THE SABR CHRONICLES 1. Origins. Fixed income

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tunately, we do care a great deal because of hedging: Suppose we were short, say, a $1 million delta of low strike options, and long $1 million delta in high strike ...
THE SABR CHRONICLES PATRICK S. HAGAN

1. Origins. Fixed income desks orginally used Black’s model,  =   to price derivatives. This allowed them to use dynamic hedging schemes to minimize their risks. But to match the market price required a different implied volatility  for each different strike . If we asked our trader to price an at-the-money option, we might get a price of, say,  = 20%, while for a high strike option on the same asset, the trader might charge 30% , and for a low strike option, the price might be 40%. This is not an oddity of our trader: every other trader on the street would offer the same implied volatilities for the same strikes. This means that we are using different models for different options on the same underlying asset  : for some options we are modeling  () as log normal Brownian motion with a tame volatility of 20%; for other options we are modeling the same  () as log normal Brownian motion with a wilder volatility of 40%. Clearly there is something is wrong with this approach. So the next question is: Do we care? Unfortunately, we do care a great deal because of hedging: Suppose we were short, say, a $1 million delta of low strike options, and long $1 million delta in high strike options. Are we delta neutral, or do we have to buy or sell more of the underlying to be neutral? There is no way to tell because we are using a different model for each strike, so we don’t know whether we can consolidate the risks by simply adding them together. This is a critical issue, because the only way this business works is by first consolidating the risk of all options on the same underlying, and then hedging the residual risk. If we had to hedge each trade seperately, we’d quickly chew up any profit in trading costs. Since a trader’s first line of defense is a flatish book, the residual is usually much smaller than the total notionals. Similarly we could be short a million vega of low strike options and long a million vega of high strike options. Are we vega neutral, or do we have to buy or sell some options to achieve vega neturality? Said another way, if we are bumping at-the-money vols from 20% to 21% to calculate vega risk, should we bump high strike vols from 30% to 31%? Or from 30% to 315%? And should we bump the low strike vols from 40% to 41% or from 40% to 42%? I.e., to consolidate our vega risks, should we calculate vega risks by using a parallel shift, and then add the individual vega risks together, or should we use a proportional shift and then add the risks together? Or maybe some other shift. Since we’re using a different model for each strike, we cannot know. 2. Local vol models. The only way out of this dilemma is to develop a model which self-consistently prices options for all strikes . This problem came to a head in the late eighties and early nineties, where it was tackled by Dupire, Derman, and Kani among others. See, e.g., [1], [2] and references cited within. With Martingale pricing theory, these researchers knew that the forward had to be a Martingale, so  =  (∗)  for some coefficient . They then concluded that Black had been too bold in assuming that  (∗) was a constant times  ; instead they only assumed that  was Markovian, so that it is some explicit function of  and  , which they wrote as  (  )  . This yielded their new local volatility model  =  (  )   Their key insight was that instead of guessing a priori what the ‘local volatilty’  (  ) is, one can just let the market determine it. By selecting  (  ) appropriately, the model can match not only the market’s 1

implied volatily smiles –  (  ) for all different strikes  at a given  , but it can match the entire volatility surface –  (  ) for all different strikes  and expiries  . The calibration step, selecting  (  ) so the model predicts the observed volatility surface, is non-trivial, but manageable with the tools built by Dupire among others. Once the local volatility model is calibrated to the market’s volatility suface, the model works for all strikes without any “adjustments.” So risks can be consolidated by summing the exposure at each strike, and only the net exposure need be hedged. So, case closed and job done. Maybe. As local volatility models became pervasive, fixed income and FX traders began noticing that after calibrating to the market’s implied volatility surface, stepping foward in time often led to peculiar implied volatility surfaces. I.e., the current volatility surface fits the market perfectly, but the forward volatilty surfaces were highly contrived. This is more serious then it sounds, since running fixed income vol and exotic books is largely an exercise in managing one’s forward vol exposure. Part of the reason for this distortion is that when the local volatility model is calibrated, it predicts that the underlying  and the smile  (  ) move in opposite directions: If the forward  decreases, the model predicts that the volatility smile shifts to higher strikes , and if  increases the volatility shifts to lower strikes . In reality, the smile and the underlying almost always move in the same direction. Since hedges are based on the model’s predictions, this difference destabilizes the hedges, leading to hedging “chatter.” The net position has to be re-hedged frequently, which is expensive. 3. The standard SABR model. The SABR model was developed to fix the forward volatity problem. Since one factor Markovian models would lead straight back to the local volatility model, we decided that there must be some other factor creating the volatility smile, that we needed some other stochastic factor. Volatility was a promising candidate: The value of at-the-money options is almost linear in the volatility, so the gamma (second derivative) with respect to the volatility is nearly zero for options with  =  (0). However, the gamma with respect to the volatility (vol gamma) increases strongly as the strike  moves away from  =  (0)  either higher or lower. I.e., vol gamma is roughly a parabola in  −  (0). But with a positive gamma with respect to the volatility, any movements in the volatility increase the value of the option. Just as in the original Black-Scholes argument, the options’s carry must be increased by vol gamma times the square of the volatility of the volatility to avoid arbitrage. Since vol gamma is roughly a parabola, this effect increases option prices for low and high strikes, but only minimally changes at-the-money options. I.e., stochastic volatility should lead to a smile in the option’s implied vol. So the question becomes, is stochastic vol the reason for the observed market smiles, or is the stochastic vol effect too small? Since time was of the essence, not atypical in banking, we tried to create the simplest possible model that had a chance of working, which is probably the right approach to modeling anyway. This led to the SABR model[3]  =  ( ) 1   = 2  where 1 1 =  We left the function  ( ) general because there was no agreement on whether we should use  ( ) =  (a stochastic log normal model),  ( ) = 1 (a stochastic normal model), or maybe  ( ) =   for some intermediate exponent . So  ( ) was arbitrary in the analysis, and specialized to   in the final formulas. The volatility  was log normal since this seemed to be the simplest positive process. Correlation between  and  was included because it is pounded into fixed income traders that as rates go up, vols come down, and as rates go down, vols go up, suggesting a negative correlation. 2

Our analysis of the SABR model was focussed on estimating the size of the stochastic vol effect, trying to determine if it is large enough to be the cause of the observed market smiles. Here we got lucky, as our singular perturbation techniques led to explicit asymptotic formulas for the implied volatility of European options[3]. These implied volatility formulas show that SABR predicts realistic volatility smiles. In fact, these formulas are normally treated as if they are exact1 , which makes the SABR model easy to implement, and makes the SABR model easy to calibrate to actual market smiles. This all helped make the SABR model very popular. The SABR model is very successful at matching volatility smiles –  (  ) as a function of  at a specific date  . However, volatility smiles  (  ) at different expiries  generally require different values of the SABR parameters  =  (0), , and . I.e., the standard SABR model cannot self consistently model volatility surfaces. 4. The effective forward equation and arb free SABR. The explicit implied vol formulas can exhibit arbitrage at extremely low strikes, even though the SABR model is exactly arbitrage free. This motivated us to re-analyze the SABR model[4] using the ideas of Balland and Tan[5]. The probability density  (  ) of the SABR model at time  obeys a forward Kolmogorov equation, which is a partial differential equation with two spatial variables,  and . By analyzing the forward equation we found that the marginal density, Z ∞  (  ) ≡ (  ) 0

satisfies its own “effective forward equation”  = where  =  (0) {1 + · · · } and

¡£ 2 ¢ ¤  +  +  2  2  2 ( )    

 ( ) =

Z

 

 0   ( 0 )

The reduction from two to one spatial variables isn’t exact, but has the same asymptotic accuracy as the implied vol formulas. Options have positive gamma (second derivative) with respect to both the underlying  and the volatility , so on average options gain value each day due to movements in both quantities. To be priced fairly, an option’s daily carry must equal these expected ¡ gains. This ¢ yields a striking interpretation of the effective forward equation: there  is the daily carry, 2  2 ( )    is the expected daily gain due to movements in ¡ ¢  , and  2  2  2 ( )    is an asymptotically-accurate estimate of the expected daily gain due to movements in . The middle term  arises because correlation entangles the two gammas together. Analyzing the effective forward equation provides another avenue into the SABR model. Rather than analyze the SABR model directly, one can analyze the effective forward equation. This is simpler since the effective forward equation has only one spatial variable and has time-independent coefficients. By analyzing the effective forward equation, a new explicit formula for the implied volatilities of European options is derived in [6]. Under moderate conditions, the new formula is essentially identical to the original SABR formula, but under extreme conditions, the new formula appears to be significantly better. 1 Apparently, rather than treat these formulas as accurate approximations of the implied volatilty of the SABR model, they are viewed as the exact implied volatilities for some other arbitrage free model which is accurately approximated by the SABR model.

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5. Dynamic SABR and volatility surfaces. The effective forward equation can be viewed another way: The factor 2 +  +  2  2 looks a lot like a Taylor expansion in , the volatility of volatility. If so, then maybe many other (all other?) stochastic volatility models obey the same effective forward equation. If we could show this, then the analysis of the effective forward equation in [6] would immediately yield the same closed form implied volatility formulas for these other models as for the SABR model itself. I.e., these other models would have to exhibit the same smiles as the SABR model. To check this, we analyzed the dynamic (time-dependent) SABR model in this issue[7], and immediately ran into an obstacle. There we found that the effective forward equation is of the form ¡£ ¢ ¤  =  ( ) +  ( )  +  ( )  2  2 ( )    

and we derived  ( ),  ( ),  ( ) in terms of the model parameters. The time dependence of the coefficients is a serious problem: It is not clear a priori whether the effective forward equations with time-dependent coefficients have implied volatility smiles similar to effective forward equations with time-independent coefficients. The variation of the coefficients with  may leave remnants and artifacts in the smiles. To settle this issue, an effective media analysis is developed in this issue[8]. Effective media theory is a mathematical approach in which one seeks to replace fine-scale structures by equivalent macroscopic parameters. In our case, for any specifc  , we derived constant coefficients  ¯, ¯, ¯ such that the solution  (   ) of the effective forward equation with these constant coefficients is the same as the solution with  ( ),  ( ),  ( ). Since the probability densities are the same at  , so are the European option values and implied vols for this date  . Thus, the implied volatility smiles for the dynamic SABR model are the same as the smiles of the original SABR model. In [7] this approach is used to obtain the effective constant SABR parameters  ¯,  ¯, ¯ in terms of the original, time-dependent parameters for each  . Since these parameters are averages – albeit unusual ones – of the time-dependent parameters, they are generally different for different expiries  . There we also show how to calibrate the dynamic SABR model to the entire volatility surface by first calibrating to a set of market smiles at different expiries to obtain the standard constant SABR parameters  ,  ,   for each expiry  , and then using a bootstrap method to obtain piecewise constant dynamic SABR parameters which yield this set of SABR parameters. This means that calibrating to a volatility surface no longer requires using an unphysical local volatility model nor laborious numerical computations. 6. And beyond .... We now have a process for finding explicit implied volatility smiles for stochastic volatilty models. The first, and most difficult, step is to analyze the stochastic model to obtain the effective forward equation ¡£ ¢ ¤  =  ( ) +  ( )  +  ( )  2  2 ( )    

and to obtain the coefficients  ( ),  ( ),  ( ) in terms of the original model parameters. The effective media results can then be used to replace these with constant coefficients  ¯, ¯, ¯. for each  . Finally, the implied volatility formulas are derived from the analysis of the effective equation with constant coefficients. This shows that the implied volatility formulas are identical to the standard SABR model’s formulas, and determines the effective constant SABR parameters , ,  in terms of  ¯, ¯, ¯, and thus in terms of the original model parameters. This process has been applied to the mean reverting SABR model[9], the general Heston model[10], and many others[11]-[14]. For each model the process shows that the model’s implied volatility smiles are the same as the smiles of the standard, constant coefficient SABR model, and determines the effective SABR parameters , ,  for each  in terms of the original model parameters. This also explains why the SABR implied volatility formulas match the market smiles in so many markets: It’s a quirk of fate, because if we had chosen to first analyze the Heston model, then we would have obtained the same set of smiles, but they’d be known as the Heston implied volatility formulas. 4

REFERENCES [1] Dupire, B. (1994) Pricing with a smile, Risk 7 1994: 18-20. [2] Derman, E. and Kani, I. (1994) Riding a smile, Risk 7 1994: 139-145. [3] Hagan, P. S., Kumar, D., Lesniewski, A.S., and Woodward, D.E. (2002). Managing smile risk. Wilmott Magazine, 2002: 84-108. [4] Hagan, P. S., Kumar, D., Lesniewski, A.S., and Woodward, D.E. (2013). Arbitrage Free SABR. Wilmott Magazine, January, 2013: 1-16. [5] Balland, P. and Q. Tran, Q. (2013). SABR goes normal. Risk Magazine, 2013: 76-81. [6] Hagan, P. S., Kumar, D., Lesniewski, A.S., and Woodward, D.E. (2016). Universal smiles. Wilmott Magazine, July 2016: 40-55. [7] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E. (2017). Managing vol surfaces. Wilmott Magazine, this issue. [8] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E. (2017). Effective media analysis for stochastic volatility models. Wilmott Magazine, this issue. [9] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E. (2017). Effective forward equations for the mean reverting SABR model, submitted [10] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E. (2017). Effective forward equations for the generalized Heston model, submitted [11] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E. (2017). Effective forward equations for exponential stochastic volatility model , working paper [12] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E. (2017). Self consistent cross FX smile models, working paper [13] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E. (2017). Self consistent smile models for baskets and spreads, working paper [14] Hagan, P. S., Lesniewski, A.S., and Woodward, D.E. (2017). Implied volatility smiles for ZABR-type models, working paper

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