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Option price: volatility expansion in .NET Generation qr bidimensional barcode in .NET Option price: volatility expansion

3.14 Option price: volatility expansion generate, create qr code iso/iec18004 none on .net projects UCC-14 volatility being typicall visual .net qr codes y of the order of 10 2 /year. One can, hence, de ne a generic and rapidly convergent expansion of the option price in powers of the volatility.

Consider an equity, interest rate, or coupon bond option that has a payoff function P , which matures at time t with strike price K and is given by. P = [S(t ) K]+ S(t ) is the nancial in .net framework qr codes strument on which the option is written. The price of the option at earlier time t0 , using the forward bond numeraire, is given by Eq.

(3.8) as C(t0 , t , K) = B(t0 , t )E[S(t ) K]+ Let S(t0 ) be the price of the instrument at time t0 : one expects that S(t ) S(t0 ), up to factors depending on drift, is of the order of volatility since all uctuations away from the initial value are due to nonzero volatility. One has the following C(t0 , t , K) = B(t0 , t )E[S(t ) K]+ = B(t0 , t )E[V K]+ V = S(t ) S(t0 ); K = K S(t0 ) (3.

62). The (random) quantity V = S(t ) S(t0 ), up to factors of drift that will be accounted for, has an order of magnitude value equal to O( ), the volatility of S(t). The option price will be obtained in powers of V , which in turn, after the expectation value is taken, will lead to the option price as a power series in . Using the representation of the Dirac-delta function given in Eq.

(A.7) (Q) = 1 2 . + . d ei Q (3.63). yields the following expression for the payoff function P = S(t ) K + + dQ (V Q) Q K dQ d i (V Q) Q K e 2 ei (V Q) Q K e i Q Q K 1 + i V + in i2 2 2 V + n V n + O( n+1 ) 2 n! (3.64). Options and option theory Note V is the only random quantity in the payoff function and is an effective potential for option pricing. One has the following expansion for the option price C(t0 , t , T , K) =E P B(t0 , t ) e i Q Q K. Q, + (3.65) E[1] + i E[V ] + i n i2 2 E[V 2 ] + n E[V (n) ] 2 n!. e i Q Q K 1 in C0 + i C1 2 C2 + + n Cn + O( n+1 ) 2 n!. To evaluate the option pr QR Code JIS X 0510 for .NET ice to an accuracy of O( n ), one needs to evaluate the coef cients C0 = E[1] C1 = E[V ] : D1 = C1 /C0 C2 = E[V 2 ] : D2 = C2 /C0 . .

. Cn = E[V n ] : Dn = Cn /C0 For most cases, the option price is obtained by evaluating the coef cients C0 , C1 and C2 . Eq.

(3.65) yields a cumulant expansion that, to second order and for D1 = C1 /C0 and D2 = C2 /C0 , gives the following approximate option price C(t0 , t , T , K) B(t0 , t ) = C0. (3.66) (3.67) (3.

68). e i Q Q K 1 2 exp i D1 2 (D2 D 1 ) 2. e i Q Q + D1 K 1 2 exp 2 (D2 D1 ) 2. 2 D2 D1 + O( 3 ). C(t0 , t , T , K) 1 = C0 I (X) B(t0 , t ) 2 K D1 2 D2 D1 + (3.69). The function I (X) is giv VS .NET QR-Code en in terms of the error function N (u) as follows X= ; K = K S(t0 ). I (X) =. dQ(Q X)+ e 2 Q = e 2 X + 1 2 X(N (X) 1); N (u .NET QR Code JIS X 0510 ) = 2 . u . dQe 2 Q (3.70). 3.15 Derivatives and the real economy The asymptotic behavior o f the error function N (u) yields the following limits I (X) = 1 + O(X2 ) e. 1 X2 2. 2 + O(e X ). X 0 (3.71) X >> 0 The option s price, for X 0, is the following 1 C(t0 , t , K) B(t0 , t )C0 2 1 2 D2 D1 (K D1 ) + O(X2 ) (3.72) 2. The coef cients that dete visual .net QR Code 2d barcode rmine the option price C(t0 , t , K) have the following intuitive interpretation..

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