Lecture 3 Supplement Annoying Proof of a Useful Result

Proof1 that E(ea+bx̃) = ea+b+½Var(x̃)
I. The Normal Distribution The probability density function of a normal distribution is
1 2 v
1 y     e 2 v 
2

which has the property, for any y and v, that
 



1 2 v

1 y     e 2 v 

2

dy  1

II.

Fun with Completing the Square! Now suppose a  b~ is distributed N a  b , b 2 2 . x





Let’s find the because a  bx  a  b  bx   

expected value of e
E e a bx 

a b~ x


. a  b~  a  b 2 x 1 2 b



~



1 2 b



e

a  b~ x

e



 dx
2

 1 a  b  e  2 b   e a  b







~ e bx   e

 bx     1  2  b 

 dx    
2

  1  2 b  







1  bx      b ~     2  x e 2b  e  b 

2b 2 2
2 2

 dx    

 1  e a b   2 b 

   1  b 2  x   2  2b  x   b 2 2   2 2 e  2b   







 dx   

Larger view

Ee



a  b~ x



 1  e a  b   2 b 

   1  b 2  x   2  2b  x   b 2 2    2b 2 2  e 







 dx   

1

Why? We use this result a lot in finance. Let’s show that it is true.

1

MSF 501 – Math with Financial Applications

Lecture 3 Supplement Annoying Proof of a Useful Result

We know that  p  q 2  p 2  2 pq  q 2 . We can see that the expression in the { } could be expressed as a squared term by defining p  bx    , and q  b 2 2 , and also adding and subtracting q 2  b 4 4 within the curly brackets.

Ee



a  b~ x



 1 a  b  e  2 b   1 a  b  e  2 b 
  b 4 4 2b 
2 2

   1  b 2  x   2  2b  x   b 2 2   2 2 e 2 b   

 





 dx   

Complete the square

   1  b 2  x   2  2b 