THE BLACK SCHOLES FORMULA
|
|
- Angelina Robbins
- 8 years ago
- Views:
Transcription
1 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
2 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 σ2 St 2 2 C rc = 1.5) S2 CT, S) = [S K] ) 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
3 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
4 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 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 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
5 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 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 ) 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
6 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) 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
7 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
8 factor f moneyne, m a Fig 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 of [1]) 8
9 give Y t = p t Cu, S u )du + S Cu, S u )κ u S u dw u [,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 ) 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
10 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 ) [3] N. El Karoui, M. Jeanblanc-Picqué and S.E. Shreve, Robutne of the Black and Schole formula, Mathematical Finance ) [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
Chapter 10 Stocks and Their Valuation ANSWERS TO END-OF-CHAPTER QUESTIONS
Chapter Stoc and Their Valuation ANSWERS TO EN-OF-CHAPTER QUESTIONS - a. A proxy i a document giving one peron the authority to act for another, typically the power to vote hare of common toc. If earning
More informationMECH 2110 - Statics & Dynamics
Chapter D Problem 3 Solution 1/7/8 1:8 PM MECH 11 - Static & Dynamic Chapter D Problem 3 Solution Page 7, Engineering Mechanic - Dynamic, 4th Edition, Meriam and Kraige Given: Particle moving along a traight
More informationMBA 570x Homework 1 Due 9/24/2014 Solution
MA 570x Homework 1 Due 9/24/2014 olution Individual work: 1. Quetion related to Chapter 11, T Why do you think i a fund of fund market for hedge fund, but not for mutual fund? Anwer: Invetor can inexpenively
More informationMSc Financial Economics: International Finance. Bubbles in the Foreign Exchange Market. Anne Sibert. Revised Spring 2013. Contents
MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market Anne Sibert Revied Spring 203 Content Introduction................................................. 2 The Mone Market.............................................
More informationThe Black-Scholes Formula
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the
More informationCS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options
CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common
More informationOption Pricing. 1 Introduction. Mrinal K. Ghosh
Option Pricing Mrinal K. Ghosh 1 Introduction We first introduce the basic terminology in option pricing. Option: An option is the right, but not the obligation to buy (or sell) an asset under specified
More informationQueueing systems with scheduled arrivals, i.e., appointment systems, are typical for frontal service systems,
MANAGEMENT SCIENCE Vol. 54, No. 3, March 28, pp. 565 572 in 25-199 ein 1526-551 8 543 565 inform doi 1.1287/mnc.17.82 28 INFORMS Scheduling Arrival to Queue: A Single-Server Model with No-Show INFORMS
More informationUnit 11 Using Linear Regression to Describe Relationships
Unit 11 Uing Linear Regreion to Decribe Relationhip Objective: To obtain and interpret the lope and intercept of the leat quare line for predicting a quantitative repone variable from a quantitative explanatory
More informationSenior Thesis. Horse Play. Optimal Wagers and the Kelly Criterion. Author: Courtney Kempton. Supervisor: Professor Jim Morrow
Senior Thei Hore Play Optimal Wager and the Kelly Criterion Author: Courtney Kempton Supervior: Profeor Jim Morrow June 7, 20 Introduction The fundamental problem in gambling i to find betting opportunitie
More informationAdditional questions for chapter 4
Additional questions for chapter 4 1. A stock price is currently $ 1. Over the next two six-month periods it is expected to go up by 1% or go down by 1%. The risk-free interest rate is 8% per annum with
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationPricing of an Exotic Forward Contract
Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan E-mail: {akahori,
More informationLectures. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. No tutorials in the first week
Lectures Sergei Fedotov 20912 - Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics
More informationv = x t = x 2 x 1 t 2 t 1 The average speed of the particle is absolute value of the average velocity and is given Distance travelled t
Chapter 2 Motion in One Dimenion 2.1 The Important Stuff 2.1.1 Poition, Time and Diplacement We begin our tudy of motion by conidering object which are very mall in comparion to the ize of their movement
More informationPartial optimal labeling search for a NP-hard subclass of (max,+) problems
Partial optimal labeling earch for a NP-hard ubcla of (max,+) problem Ivan Kovtun International Reearch and Training Center of Information Technologie and Sytem, Kiev, Uraine, ovtun@image.iev.ua Dreden
More informationA technical guide to 2014 key stage 2 to key stage 4 value added measures
A technical guide to 2014 key tage 2 to key tage 4 value added meaure CONTENTS Introduction: PAGE NO. What i value added? 2 Change to value added methodology in 2014 4 Interpretation: Interpreting chool
More informationFour Points Beginner Risk Managers Should Learn from Jeff Holman s Mistakes in the Discussion of Antifragile arxiv:1401.2524v1 [q-fin.
Four Point Beginner Rik Manager Should Learn from Jeff Holman Mitake in the Dicuion of Antifragile arxiv:1401.54v1 [q-fin.gn] 11 Jan 014 Naim Nichola Taleb January 014 Abtract Uing Jeff Holman comment
More informationA note on profit maximization and monotonicity for inbound call centers
A note on profit maximization and monotonicity for inbound call center Ger Koole & Aue Pot Department of Mathematic, Vrije Univeriteit Amterdam, The Netherland 23rd December 2005 Abtract We conider an
More information1 The Black-Scholes model: extensions and hedging
1 The Black-Scholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes
More informationEXP 481 -- Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0
EXP 481 -- Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the
More informationRISK MANAGEMENT POLICY
RISK MANAGEMENT POLICY The practice of foreign exchange (FX) rik management i an area thrut into the potlight due to the market volatility that ha prevailed for ome time. A a conequence, many corporation
More informationHow To Price A Call Option
Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min
More informationTwo Dimensional FEM Simulation of Ultrasonic Wave Propagation in Isotropic Solid Media using COMSOL
Excerpt from the Proceeding of the COMSO Conference 0 India Two Dimenional FEM Simulation of Ultraonic Wave Propagation in Iotropic Solid Media uing COMSO Bikah Ghoe *, Krihnan Balaubramaniam *, C V Krihnamurthy
More informationTRADING rules are widely used in financial market as
Complex Stock Trading Strategy Baed on Particle Swarm Optimization Fei Wang, Philip L.H. Yu and David W. Cheung Abtract Trading rule have been utilized in the tock market to make profit for more than a
More informationMartingale Pricing Applied to Options, Forwards and Futures
IEOR E4706: Financial Engineering: Discrete-Time Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the
More informationOn Market-Making and Delta-Hedging
On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing What to market makers do? Provide
More informationLecture 6 Black-Scholes PDE
Lecture 6 Black-Scholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the risk-neutral measure Q by If the contingent
More informationLinear Momentum and Collisions
Chapter 7 Linear Momentum and Colliion 7.1 The Important Stuff 7.1.1 Linear Momentum The linear momentum of a particle with ma m moving with velocity v i defined a p = mv (7.1) Linear momentum i a vector.
More informationProject Management Basics
Project Management Baic A Guide to undertanding the baic component of effective project management and the key to ucce 1 Content 1.0 Who hould read thi Guide... 3 1.1 Overview... 3 1.2 Project Management
More informationwhere N is the standard normal distribution function,
The Black-Scholes-Merton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at
More informationMorningstar Fixed Income Style Box TM Methodology
Morningtar Fixed Income Style Box TM Methodology Morningtar Methodology Paper Augut 3, 00 00 Morningtar, Inc. All right reerved. The information in thi document i the property of Morningtar, Inc. Reproduction
More informationDoes Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial
More informationLinear energy-preserving integrators for Poisson systems
BIT manucript No. (will be inerted by the editor Linear energy-preerving integrator for Poion ytem David Cohen Ernt Hairer Received: date / Accepted: date Abtract For Hamiltonian ytem with non-canonical
More informationAssessing the Discriminatory Power of Credit Scores
Aeing the Dicriminatory Power of Credit Score Holger Kraft 1, Gerald Kroiandt 1, Marlene Müller 1,2 1 Fraunhofer Intitut für Techno- und Wirtchaftmathematik (ITWM) Gottlieb-Daimler-Str. 49, 67663 Kaierlautern,
More informationThe Cash Flow Statement: Problems with the Current Rules
A C C O U N T I N G & A U D I T I N G accounting The Cah Flow Statement: Problem with the Current Rule By Neii S. Wei and Jame G.S. Yang In recent year, the tatement of cah flow ha received increaing attention
More informationName: SID: Instructions
CS168 Fall 2014 Homework 1 Aigned: Wedneday, 10 September 2014 Due: Monday, 22 September 2014 Name: SID: Dicuion Section (Day/Time): Intruction - Submit thi homework uing Pandagrader/GradeScope(http://www.gradecope.com/
More informationA Note on Profit Maximization and Monotonicity for Inbound Call Centers
OPERATIONS RESEARCH Vol. 59, No. 5, September October 2011, pp. 1304 1308 in 0030-364X ein 1526-5463 11 5905 1304 http://dx.doi.org/10.1287/opre.1110.0990 2011 INFORMS TECHNICAL NOTE INFORMS hold copyright
More informationIs Mark-to-Market Accounting Destabilizing? Analysis and Implications for Policy
Firt draft: 4/12/2008 I Mark-to-Market Accounting Detabilizing? Analyi and Implication for Policy John Heaton 1, Deborah Luca 2 Robert McDonald 3 Prepared for the Carnegie Rocheter Conference on Public
More informationBlack-Scholes Option Pricing Model
Black-Scholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,
More informationARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic
More informationBidding for Representative Allocations for Display Advertising
Bidding for Repreentative Allocation for Diplay Advertiing Arpita Ghoh, Preton McAfee, Kihore Papineni, and Sergei Vailvitkii Yahoo! Reearch. {arpita, mcafee, kpapi, ergei}@yahoo-inc.com Abtract. Diplay
More information7: The CRR Market Model
Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The Cox-Ross-Rubinstein
More informationProfitability of Loyalty Programs in the Presence of Uncertainty in Customers Valuations
Proceeding of the 0 Indutrial Engineering Reearch Conference T. Doolen and E. Van Aken, ed. Profitability of Loyalty Program in the Preence of Uncertainty in Cutomer Valuation Amir Gandomi and Saeed Zolfaghari
More informationHealth Insurance and Social Welfare. Run Liang. China Center for Economic Research, Peking University, Beijing 100871, China,
Health Inurance and Social Welfare Run Liang China Center for Economic Reearch, Peking Univerity, Beijing 100871, China, Email: rliang@ccer.edu.cn and Hao Wang China Center for Economic Reearch, Peking
More informationThe Black-Scholes-Merton Approach to Pricing Options
he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining
More informationCHARACTERISTICS OF WAITING LINE MODELS THE INDICATORS OF THE CUSTOMER FLOW MANAGEMENT SYSTEMS EFFICIENCY
Annale Univeritati Apuleni Serie Oeconomica, 2(2), 200 CHARACTERISTICS OF WAITING LINE MODELS THE INDICATORS OF THE CUSTOMER FLOW MANAGEMENT SYSTEMS EFFICIENCY Sidonia Otilia Cernea Mihaela Jaradat 2 Mohammad
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationA Life Contingency Approach for Physical Assets: Create Volatility to Create Value
A Life Contingency Approach for Phyical Aet: Create Volatility to Create Value homa Emil Wendling 2011 Enterprie Rik Management Sympoium Society of Actuarie March 14-16, 2011 Copyright 2011 by the Society
More informationControl of Wireless Networks with Flow Level Dynamics under Constant Time Scheduling
Control of Wirele Network with Flow Level Dynamic under Contant Time Scheduling Long Le and Ravi R. Mazumdar Department of Electrical and Computer Engineering Univerity of Waterloo,Waterloo, ON, Canada
More informationOnline story scheduling in web advertising
Online tory cheduling in web advertiing Anirban Dagupta Arpita Ghoh Hamid Nazerzadeh Prabhakar Raghavan Abtract We tudy an online job cheduling problem motivated by toryboarding in web advertiing, where
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationA SNOWBALL CURRENCY OPTION
J. KSIAM Vol.15, No.1, 31 41, 011 A SNOWBALL CURRENCY OPTION GYOOCHEOL SHIM 1 1 GRADUATE DEPARTMENT OF FINANCIAL ENGINEERING, AJOU UNIVERSITY, SOUTH KOREA E-mail address: gshim@ajou.ac.kr ABSTRACT. I introduce
More informationUnderstanding Options and Their Role in Hedging via the Greeks
Understanding Options and Their Role in Hedging via the Greeks Bradley J. Wogsland Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996-1200 Options are priced assuming that
More informationSolution of the Heat Equation for transient conduction by LaPlace Transform
Solution of the Heat Equation for tranient conduction by LaPlace Tranform Thi notebook ha been written in Mathematica by Mark J. McCready Profeor and Chair of Chemical Engineering Univerity of Notre Dame
More informationCaput Derivatives: October 30, 2003
Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor
More informationStochastic House Appreciation and Optimal Subprime Lending
Stochatic Houe Appreciation and Optimal Subprime Lending Tomaz Pikorki Columbia Buine School tp5@mail.gb.columbia.edu Alexei Tchityi NYU Stern atchity@tern.nyu.edu February 8 Abtract Thi paper tudie an
More informationChapter 13 The Black-Scholes-Merton Model
Chapter 13 The Black-Scholes-Merton Model March 3, 009 13.1. The Black-Scholes option pricing model assumes that the probability distribution of the stock price in one year(or at any other future time)
More informationHUMAN CAPITAL AND THE FUTURE OF TRANSITION ECONOMIES * Michael Spagat Royal Holloway, University of London, CEPR and Davidson Institute.
HUMAN CAPITAL AND THE FUTURE OF TRANSITION ECONOMIES * By Michael Spagat Royal Holloway, Univerity of London, CEPR and Davidon Intitute Abtract Tranition economie have an initial condition of high human
More informationMathematical Finance
Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationBi-Objective Optimization for the Clinical Trial Supply Chain Management
Ian David Lockhart Bogle and Michael Fairweather (Editor), Proceeding of the 22nd European Sympoium on Computer Aided Proce Engineering, 17-20 June 2012, London. 2012 Elevier B.V. All right reerved. Bi-Objective
More informationChapter 10 Velocity, Acceleration, and Calculus
Chapter 10 Velocity, Acceleration, and Calculu The firt derivative of poition i velocity, and the econd derivative i acceleration. Thee derivative can be viewed in four way: phyically, numerically, ymbolically,
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science
aachuett Intitute of Technology Department of Electrical Engineering and Computer Science 6.685 Electric achinery Cla Note 10: Induction achine Control and Simulation c 2003 Jame L. Kirtley Jr. 1 Introduction
More informationHedging Options In The Incomplete Market With Stochastic Volatility. Rituparna Sen Sunday, Nov 15
Hedging Options In The Incomplete Market With Stochastic Volatility Rituparna Sen Sunday, Nov 15 1. Motivation This is a pure jump model and hence avoids the theoretical drawbacks of continuous path models.
More informationReview of Basic Options Concepts and Terminology
Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some
More information6. Friction, Experiment and Theory
6. Friction, Experiment and Theory The lab thi wee invetigate the rictional orce and the phyical interpretation o the coeicient o riction. We will mae ue o the concept o the orce o gravity, the normal
More informationSoftware Engineering Management: strategic choices in a new decade
Software Engineering : trategic choice in a new decade Barbara Farbey & Anthony Finkeltein Univerity College London, Department of Computer Science, Gower St. London WC1E 6BT, UK {b.farbey a.finkeltein}@ucl.ac.uk
More informationRisk Management for a Global Supply Chain Planning under Uncertainty: Models and Algorithms
Rik Management for a Global Supply Chain Planning under Uncertainty: Model and Algorithm Fengqi You 1, John M. Waick 2, Ignacio E. Gromann 1* 1 Dept. of Chemical Engineering, Carnegie Mellon Univerity,
More informationCh 7. Greek Letters and Trading Strategies
Ch 7. Greek Letters and Trading trategies I. Greek Letters II. Numerical Differentiation to Calculate Greek Letters III. Dynamic (Inverted) Delta Hedge IV. elected Trading trategies This chapter introduces
More informationIntroduction to the article Degrees of Freedom.
Introduction to the article Degree of Freedom. The article by Walker, H. W. Degree of Freedom. Journal of Educational Pychology. 3(4) (940) 53-69, wa trancribed from the original by Chri Olen, George Wahington
More informationOn Black-Scholes Equation, Black- Scholes Formula and Binary Option Price
On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.
More informationStochastic House Appreciation and Optimal Mortgage Lending
Stochatic Houe Appreciation and Optimal Mortgage Lending Tomaz Pikorki Columbia Buine School tp2252@columbia.edu Alexei Tchityi UC Berkeley Haa tchityi@haa.berkeley.edu December 28 Abtract We characterize
More informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationFEDERATION OF ARAB SCIENTIFIC RESEARCH COUNCILS
Aignment Report RP/98-983/5/0./03 Etablihment of cientific and technological information ervice for economic and ocial development FOR INTERNAL UE NOT FOR GENERAL DITRIBUTION FEDERATION OF ARAB CIENTIFIC
More information4. Option pricing models under the Black- Scholes framework
4. Option pricing models under the Black- Scholes framework Riskless hedging principle Writer of a call option hedges his exposure by holding certain units of the underlying asset in order to create a
More informationSocially Optimal Pricing of Cloud Computing Resources
Socially Optimal Pricing of Cloud Computing Reource Ihai Menache Microoft Reearch New England Cambridge, MA 02142 t-imena@microoft.com Auman Ozdaglar Laboratory for Information and Deciion Sytem Maachuett
More informationNumerical methods for American options
Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment
More informationIntroduction to Stochastic Differential Equations (SDEs) for Finance
Introduction to Stochastic Differential Equations (SDEs) for Finance Andrew Papanicolaou January, 013 Contents 1 Financial Introduction 3 1.1 A Market in Discrete Time and Space..................... 3
More informationOptions: Valuation and (No) Arbitrage
Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial
More informationThe Black-Scholes pricing formulas
The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More informationJung-Soon Hyun and Young-Hee Kim
J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL Jung-Soon Hyun and Young-Hee Kim Abstract. We present two approaches of the stochastic interest
More information第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model
1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American
More informationLecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The Black-Scholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The Black-Scholes-Merton Model
More informationAn exact formula for default swaptions pricing in the SSRJD stochastic intensity model
An exact formula for default swaptions pricing in the SSRJD stochastic intensity model Naoufel El-Bachir (joint work with D. Brigo, Banca IMI) Radon Institute, Linz May 31, 2007 ICMA Centre, University
More informationReview of Multiple Regression Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised January 13, 2015
Review of Multiple Regreion Richard William, Univerity of Notre Dame, http://www3.nd.edu/~rwilliam/ Lat revied January 13, 015 Aumption about prior nowledge. Thi handout attempt to ummarize and yntheize
More informationQueueing Models for Multiclass Call Centers with Real-Time Anticipated Delays
Queueing Model for Multicla Call Center with Real-Time Anticipated Delay Oualid Jouini Yve Dallery Zeynep Akşin Ecole Centrale Pari Koç Univerity Laboratoire Génie Indutriel College of Adminitrative Science
More informationGlobal Imbalances or Bad Accounting? The Missing Dark Matter in the Wealth of Nations. Ricardo Hausmann and Federico Sturzenegger
Global Imbalance or Bad Accounting? The Miing Dark Matter in the Wealth of Nation Ricardo Haumann and Federico Sturzenegger CID Working Paper No. 124 January 2006 Copyright 2006 Ricardo Haumann, Federico
More informationBlack-Scholes model: Greeks - sensitivity analysis
VII. Black-Scholes model: Greeks- sensitivity analysis p. 1/15 VII. Black-Scholes model: Greeks - sensitivity analysis Beáta Stehlíková Financial derivatives, winter term 2014/2015 Faculty of Mathematics,
More information1 Introduction. Reza Shokri* Privacy Games: Optimal User-Centric Data Obfuscation
Proceeding on Privacy Enhancing Technologie 2015; 2015 (2):1 17 Reza Shokri* Privacy Game: Optimal Uer-Centric Data Obfucation Abtract: Conider uer who hare their data (e.g., location) with an untruted
More information2. METHOD DATA COLLECTION
Key to learning in pecific ubject area of engineering education an example from electrical engineering Anna-Karin Cartenen,, and Jonte Bernhard, School of Engineering, Jönköping Univerity, S- Jönköping,
More informationHow To Understand The Hort Term Power Market
Short-term allocation of ga network and ga-electricity input forecloure Miguel Vazquez a,, Michelle Hallack b a Economic Intitute (IE), Federal Univerity of Rio de Janeiro (UFRJ) b Economic Department,
More informationChapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.
Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of risk-adjusted discount rate. Part D Introduction to derivatives. Forwards
More informationA Duality Model of TCP and Queue Management Algorithms
A Duality Model of TCP and Queue Management Algorithm Steven H. Low CS and EE Department California Intitute of Technology Paadena, CA 95 low@caltech.edu May 4, Abtract We propoe a duality model of end-to-end
More informationJanuary 21, 2015. Abstract
T S U I I E P : T R M -C S J. R January 21, 2015 Abtract Thi paper evaluate the trategic behavior of a monopolit to influence environmental policy, either with taxe or with tandard, comparing two alternative
More informationBlack-Scholes and the Volatility Surface
IEOR E4707: Financial Engineering: Continuous-Time Models Fall 2009 c 2009 by Martin Haugh Black-Scholes and the Volatility Surface When we studied discrete-time models we used martingale pricing to derive
More informationTowards Control-Relevant Forecasting in Supply Chain Management
25 American Control Conference June 8-1, 25. Portland, OR, USA WeA7.1 Toward Control-Relevant Forecating in Supply Chain Management Jay D. Schwartz, Daniel E. Rivera 1, and Karl G. Kempf Control Sytem
More informationPiracy in two-sided markets
Technical Workhop on the Economic of Regulation Piracy in two-ided market Paul Belleflamme, CORE & LSM Univerité catholique de Louvain 07/12/2011 OECD, Pari Outline Piracy in one-ided market o Baic model
More informationPhysics 111. Exam #1. January 24, 2014
Phyic 111 Exam #1 January 24, 2014 Name Pleae read and follow thee intruction carefully: Read all problem carefully before attempting to olve them. Your work mut be legible, and the organization clear.
More informationScheduling of Jobs and Maintenance Activities on Parallel Machines
Scheduling of Job and Maintenance Activitie on Parallel Machine Chung-Yee Lee* Department of Indutrial Engineering Texa A&M Univerity College Station, TX 77843-3131 cylee@ac.tamu.edu Zhi-Long Chen** Department
More informationOption pricing. Vinod Kothari
Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate
More information