Takada
Subtype of Build ApproxThe Takada optimization (actually approximation) method is used as described at here
Applies only when the effective rate r over an accrual period (T₁ , T₂) is defined as the arithmetic average of the overnight rates rᵢ observed over that period.
r = (1/Δ)Σ(rᵢδᵢ)
where
the index i enumerates the business days (value days) in the period (T₁ , T₂) when the respective overnight rates rᵢ are observed.
the date T₂ is the end of the accruing period of the last observed overnight rate => the last i refers to the business day before T₂
Σ denotes the sum over the index i
Δ is the daycount fraction of the interval (T₁ , T₂)
δᵢ is the daycount fraction of the iᵗʰ interval between two consecutive business days.
The Takada optimization is the following replacement:
Σ(rᵢδᵢ) -> ln[DF(T₁)/DF(T₂)]
which holds in an approximate sense because δᵢ are very small.
The proof is very simple:
Let DFᵢ the discount factor for maturity equal to the iᵗʰ business day in the interval (T₁ , T₂)
Assume also that i starts with 1 and there exist n business days on which the overnight rate is observed.
Since the product rᵢδᵢ is very small, it follows that (applying exp and ln in sequence and using Π for product):
Σ(rᵢδᵢ)
= ln[ exp[ Σ(rᵢδᵢ) ] ]
= ln[ Π[ exp(rᵢδᵢ) ] ]
= ln[ Π(1 + rᵢδᵢ) ] (due to rᵢδᵢ being very small)
= ln[ Π(DFᵢ/DFᵢ₊₁) ]
= ln(DF₁/DFₙ₊₁)(due to cross cancellations when the product is expanded)
= ln[DF(T₁)/DF(T₂)]
The last equality holds because the first business day is T₁ and the (n+1)ᵗʰ business day is T₂
This optimization requires the generation of only the two dates T₁ and T₂
For technical reasons, a small set of dates is generated and can be seen as part of the instrument's cash flows in the wizard's Browse Area that is still far smaller than the set of all business dates in the period (T₁ , T₂)