Formula for Final Exam

Formula for Final Exam - Results on the normal distribution...

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Results on the normal distribution: Let Z be a normal random variable N ( m,σ 2 ) with mean m and variance σ 2 (under P ). Then, we have that E ± e Z I { e Z <a } ² = exp ³ m + σ 2 2 ´ N ³ ln( a ) - m - σ 2 σ ´ , E ± e Z I { e Z >a } ² = exp ³ m + σ 2 2 ´ N ³ m + σ 2 - ln a σ ´ , where N ( z ) = P {N (0 , 1) z } , and its moment-generating function is given by t 7→ E ± e tZ ² = e mt + σ 2 t 2 / 2 . Itˆ o’s formula: Let X be an Itˆ o process such that dX t = u ( t,X t ) dt + v ( t,X t ) dW t , with X 0 = x 0 . If g ( t,x ) is a function in C 1 , 2 ([0 , ) × R ), then Y t = g ( t,X t ) is also an Itˆo process and it is such that dY t = ∂g ∂t ( t,X t ) dt + ∂g ∂x ( t,X t ) dX t + 1 2 2 g ∂x 2 ( t,X t ) ( dX t ) 2 , where the following rules hold: dt · dt = dt · dW t = dW t · dt = 0 and dW t · dW t = dt . Black-Scholes PDE: For a European derivative with payoff g ( S T ) and time- t price F ( t,S t ), the determistic function F ( t,x ) is the solution to - rF ( t,x ) + rxF x ( t,x ) + F t ( t,x ) + 1 2 σ 2 x 2 F xx ( t,x ) = 0 , with the boundary condition F ( T,x ) = g ( x ) . Black-Scholes formula:
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