I am solving a problem in which following payoff is provided:
With $S_0=100$ and $T=8$. Looking at the payoff it seems obvious that it is replicated with two european put options ($K=100$ and $K=150$) and an european call option ($K=200$), therefore:
$V_t=[100-S_t]^{+}+[150-S_t]^{+}+[S_t-200]^{+}$
And hence the initial value needed to replicate the derivative is $V_0=50$. However, in the solution of the problem it says that the initial value is 41.4. Where am I wrong?
The 50 value you compute is the payoff of your structure if the underlying is worth 100 at maturity. The initial value is the expected payoff of your trade, then, given the decomposition in call and puts, and given the replication principle, it should be: $V_t = Put_0(100,8) + Put_0(150,8) + Call_0(200,8)$, with $Put_0(k,T)$ and $Call_0(k,T)$ the price at time $0$ of calls and puts of strikes $k$ and maturity $T$.
Either prices of calls and puts are also provided in your exercise, otherwise, you need additional information like volatilities and, as mentioned by rubikiscube09, interest rates, to discount payoffs.
