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On the second question, you have the choice to pay (S_t - K) at t or (S_T - K) at T. The value at t of deciding to pay now versus later is: Value at t of paying (S_t - K) at t - Value at t of not paying (S_T - K) at T. = -(S_t - K) + exp(-r(T-t)) (F(t,T) - K) where F(t,T) is the forward price of the index. Now F(t,T) = S_t exp(r-d)(T-t) where d is ...


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 ...


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 $\sigma=0.$


I think (1) is the issue. You need to compare market normal vols to normal vols implied by the sabr model. (2) is not the issue - these vols look reasonable. By the way we express normal vols in bp per annum, not percent!

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