Saturday, December 10, 2011

N(d2) interpreted as a (Risk Neutral) probability

Under RN measure, underlier price process is a GBM
  dS = r S     dt +     σ S    dW
denoting L := log S
  dL = ( r - ½ σ2 )  dt +     σ    dW         ...... BM, not GBM. No “L” on the RHS !
Therefore, underlier terminal price has lognormal distribution
  log ST ~ N(logS0 + T(r - ½ σ2)    ,    Tσ2 )  or N (mean, std^2)
Now, Pr(ST > K) simply translates to Pr(log ST > log K)  .... normal distro math! It’s now straightforward to standardize the condition to
  Pr ( (log ST – mean)/std  > (log K – mean)/std )
=Pr( z > [log K - logS0 - T(r - ½ σ2)] / σ sqrt(T) ) ..... which by definition is
=1 - N ( [log K - logS0 - T(r - ½ σ2)] / σ sqrt(T) )

Now be careful. 1-N(a) = N(-a), so that Pr() becomes
N([    -  [log K - logS0 - T(r - ½ σ2)] / σ sqrt(T) ) = N(d2)
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For d1, recognize that under Share-measure,
dS = (r+σ2) S     dt +     σ S    dW
dL = ( r +σ2 - ½ σ2 )dt +σdW  ...simplifying to
dL = ( r        + ½ σ2)dt +σdW  ...differs from earlier formula (RN measure) only in the “+” sign

log ST ~ N(logS0 + T(r   +   ½ σ2)    ,    Tσ2 ) ...... notice the “+” sign in front of ½ σ2

Therefore under share-measure, Pr (ST > K) = N(d1)

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