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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.

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it certainly works best at the money. Why? I think it comes from the fact that Black's formula is approximately linear at the money. The approximation $$ \frac{1}{\sqrt{2\pi}} \operatorname{SR} \sigma \sqrt{T} A, $$ with $A$ the annuity is remarkably good.

One way of deducing these formulas is to do an asymptotic/Taylor expansion about $\sigma=0.$

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Thank you for your answer @MarkJoshi. I followed you advice and achieved in deriving the approximation formula. However, I can not fully understand why the fact that Black's formula is linear in $\sigma$ for ATM strikes causes the Rebonato approximation only to be accurate for ATM strikes and not OTM and ITM strikes.

I would be grateful if somebody can point me in the right direction

My foundings:

I achieved in deriving the approximation as follows: Defining the annuity factor as $$A_{0,N}\left(t\right):=\sum_{n=0}^{N-1}\tau_nP\left(t,T_{n+1}\right)$$ and the swap rate as $$S_{0,N}\left(t\right):=\frac{\sum_{n=0}^{N-1}\tau_nP\left(t,T_{n+1}\right)L_n\left(t\right)}{A_{0,N}\left(t\right)}$$ the Black formula can be written as $$Black\left(0\right)=A_{0,N}\left(0\right)\left[S_{0,N}\left(0\right)N\left(d_1\right)-KN\left(d_2\right)\right]$$ where $$d_1=\frac{\log\left(S_{0,N}\left(0\right)/K\right)+\frac{\sigma^2T}{2}}{\sigma\sqrt{T}}$$ and $$d_2=d_1-\sigma\sqrt{T}$$ Considering the ATM strike $K=S_{0,N}\left(0\right)$, I obtain: $$Black\left(0\right)=A_{0,N}\left(0\right)\left[S_{0,N}\left(0\right)\left(N\left(\frac{1}{2}\sigma\sqrt{T}\right)-N\left(-\frac{1}{2}\sigma\sqrt{T}\right)\right)\right]$$ $$=A_{0,N}\left(0\right)\left[S_{0,N}\left(0\right)\left[N\left(0\right)+\frac{N'\left(0\right)}{1!}\left(\frac{1}{2}\sigma\sqrt{T}\right)+\frac{N''\left(0\right)}{2!}\left(\frac{1}{2}\sigma\sqrt{T}\right)^2+O\left(\sigma^3T^{3/2}\right)-\left(N\left(0\right)+\frac{N'\left(0\right)}{1!}\left(-\frac{1}{2}\sigma\sqrt{T}\right)+\frac{N''\left(0\right)}{2!}\left(-\frac{1}{2}\sigma\sqrt{T}\right)^2+O\left(\sigma^3T^{3/2}\right)\right)\right]\right]$$ $$=A_{0,N}\left(0\right)\left[S_{0,N}\left(0\right)\left[N'\left(0\right)\left(\frac{1}{2}\sigma\sqrt{T}\right)-N'\left(0\right)\left(-\frac{1}{2}\sigma\sqrt{T}\right)+O\left(\sigma^3T^{3/2}\right)\right]\right]$$ $$=A_{0,N}\left(0\right)\left[S_{0,N}\left(0\right)\sigma\sqrt{T}N'\left(0\right)+O\left(\sigma^3T^{3/2}\right)\right]$$ $$=A_{0,N}\left(0\right)\left[S_{0,N}\left(0\right)\frac{\sigma\sqrt{T}}{\sqrt{2\pi}}+O\left(\sigma^3T^{3/2}\right)\right]$$ $$\approx \frac{1}{\sqrt{2\pi}}A_{0,N}\left(0\right)S_{0,N}\left(0\right)\sigma\sqrt{T}$$ where the second equality follows by the Taylor expansion of $N$ around $0$ and the approximation holds for small values of $\sigma\sqrt{T}$.

I checked the accuracy of this approximation in Matlab for an ATM 1Y6Y swaption with an underlying paying swap, see figure below. Comparision of Black's Formula and Approximation

As can be seen from the figure, the approximation works well for small values of $\sigma$.

For ITM and OTM strikes, Black's formula is clearly not linear in $\sigma$ but exhibits a convex shape. I examined the behavior of Black's formula for OTM strikes in Matlab and obtained the following results:

OTM Strikes

As the strike increases and the swaption gets more out of the money, Black's formula becomes more convex.

In the same picture I also plotted, for different strikes, the price of the 1Y6Y swaption in terms of the Rebonato approximation volatility and in terms of the implied volatility obtained by a Monte Carlo evaluation. For the ATM strike, the Rebonato volatility and the MC implied volatility are nearly the same. For increasing strikes, the difference between the two types of volatility increases as well, meaning that the Rebonato approximation is less accurate for strikes away from the at-the-money level.

Question 1 How can this increasing difference (in the strike direction) between the MC implied volatility and the Rebonato volatility be linked to the fact that Black's formula is linear for ATM strikes?

Furthermore, I noticed that the Rebonato approximation becomes less accurate (difference between Rebonato approximation and MC implied volatility increases) for longer-dated swaptions even for the ATM strike. I performed the same test as above for a 5Y10Y swaption and obtained the following results:

enter image description here

As can be seen from the figure, there is a significant difference between the MC implied volatility and the Rebonato volatility for the ATM strike. Comparing the MC price and the price obtained by inserting the Rebonato volatility in Blck, the difference between the prices is 12.72 basis points.

Question 2 Is it consistent with the theory that the Rebonato approximation is less accurate for long-dated ATM swaptions? What is an acceptable difference? Is 12.72 basis points acceptable? Does this mean that Rebonato approximation is only accurate for ATM strikes and short/medium dated swaptions?

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