ST6: April 2012 Question 7 part ii

[WhiteNessDukeNess]

Please can anyone help me to understand the examiners solution to the 'ST6: April 2012 Question 7 part ii' for finding X?

The examiners report states that:

At time 0
The value of the youngest’s share = value of put + 10,000* Exp(...)* Prob(S5 < X)

Please can anyone explain why the “value of put” is used and not “value of put* Prob(S5<3000)”?

Thanks

 

[Mike Lewry]

Please can anyone help me to understand the examiners solution to the 'ST6: April 2012 Question 7 part ii' for finding X?

The examiners report states that:

At time 0
The value of the youngest’s share = value of put + 10,000* Exp(...)* Prob(S5 < X)

Please can anyone explain why the “value of put” is used and not “value of put* Prob(S5<3000)”?

Thanks

The value of the put already takes into account the fact that we must be below 3000, which is a stronger condition than being below X (since X>3000).

Given there's a payoff (ie S5<3000) the portfolio will be given to the youngest son (since 3000<X) and so there's no need to multiply the value of the put by a further probability.

It might help to note that the put can never be of any value to the other brothers (there's no payout for X<S5<Y or for S5>Y) and so the full value must belong to the youngest son.