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\ displaystyle d _ 1 = \ frac \ ln \ frac S K +(r-q+\ sigma 2/ 2)T \ sigma \ sqrt T, \, d _ 2 =d _ 1 -\ sigma \ sqrt T. \, This pays out one unit of money if the area is above the strike at maturity. Its value now is given by C = e r T (d 2 ). \ displaystyle C=e -rT \ Phi (d _ 2 ). \, This pays one unit of cash if the spot is listed below the strike at maturity.

\ displaystyle P=e -rT \ Phi (-d _ 2 ). \, This pays one unit of property if the area is above the strike at maturity. Its value now is offered by C = S e q T (d 1 ). \ displaystyle C=Se -qT \ Phi (d _ 1 ). \, This pays one unit of asset if the area is below the strike at maturity.

\ displaystyle P=Se -qT \ Phi (-d _ 1 ). \, American binary put with K = 100, r = , = , T = 1. An American option gives the holder the right to work out at any point up to and consisting of the expiry time T \ displaystyle T. That is, representing by K \ displaystyle K the strike price, if K S \ displaystyle K \ geq S (resp.

call) deserves precisely one system. Let a = 1 ln (K/ S ), = r q 2, and b = 2 + 2 r. \ displaystyle a= \ frac 1 \ sigma \ ln(K/S) ext , \ xi = \ frac r-q \ sigma - \ frac \ sigma 2 ext , and b= \ sqrt \ xi 2 +2 r. \, The rate of a cash-or-nothing American binary put (resp.

K S \ displaystyle K S) and time-to-expiry T \ displaystyle T is 1 2 e a (b) 1 + sgn (a) erf (b T a 2 T) + e 2 a b [1 sgn (a) erf (b T + a 2 T)] \ displaystyle \ frac 1 2 e a \ left(\ xi -b \ right) \ left \ 1+\ operatorname sgn (a)\ operatorname erf \ left( \ frac bT- \ sqrt 2T \ right)+e 2ab \ left [1-\ operatorname sgn (a)\ operatorname erf \ left( \ frac bT+ \ sqrt 2T \ ideal)\ ideal] \ ideal \ \, where erf \ displaystyle \ operatorname erf signifies the mistake function and sgn \ displaystyle \ operatorname sgn represents the sign function.

If we signify by S the FOR/DOM currency exchange rate (, 1 system of foreign currency deserves 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 listed below the strike is exactly like a cash-or nothing call and put respectively.

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