Can somebody explain me if the Rebonato swaption volatility approximation formula is accurate for only ATM strikes, and if yes why? Can it also be used for ITM and OTM strikes?
My foundings:
Let $0 < T_0 < T_1 < \ldots < T_N$ be a tenor structure. Consider a payer swaption that gives the right to enter into a payer interest rate swap at $T_0$ with payments of both the floating and the fixed leg on $T_1,\ldots,T_N$. The fixed rate is set to $K$.
I have implemented the Rebonato swaption volatility approximation formula in Matlab as
$$\upsilon^{REB}= \sqrt{\frac{\sum_{n=0}^{N-1}\sum_{k=0}^{N-1}w_n\left(0\right)w_k\left(0\right)L_n\left(0\right)L_k\left(0\right)\rho_{n,k}\int_0^{T_0}\sigma_n\left(t\right)\sigma_k\left(t\right)dt}{SwapRate\left(0\right)^2}}\\
=\sqrt{\frac{\sum_{n=0}^{N-1}\sum_{k=0}^{N-1}w_n\left(0\right)w_k\left(0\right)L_n\left(0\right)L_k\left(0\right)\rho_{n,k}\int_0^{T_0}\sigma_n\left(t\right)\sigma_k\left(t\right)dt}{\sum_{n=0}^{N-1}w_n\left(0\right)L_n\left(0\right)}},$$
where $L_n\left(0\right):=L\left(0;T_n,T_{n+1}\right)$ represent the initial Libor curve and $w_n\left(0\right)$ are the weights defined as
$$w_n\left(t\right) = \frac{\tau_n P\left(t,T_{n+1}\right)}{\sum_{r=0}^{N-1} \tau_r P\left(t,T_{r+1}\right)},$$
with $\tau_n =T_{n+1}-T_n$.
The instantaneous volatilities $\sigma_n\left(t\right)$ are given by the following parametrization; $$\sigma_n\left(t\right) = \phi_n\left(a+b\left(T_n-t\right)\right)e^{-c\left(T_n-t\right)}+d.$$
To get the swaption price at time $0$, I have used this swaption approximation as an input in Black's forumla; $$V_{swaption}\left(0\right) = Black\left(K,SwapRate\left(0\right),\upsilon^{REB}\right)\\ =Black\left(K,\sum_{n=0}^{N-1}w_n\left(0\right)L_n\left(0\right),\upsilon^{REB}\right)$$
In order to access the accuracy of the Rebonato approximation formula I have compared the prices of various swaptions obtained by plugging the approximation volatility in Black (as above) and the prices obtained by a Monte Carlo evaluation doing 1000000 simulations.
I was particularly interested in the accuracy among different strikes $K$. To illustrate this, consider the 4Y10Y swaption and its corresponding ATM , ATM+1%, ATM+2% and ATM+3% strikes (ATM strike is $K=SwapRate\left(0\right)$).
My foundings were that as you move further away from the ATM strike, the approximation gets worse (difference between Monte Carlo price and price with Rebonato swaption approx volatility increases). In concrete numbers, the difference for ATM strike is 9 bp and for ATM+3% 36 bp.
I have searched in the literature for an explanation, but cannot find any. As far as I have understood, no assumptions evolving the strike are made in deriving the Rebonato formula.
Brigo and Mercurio also perform an accuracy test of the Rebonato formula in their book 'Interest Rate Models - Theory and Practice', namely:
"The results are based on a comparision of Rebonato's formula with the volatilities that plugged into Black's formula lead to the Monte Carlo prices of the corresponding at-the-money swaptions. "
Furthermore, Jäckel and Rebonato analyze in their paper 'Linking Caplet and Swaption Volatilities in a BGM/J Framework: Approximate Solutions' how well the approximation performs by comparing the ATM swaption prices obtained by the Rebonato volatility and the Monte Carlo ATM prices.
Is it coincidence that I only can find results for ATM swaptions or does Rebonato's swaption volatility approximation formula really not perform well for ITM and OTM swaptions?
Any help is appreciated. Thanks in advance.