The stochastic-alpha-beta-rho (SABR) model introduced by Hagan et al. () is Keywords: SABR model; Approximate solution; Arbitrage-free option pricing . We obtain arbitrage‐free option prices by numerically solving this PDE. The implied volatilities obtained from the numerical solutions closely. In January a new approach to the SABR model was published in Wilmott magazine, by Hagan et al., the original authors of the well-known.
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The first paper provides background about the method in general, where the second one is a nice short overview more applied to the specific situation I’m interested in. This is straight forward and can be tuned to get dsirable results.
As outlined for low strikes and logner maturities the implied density function can go negative. In the case of swaption we see low rates and have long maturities, so I would like to remove this butterfly arbitrage using the technique described in the papers above. The remaining steps are based on the second paper.
How we choose this strikes is not important for my question. How should I integrate this? Do I have to approximate it numerically, or should I use the partial derivative of the call prices? Here they suggest to recalibrate to market data using: Since they dont mention the specific formula it must be a rather trivial question, but I dont see the solution.
Then you step back and think the SABR distribution needs improvement because it is not arbitrage free. Instead you use the collocation method to replace it with its projection onto a series of normal distributions. This arbitrage-free distribution gives analytic option prices paper 2, section 3.
Q “How should I integrate” the above density? Numerically if you don’t find arbitrsge-free analytic formula. How is volatility at the strikes in the arbitrage-free distribution “depending on” its parameters?
From what is written out in sections 3. It is subsumed that these prices then via Black gives implied volatilities.
So the volatilites are a function of SARB-parameters and should exactly match the implieds from which we took the SARB if it not where for adjusting the distribution to arbitrage-freee arbitrage-free one. The solution to minimizing 3.
That way you will end up with the arbitrage-free distribution of those within this scope at least that most closely mathces the market prices. No need for simulation.
SABR volatility model – Wikipedia
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SABR volatility model