, and to indicate why a formula baed on a tylized mathematical model which i known not to be a particularly accurate repreentation of real aet price ha neverthele proved o effective in the world of option trading. Section 1 formulate the model and tate and prove the formula. A i well known, the formula can equally well be tated in the form of a partial differential equation PDE); thi i equation 1.5) below. Section 2 dicue the PDE apect of Black-Schole. Section 3 ummarize information about the option Greek, while Section 4 and 5 introduce what i actually a more ueful form of Black-Schole, uually known a the Black formula. Finally, Section 6 dicue the application of the formula in market trading. We define the implied volatility and demontrate a robutne property of Black-Schole which implie that effective hedging can be achieved even if the true price proce i ubtantially different from Black and Schole tylized model. 1. The model and formula. Let Ω, F, F t ) t R +, P) be a probability pace with a given filtration F t ) repreenting the flow of information in the market. Traded aet price are F t -adapted tochatic procee on Ω, F, P). We aume that the market i frictionle: aet may be held in arbitrary amount, poitive and negative, the interet rate for borrowing and lending i the ame, and there are no tranaction cot i.e. the bid-ak pread i zero). While there may be many traded aet in the market, we fix attention on two of them. Firtly, there i a riky aet whoe price proce S t, t R + ) i aumed to atify the tochatic differential equation ds t = μs t dt + σs t dw t 1.1) with given drift μ and volatility σ. Here w t, t R + ) i an F t )-Brownian motion. Equation 1.1) ha a unique olution: if S t atifie 1.1) then by the Itô formula d log S t = μ 1 2 σ2 )dt + σdw t, o that S t atifie 1.1) if and only if S t = S exp μ 12 ) σ2 )t + σw t. 1.2) Aet S t i aumed to have a contant dividend yield q, i.e. the holder receive a dividend payment qs t dt in the time interval [t, t + dt[. Secondly, there i a rikle aet paying interet at a fixed continuouly-compounding rate r. The exact form of thi aet i unimportant it could be a moneymarket account in which $1 depoited at time grow to $e rt ) at time t, or it could be a zero-coupon bond maturing with a value of $1 at ome time T, o that it value at t T i B t = exp rt t)). 1

Thi grow, a required, at rate r: db t = rb t dt 1.3) Note that 1.3) doe not depend on the final maturity T the ame growth rate i obtained from any zero-coupon bond) and the choice of T i a matter of convenience. A European call option on S t i a contract, entered at time and pecified by two parameter K, T ), which give the holder the right, but not the obligation, to purchae one unit of the riky aet at price K at time T >. In the frictionle market etting, an option to buy N unit of tock i equivalent to N option on a ingle unit, o we do not need to include quantity a a parameter.) If S T K the option i worthle and will not be exercied. If S T > K the holder can exercie hi option, buying the aet at price K, and then immediately elling it at the prevailing market price S T, realizing a profit of S T K. Thu the exercie value of the option i [S T K] + = maxs T K, ). Similarly, the exercie value of a European put option, conferring on the holder the right to ell at a fixed price K, i [K S T ] +. In either cae the exercie value i non-negative and, in the above model, i trictly poitive with poitive probability, o the option buyer hould pay the writer a premium to acquire it. Black and Schole [2] howed that there i a unique arbitrage-free value for thi premium. Theorem 1.1. a) In the above model, the unique arbitrage-free value at time t < T when S t = S of the call option maturing at time T with trike K i Ct, S) = e qt t) SNd 1 ) e rt t) KNd 2 ) 1.4) where N ) denote the cumulative tandard normal ditribution function and Nx) = 1 2π x e 1 2 y2 dy d 1 = logs/k) + r + σ2 /2)T t) σ, T t d 2 = d 1 σ T t. b) The function Ct, S) may be characterized a the unique C 1,2 olution 1 of the Black-Schole partial differential equation PDE) olved backward in time with the terminal boundary condition t c) The value of the put option with exercie time T and trike K i + rs S + 1 2 σ2 St 2 2 C rc = 1.5) S2 CT, S) = [S K] +. 1.6) P t, S) = e rt t) KN d 2 ) e qt t) SN d 1 ). 1.7) To prove the theorem, we are going to how that the call option value can be replicated by a dynamic trading trategy inveting in the aet S t and in the zero-coupon bond B t = e rt t). A trading trategy i pecified by an initial capital x and a pair of adapted procee α t, β t repreenting the number of unit of S, B repectively held at time t; the portfolio value at time t i then X t = α t S t + β t B t, and by definition x = α S + β B. The trading trategy x, α, β) i admiible if i) T α2 t St 2 dt < a., ii) T β t dt < a.. iii) There exit a contant L uch that X t L for all t, a.. 1.8) 1 A two-parameter function i C 1,2 if it i once [twice] continuouly differentiable in the firt[econd] argument. 2

The gain from trade in [, t] i α u ds u + β u db u + qα u S u du, where the firt integral i an Itô tochatic integral. Thi i the um of the accumulated capital gain/loe in the two aet plu the total dividend received. The trading trategy i elf-financing if α t S t + β t B t α S β B = α u ds u + qα u S u du + β u db u, implying that the change in value over any interval in portfolio value i entirely due to gain from trade the accumulated increment in the value of the aet in the portfolio plu the total dividend received). We can alway create elf-financing trategie by fixing α, the invetment in the riky aet, and inveting all reidual wealth in the bond. Indeed, the value of the riky aet holding at time t i α t S t, o if the total portfolio value i X t we take β t = X t α t S t )/Bt). The portfolio value proce i then defined implicitly a the olution of the SDE dx t = α t ds t + qα t S t dt + β t db t = α t ds t + qα t S t dt + X t α t S t )r dt = rx t dt + α t S t σθdt + dw t ), 1.9) where θ = μ r + q)/σ. Thi trategy i alway elf-financing ince X t i by definition the gain from trade proce, while the value i αs + βb = X. Proof of Theorem 1.1). The key tep i to put the wealth equation 1.9) into a more convenient form by change of meaure. Define a meaure Q, the o-called rik-neutral meaure on Ω, F T ) by the Radon-Nikodým derivative dq dp = exp θw T 1 2 θ2 T ). The right-hand ide ha expectation 1, ince w T N, T ).) Expectation with repect to Q will be denoted E Q. By the Giranov theorem, ˇw = w t + θt i a Q-Brownian motion, o that from 1.1) the SDE atified by S t under Q i ds t = r q)s t dt + σs t d ˇw t 1.1) o that for t < T S T = S t exp r q 1 ) 2 σ2 )T t) + σ ˇw T ˇw t ). 1.11) Applying the Itô formula and equation 1.9) we find that, with X t = e rt X t and S t = e rt S t, d X t = α t St σd ˇw t, 1.12) Thu e rt X t i a Q-local martingale under condition 1.8)i). Let hs) = [S K] + and uppoe there exit a replicating trategy, i.e. a trategy x, α, β) with value proce X t contructed a in 1.9) uch that X T = hs T ) a.. Suppoe alo that α t atifie the tronger condition Then X t i a Q-martingale, and hence for t < T T E Q αt 2 St 2 dt <. 1.13) X t = e rt t) E Q [hs T ) F t ] 1.14) 3

and in particular x = e rt E Q [hs T )]. 1.15) Now S t i a Markov proce, o the conditional expectation in 1.14) i a function of S t, and indeed we ee from 1.11) that S T i a function of S t and the increment ˇw T ˇw t ) which i independent of F t. Writing ˇw T ˇw t ) = Z T t where Z N, 1), the expectation i imply a 1-dimenional integral with repect to the normal ditribution. Hence X t = Ct, S t ) where Ct, S) = e rt t) 2π hs expr q σ 2 /2)T t) σx T t))e 1 2 x2 dx. 1.16) Straightforward calculation how that thi integral i equal to the cloed-form expreion at 1.4). The argument o far how that if there i a replicating trategy the initial capital required mut be x = C, S ) where C i defined by 1.16). It remain to identify the trategy x, α, β) and to how that it i admiible. Let u temporarily take for granted the aertion of part b) of the theorem; thee will be proved in Propoition 2.1 below, where we will alo how that / S)t, S) = e qt t) Nd 1 ), o that in particular < / S < 1. The replicating trategy i A = x, α, β) defined by x = C, S ), α t = S t, S t), β t = 1 rb t t + 1 2 σ2 St 2 2 C S 2 qs t S Indeed, uing the PDE 1.5) we find that X t = α t S t + β t B t = Ct, S t ), o that A i replicating and alo X t, o that condition 1.8)iii) i atified. From 1.11) S 2 t = S 2 exp2r 2q σ 2 )t + 2σ ˇw t ), o that E Q [St 2 ] = exp2r 2q+σ 2 )t). Since e rt t) T / S < 1, thi how that E Q α2 t St 2 dt <, i.e. condition 1.13) i atified. Since β t i, almot urely, a continuou function of t it atifie 1.8)ii). Thu A i admiible. Finally, the gain from trade in an interval [, t] i α u ds u + qα u S u du + β u db u = = S ds + dc = Ct, S t ) C, S ). t + 1 2 σ2 S 2 t ). 2 ) C S 2 du 1.17) We obtain the right-hand ide of 1.17) from the definition of α, β, and it turn out to be jut the Itô formula applied to the function C.) Thi confirm the elf-financing property and complete the proof. Finally, part c) of the theorem follow from the model-free put-call parity relation C P = e qt t) S e rt t) K and ymmetry of the normal ditribution: N x) = 1 Nx). The replicating trategy derived above i known a delta hedging: the number of unit of the riky aet held in the portfolio i equal to the Black-Schole delta Δ = / S. So far, we have concentrated entirely on the hedging of call option. We conclude thi ection by howing that, with the cla of trading trategie we have defined, there are no arbitrage opportunitie in the Black-Schole model. Theorem 1.2. There i no admiible trading trategy in a ingle aet and the zero-coupon bond that generate an arbitrage opportunity, in the Black-Schole model. Proof. Suppoe X t i the portfolio value proce correponding to an admiible trading trategy x, α, β). There i an arbitrage opportunity if x = and, for ome t, X t a.. and P[X t > ] >, or equivalently E[X t ] >. Thi i the P-expectation, but E[X t ] > E Q [ X t ] > ince P and Q are equivalent meaure and e rt >. From 1.12), Xt i a Q-local martingale which, by the definition of admiibility, i bounded below by a contant L. It follow that X t i a upermartingale, o if x = then E Q [ X t ] for any t. So no arbitrage can arie from the trategy, α, β). 4

2. The Black-Schole partial differential equation. Propoition 2.1. a) The Black-Schole PDE 1.5) with boundary condition 1.6) ha a unique C 1,2 olution, given by 1.4). b) The Black-Schole delta Δt, S) i given by Δt, S) = S Ct, S) = e qt t) Nd 1 ). 2.1) Proof. It can with ome pain be directly checked that Ct, S) defined by 1.4) doe atify the Black-Schole PDE 1.5), 1.6), and a further calculation not quite a imple a it appear at firt ight!) give the formula 2.1) for the Black-Schole delta. It i however enlightening to take the orginal route of Black and Schole and relate the 1.5) to a impler equation, the heat equation. Note from the explicit expreion 1.11) for the price proce under the rik neutral meaure that, given the tarting point S t, there i a 1-1 relation between S T and the Brownian increment ˇw T ˇw t. We can therefore alway expre thing interchangeably in S coordinate or in ˇw coordinate. In fact we already made ue of thi in deriving the integral price expreion 1.16). Here we proceed a follow. For fixed parameter S, r, q, σ, define the function φ : R + R R + and u : [, T [ R R + by φt, x) = S exp r q 1 ) 2 σ2 )t + σ x and ut, x) = Ct, φt, x)). Note that the invere function ψt, ) = φ 1 t, ) i.e. the olution for x of the equation = φt, x)) i ψt, ) = 1 ) log r q 1 ) σ 2 σ2 )t. A direct calculation how that C atifie 1.5) if and only if u atifie the heat equation S u t + 1 2 u r u =. 2.2) 2 x2 If W t i Brownian motion on ome probability pace and u i a C 1,2 function then an application of the Itô formula how that u de rt ut, W t )) = e rt t + 1 2 ) u 2 x 2 ru rt u dt + e x dw t. If u atifie 2.2) with boundary condition ut, x) = gx) and T ) 2 u E x t, W t) dt < 2.3) then the proce t e rt ut, W t ) i a martingale o that, with E t,x denoting the conditional expectation given W t = x, e rt ut, x) = E t,x [e rt ut, W T )] = E t,x [e rt gw T )]. Since W T Nx, T t), thi how that u i given by t) ) e rt 1 ut, x) = gy) exp y x)2 dy. 2.4) 2πT t) 2T t) A ufficient condition for 2.3) i 1 2πT g 2 y)e y2 /2T dy <. 5

Delta Δ S e qτ Nd 1 ) Gamma Γ 2 C S 2 e qτ N d 1) Sσ τ Theta Θ τ e qτ SN d 1)σ 2 + qe qτ SNd τ 1 ) rke rτ Nd 2 ) Rho P r Kτe rτ Nd 2 ) Vega Υ σ e qτ S τn d 1 ) Table 3.1 Black-Schole rik parameter In our cae the boundary condition i gx) = [φt, x) K] + < φt, x) and thi condition i eaily checked. Hence 2.2) with thi boundary condition ha unique C 1,2 olution 2.4), implying that the invere function Ct, S) = ut, ψt, S)) given by 1.16) i the unique C 1,2, olution of 1.5) a claimed. 3. Hedge parameter. Bringing in all the parameter, the Black-Schole formula 1.4) i a 5-parameter function Ct, S) = Cτ, S, K, r, σ), where τ = T t i the time to maturity. For rikmanagement purpoe it i important to know the enitivitie of the option value to change in the parameter. The conventional hedge parameter or Greek are given in Table 3.1. There are light notational problem in that vega i not the name of a Greek letter here we have ued upper-cae upilon, but thi i not necearily a conventional choice) and upper-cae rho coincide with Latin P, o thi parameter i uually written ρ, riking confuion with correlation parameter. The expreion in the right-hand column are readily obtained from the enitivity parameter 5.3) and 5.4) of the univeral Black Formula introduced in Section 5 below. Delta i, of coure, the Black-Schole hedge ratio. Gamma meaure the convexity of C and i at it maximum when the option i cloe to being at-the-money. Since gamma i the rate of change of delta, frequent rebalancing of the hedge portfolio will be required in area of high gamma. Theta i defined a / τ and i generally negative a can be een from the table, it i alway negative for a call option on an aet with no dividend). It repreent the time decay in the option value a the maturity time i reduced, i.e. real time advance. A regard rho, it i not immediately obviou, without doing the calculation, what it ign will be: on the one hand, increaing r increae the forward price, puhing a call option further into the money, while on the other hand increaed r implie heavier dicounting, reducing option value. A can be een from the table, the firt effect win: rho i alway poitive. Vega i in ome way the mot important parameter, ince a key rik in managing book of traded option i vega rik, and in Black-Schole thi i completely outide the model. Bringing it back inide the model i the ubject of tochatic volatility. An extenive dicuion of the rik parameter and their ue can be found in Hull [6]. 4. The Black forward option formula. The 5-parameter repreentation Cτ, S, K, r, σ) i not the bet parametrization of Black-Schole. For the aet S t with dividend yield q the forward price at time t for delivery at time T i F t, T ) = S t e r q)t t) thi i a model-free reult, not related to the Black-Schole model). We can trivially re-expre the price formula 1.4) a Ct, S t ) = Bt, T )F t, T )Nd 1 ) KNd 2 )) 4.1) with d 1 = logf t, T )/K) + 1 2 σ2 T t) σ, d 2 = d 1 σ T t, T t where Bt, T ) = e rt t) i the zero-coupon bond value or dicount factor from T to t. There i, however, far more to thi than jut a change of notation. Firtly, the continuouly-compounding rate r i not market data. What i market data at time t i the et of dicount factor Bt, t ) for t > t. We ee from 4.1) that r play two ditinct role in Black-Schole: it appear in the computation of the 6

forward price F and the dicount factor B. But both of thee are more fundamental than r itelf and are in fact market data which, a 4.1) how, can be ued directly. A further advantage i that the exact mechanim of dividend payment i not important, a long a there i an unambiguouly-defined forward price. Formula 4.1) i known a the Black formula and i the mot ueful verion of Black-Schole, being widely applied in connection with FX foreign exchange) and interet-rate option a well a dividend-paying equitie. Fundamentally, it relate to a price model in which the price i expreed in the rik-neutral meaure a S t = F, t)m t where M t i the exponential martingale M t = exp σ ˇw t 1 ) 2 σ2 t, 4.2) which i equivalent to 1.11). Thi model accord with the general fact that, in a world of determinitic interet rate, the forward price i the expected price in the rik-neutral meaure, i.e. the ratio S t /F, t) i a poitive martingale with expectation 1. The exponential martingale 4.2) i the implet continuou-path proce with thee propertie. 5. A univeral Black formula. The parametrization of Black-Schole can be further compreed, a follow. Firt, note that σ and τ = T t) do not appear eparately, but only in the combination a = σ T t, where a 2 i ometime known a the operational time. Next, define the moneyne m a mt, T ) = K/F t, T ), and define da, m) = a 2 log m a o that d 1 = dσ T t, K/F t, T )).) Then the Black formula 4.1) become where C = BF fa, m), 5.1) fa, m) = Nda, m)) mnda, m) a). 5.2) Now BF i the price of a zero-trike call, or equivalently the price to be paid at time t for delivery of the aet at time T. Formula 5.1) ay that the price of the K-trike call i the model-free) price of the zero-trike call modified by a factor f that depend only on the moneyne and operational time. We call f the univeral Black-Schole function, and a graph of it i hown in Figure 5. With N = dn/dx and d = da, m) we find that mn d a) = N d) and hence obtain the following very imple expreion for the firt-order derivative: f a a, m) = N d), 5.3) f a, m) = Nd a). m 5.4) In particular, f/ a > and f/ m < for all a, m. Thi minimal parametrization of Black-Schole i i ued in tudie of tochatic volatility, ee for example Gatheral [5]. 6. Implied volatility and market trading. So far, our dicuion ha been entirely within the Black-Schole model. What happen if we attempt to ue Black-Schole delta hedging in real market trading? Thi quetion ha been conidered by everal author, including El Karoui et al [3] and Fouque et al [4], though neither of thee dicue the effect of jump in the price proce. In the univeral price formula 5.1) the parameter B, F, m are market data, o we can regard the formula a a mapping a p = BF fa, m) from a to price p [B[F K] +, BF ). In a traded option market, p i market data but mut lie in the tated interval, ele there i a tatic arbitrage 7

1..9.8.7.6 factor f.5.4.3.2.1...2.4.6.8 1 1.2 moneyne, m 1.4 1.6 1.8 2..2.4.6 1.8 a Fig. 5.1. The univeral Black-Schole function opportunity). In view of 5.3), fa, m) i trictly increaing in a and hence there i a unique value a = âp) uch that p = BF fâp), m). The implied volatility i ˆσp) = âp)/ T t. If the underlying price proce S t actually wa geometric Brownian motion 1.1) then ˆσ would be the ame, and equal to the volatility σ, for call option of all trike and maturitie. Of coure, thi i never the cae in practice ee The Volatility Surface for a dicuion. Here we retrict ourelve to examining what happen if we naïvely apply the Black-Schole delta-hedge when in reality the underlying proce i not geometric Brownian motion. Specifically, we aume that the true price model, under meaure P, i S t = S + η t S t dt + κ t S t dw t + S t v t z)μdt, dz) 6.1) [,t] R where μ i a finite-activity Poion random meaure, o that there i a finite meaure ν on R uch that μ[, t] A) νa)t μ π)[, t] A) i a martingale for each A BR). η, κ, v are predictable procee. Aume that η, κ and v are uch that the olution to the SDE 6.1) i well-defined and moreover that almot urely v t z) > 1 o S t > almot urely. Thi i a very general model including path-dependent coefficient, tochatic volatility and jump. Reader unfamiliar with jump diffuion model can et μ = ν = π = below, and refer to the lat paragraph of thi ection for comment on the effect of jump. Conider the cenario of elling at time a European call option at implied volatility ˆσ, i.e. for the price p = CT, S, K, r, ˆσ) and then following a Black-Schole delta-hedging trading trategy baed on contant volatility ˆσ until the option expire at time T. A uual, we hall denote Ct, ) = CT t,, K, r, ˆσ), o that the hedge portfolio, with value proce X t, i contructed by holding α t := S Ct, S t ) unit of the riky aet S, and the remainder β t := 1 B t X t α t S t ) unit in the rikle aet B a unit notional zero coupon bond). Thi portfolio, initially funded by the option ale o X = p), define a elf-financing trading trategy. Hence the portfolio value proce X atifie the SDE X t = p + S Cu, S u )η u S u du + + S Cu, S u )S u v u z)μdu, dz) + [,t] R S Cu, S u )κ u S u dw u X u S Cu, S u )S u )rdu. Now define Y t = Ct, S t ), o that in particular Y = p. Applying the Itô formula Lemma 4.4.6 of [1]) 8

give Y t = p + + + t Cu, S u )du + S Cu, S u )κ u S u dw u + 1 2 [,t] R S Cu, S u )η u S u du 2 SSCu, S u )κ 2 us 2 u du Cu, Su 1 + v u z))) Cu, S u ) ) μdt, dz). Thu the hedging error proce defined by Z t := X t Y t atifie the SDE Z t = rx u du rsu S Cu, S u ) + t Cu, S u ) + 1 2 κ2 usu 2 SSCu, 2 S u ) ) du Cu, Su 1 + v u z))) Cu, S u ) S Cu, S u )S u v u z) ) μdu, dz) [,t] R = rz u du + 1 Γu, S u )S 2 u ˆσ 2 2 κ 2 u)du Cu, Su 1 + v u z))) Cu, S u ) S Cu, S u )S u v u z) ) μdu, dz), [,t] R 6.2) where Γt, S t ) = 2 SS Ct, S t), and the lat equality follow from the Black-Schole PDE. Therefore the final difference between the hedging trategy and the required option payout i given by Z T = X T [S T K] + T = 1 2 e rt t) S 2 t Γt, S t )ˆσ 2 κ 2 t )dt [,T ] R e rt t) 1 ɛ Γt, S t 1 + ɛ v t z)))v 2 t z)s 2 u dɛ dɛ ) πdt, dz) M T 6.3) where M T i the terminal value of the martingale M t = e rt t) 1 ɛ Γt, S t 1 + ɛ v t z)))vt 2 z)su dɛ 2 dɛ ) μ π)dt, dz). [,T ] R Equation 6.3) i a key formula, a it how that ucceful hedging i quite poible even under ignificant model error. Without ome robutne property of thi kind, it i hard to imagine that the derivative indutry could exit at all, ince hedging under realitic condition would be impoible. Conider firt the cae μ, where S t ha continuou ample path and the lat two term in 6.3) vanih. Then ucceful hedging depend entirely on the relationhip between the implied volatility ˆσ and the true local volatility κ t. Note from Table 3.1 that Γ t >. If we, a option writer, are lucky and ˆσ 2 βt 2 a.. for all t then the hedging trategy make a profit with probability one even though the true price model i ubtantially different from the aumed model 1.1). On the other hand if we underetimate the volatility, we will conitently make a lo. The magnitude of the the profit or lo depend on the option convexity Γ. If Γ i mall then hedging error i mall even if the volatility ha been groly mi-etimated. For the option writer, jump in either direction are unambiguouly bad new. Since C i convex, ΔC > / S)ΔS, o the lat term in 6.2) i monotone decreaing: the hedge profit take a hit every time there i a jump, either upward or downward, in the underlying price. However, there i ome recoure: in 6.3), M T ha expectation zero while the penultimate term i negative. By increaing ˆσ we increae E[Z T ], o we could arrive at a ituation where E[Z T ] >, although in thi cae there i no poibility of with probability one profit becaue of the martingale term. All of thi reinforce the trader intuition that one can offet additional hedge cot by charging more upfront i.e. increaing ˆσ) and hedging at the higher level of implied volatility. 9

REFERENCES [1] D. Applebaum, Lévy Procee and Stochatic Calculu, Cambridge Univerity Pre, 24. [2] F. Black and M. Schole, The pricing of option and corporate liabilitie, J. Political Economy 81 1973) 637-654 [3] N. El Karoui, M. Jeanblanc-Picqué and S.E. Shreve, Robutne of the Black and Schole formula, Mathematical Finance 8 1998) 93-126 [4] J.-P. Fouque, G. Papanicolaou and K.R. Sircar, Derivative in financial market with tochatic volatility, Cambridge Univerity Pre 2 [5] J. Gatheral, The Volatility Surface, Wiley 26 [6] J.C. Hull, Option, Future and Other Derivative, 6th ed. Prentice Hall 25 1