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I'm reading the following two papers (first, second) which suggest a so called "stochastic collocation method" to obtain an arbitrage free volatility surface very close to an initial smile stemming from a sabr. 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.

So we observe $\sigma(K_i)$ implied normal volatilities (bpsvol) in the market for certain reltaive strikes $ATM + x$bps. $x$ can be something like $25, -50$ etc.

1. Step: Calibrate an initial Sabr Here I fit a Sabr using the Hagans paper assuming a nomral model, i.e. $\beta = 0$. Based on the original paper of Hagan $$ \sigma_N(K) = \alpha\frac{\zeta}{\hat{x}{(\zeta)}}\left(1+\frac{2-3\rho^2}{24}\nu^2\tau_{exp}\right)$$ where $\zeta = \frac{\nu}{\alpha}(f-K)$ and $\hat{x}{(\zeta)}=\log{\left(\frac{\sqrt{1-2\rho\zeta+\zeta^2}-\rho+\zeta}{1-\rho}\right)}$

This is straight forward and can be tuned to get dsirable results. In what follows we assume such a fit was successful and we were able to find a solution $(\alpha,\nu,\rho)$. 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. The idea in short is to use a cheap distribution $F_X$, easy to sample from, to sample a expansive one, $F_Y$. The goal is to find a function $g$ such that $$ F_X(x) = F_Y(g(x))$$ and $Y\overset{d}=g(X)$. They susggest to use the following approximation $$y_n \approx g_N(x_n) = \sum_{i=1}^Ny_i l(x_n)$$ a polynomial expansion. Note, $x_i$ and $y_i$ will be strike. How we choose this strikes is not important for my question.

2. Step: Survival distribution: As in the paper we want to choose $X$ to be normally distributed and $Y$ should be the survival distribution function (SDF) from the sabr model, which is given by equation 2.3 in the second paper $$G_Y(y)=1-\int_{-\infty}^y f_Y(x)dx = \int^{\infty}_y f_Y(x)dx=-\frac{\partial V_{call}(t,K)}{\partial K}\rvert_{K=y}\tag{1} $$ which then leads to $$y_n \approx g_N(x) = \sum_{i=1}^NG^{-1}_Y(G_X(\bar{x_i}))l(x)$$

Question: In the paper they say one should really integrate from $y$ to $\infty$ in $(1)$ and not use the representation $G_Y(y) = 1-F_Y(y)$. How should I integrate this? How do I find the density $f_Y$? Do I have to approximate it numerically, or should I use the partial derivative of the call prices?

Assuming I was able to get this approxmiation $g_N(x)$ the third step consists then of a recalibration of the sabr model:

Recalibration of Sabr: (section 3.5 in the second paper). Here they suggest to recalibrate to market data using: $$ \min\sum_i(\sigma(K_i)-\sigma_{g_N}(K_i))^2$$

Question But how does the form $\sigma_{g_N}$ looks like depending on $g_N$? Since they dont mention the specific formula it must be a rather trivial question, but I dont see the solution.

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If I got this right, you get a SABR model from fitting market implied volatilites (from market price via Black) to SABR volatilites (from SABR parameters via formula above). 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 3.4) not perfectly fitting the given market prices (which the SABR did).

Questions: "How do I find the [SABR probability] density?" Answer: That is given by Hagen (and your SABR-parameters). Look in Hagen's "Probability Distribution in the SABR Model of Stochastic volatility"!

Q:"Should I use the partial derivative of the call prices" for integration of the above? A: No.

Q "How should I integrate" the above density? A: Numerically if you don't find an analytic formula.

Question: How is volatility at the strikes in the arbitrage-free distribution "depending on" its parameters?

Answer: From what is written out in sections 3.4 and 3.5 the estimated SABR parameters and the collocation points gives the arbitrage-free distribution which then gives analytic call prices. 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 an arbitrage-free one.

Question: "I dont see a solution" A: The solution to minimizing 3.10 goes through finding a set of SABR-parameters close to your initial ones via a "local-search algorithm, like the Nelder-Mead". 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.

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If you want to use the normal SABR ($\beta=0$), my paper, Hyperbolic normal stochastic volatility model (arXiv | SSRN | DOI) might give you a solution. It reports an exact closed-form MC simulation scheme for the normal SABR model. Better than that, it shows that Johnson's $S_U$ distribution ($\sinh$ transformation of normal variate) is a close cousin of the normal SABR; with a slightly different pair of $\alpha$ and $\rho$ from those for SABR, you can get a very close $S_U$ distribution). So, you can use all the analytic properties of the $S_U$ distribution, including option price and cumulative distribution function. It's arbitrage free. No need for simulation.

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The answer given so far, by Mats Lind, to the first question is not in the spirit of the paper. I am referring to

Question: In the paper they say one should really integrate from $y$ to $\infty$...

What they mean is that you should not use any information on what happens from $-\infty$ to $y$, which would correspond to the standard cumulative distribution function, as the analytical SABR formula may lead to a negative implied probability density for strikes close to zero, and thus to a wrong cumulative distribution.

Instead, you should directly consider the survival distribution function, which does not suffer from this defect - in fact, if you read the papers carefully, you will notice that the left boundary strike $y_0$ is chosen so that the implied probability density is not negative for $y > y_0$.

Then, in equation (2.3), the paper use the partial derivative of the call price to compute this survival distribution directly, instead of performing some numerical integration of the probability density, which by the way, would not necessarily match the density implied by the Hagan SABR formula. The Hagan probability density expansion is not exactly the same approximation as the Hagan SABR Black volatility approximation.

"Remark (Proper calculation of the survival distribution). To determine the survival probability it is crucial not to use the representation $G_Y (y) = 1 − F_Y (y)$ but to integrate from $y$ to infinity, as presented at the right-hand side of (2.3)."

Another mistake from Mats Lind, is that the Lagrange polynomial will not exactly reproduce the original SABR Call option prices, and thus implied volatilities computed from the stochastic collocation will not match the original SABR implied volatilities. This is why the paper proposes a second step, the calibration.

Regarding the second question,

Question: But how does the form $\sigma_{g_N}$ looks like depending on $g_N$?

The paper gives a simple formula for the call option price $C(K_i)$ as a function of $g_N$. You then just use an implied volatility solver (Newton-Raphson, or better) to compute the Black implied volatility for this option price. This gives you $\sigma_{g_N}(K_i)$. The calibration consists then in a numerical least squares minimization to obtain the optimal polynomial $g_N$ that fits the original SABR (or the market) implied volatilities.

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