Please help for Q13.2 in excercises of 5th ASM manual.

[actuarialpost]

13.2 [3-F02:8] The value of currency in country M is currently the same as in country N. Let C(t) denote the difference between teh currency values in country M and N at any point in time (at time t, 1 unit of M will exchange for 1+C(t) unit of N at time t). C(t) is modeled as a Brownian motion process with drift 0 and variance parameter 0.01.

An investor in country M currently invests 1 in a risk free investment in country N that matures at 1.5 units in the currency of country N in 5 years. After the first year, 1 unit in country M is worth 1.05 in country N.

calculate the conditional probability after the first year that when the investment matures and the funds are exchanged back to country M, the investor will receive at leat 1.5 unit in the currency of country M.

My understanding:
Let N(t)=1+C(t) be the currency rate exchanged of M for N at time t. N(t) is a scaled Brownian motion as C(t) is a Brownian Motion.

N(1)=1.05, E[N(5)|N(1)=1.05]=N(1)=1.05. As 5=1+4, then VAR[N(5)|N(1)=1.05]=VAR[C[5]|C[1]]=4*VAR[C(T)]=0.04
Q: Invest 1 unit M at t=0, Probability N(5) unit of currency in N can be exchanged for >=1.5 unit M given N(1)=1.05.
I think the first step is to transform M(5)>=1.5 to be N(5) condition.

In the solution, "The currency rate must be less than 1 to get 1.05 units of currency M for 1.5 units of currency N".

Why?

Could anybod help me explain this?

Thanks a lot.

[njtp08]

I have same questions for this problem. Can anyone help please?

[Abraham Weishaus]

I find this post very confusing. What's N(5)? Is this expressed in M units? And if so, what's M(5)? M expressed in M units is by definition always equal to 1.

In my solution, everything is expressed in M units. That should answer the "Why?"

To avoid confusion, stick to C, and remember that C is expressed in M units.