The collapse of Lehman Brothers at the end of 2008 brought great turbulence to the equity and credit markets around the world. Besides the major equity markets such as S&P500, the Dollar-Yen FX market also got hit badly. But it wasn’t just the Lehman bankruptcy that hit the USD/JPY market. The trouble that was brewing in this market for a long time before Lehman kicked the bucket came to a culmination in October 2008.

It is interesting to note that this market turbulence and the extreme spikes in volatility witnessed in the USD/JPY market as well as in many major equity markets, such as S&P500, caused a major rethink amongst traders and quants as well as theoreticians of the stochastic volatility models, one of the dominant mathematical models of the decade to value financial derivatives. This would ultimately give rise to what is known as the stochastic local volatility models, or post-Heston models.

A wonderful narrative of the events of this period is provided in David DeRosa’s *Options on Foreign Exchange.*

As far as implied volatility of FX options is concerned, the period of 2007-2008 is a remarkable one. Seldom have we seen short term FX volatilities, especially the 1 month Dollar-Yen volatilities, rise so much in any other period of history. In October 2008, the Dollar-Yen volatilities exploded. The reason was not just the collapse of Lehman Brothers a month earlier; of course, that indirectly triggered events that impacted the USD/JPY market. The bigger culprit was the so called "carry trade". Ever since the U.S. Federal Reserve had started hiking rates in 2004, the Dollar-Yen carry traders had been basking in sunshine. Given the near zero interest rates for Japanese Yen, this action by the Federal Reserve, resulting in the U.S. short term rates to rise to 5.26% by June 2006, caused the interest rate differential between the USD and the JPY to widen a lot. Traders and investors would simply sell Yen – or, borrow Yen – and pay near zero interest rates and buy the Dollar or dollar denominated securities. But the party lasted only for fifteen months and with the fall of Bear Stearns and the first hint of a market panic in September 2007, the Federal Reserve reverse course abruptly. Rarely, does the Central Bank of a G7 country do such an about turn in such a short period. But as the Fed may have realized in the autumn of 2007, the times were hardly going to be normal in the coming months or years.

With the Fed abruptly lowering the short term interest rates in the U.S., the carry traders got cornered. In fact, as the Fed continued to lower short term Dollar interest rates in very quick steps, due to the ensuing financial crisis, a full-fledged panic gripped the Dollar Yen carry trade market. With shrinking interest rate differential between the USD and JPY, the carry trade wasn’t looking all that profitable.

Moreover, a far bigger worry was that the Dollar will depreciate against the Yen due to falling short term interest rates in the U.S. Investors, fearful of a Dollar crash against the Yen, started to stack up on USD Put options. More and more investors started going long (buying) on the USD put option (JPY call option). Of course, all action was centered on the short dated options, mostly one month to three months. In October 2008, the one month (implied) volatility of USD/JPY briefly touched 35%. This was a momentous event and reflective of how much panic had gripped the Dollar-Yen market.

Another indicator of volatility, known as "risk reversal" in the FX options market – equivalently called a "collar" in the equity market – also spiked up. A risk reversal, generally speaking, is a sale of a call financed by a purchase of a put. In the FX options market traders normally, quote and trade low delta risk reversals (RR). The most common product is the 25 delta RR. A 25 delta RR quote for Dollar-Yen will show the difference between the implied volatility of a 25 delta USD call option (JPY put option) and a 25 delta USD put option (JPY call option). If for example, the 25 delta RR for Dollar-Yen is 2%, it means that more investors (traders) are buying out of the money USD call options than selling these options which in turn means that Dollar is well bid. On the other hand, if a 25 delta RR quote for Dollar-Yen is, say, -3% (negative 3%) then it means more investors are buying out of the money USD put options than selling them. If a trader fears that the Dollar will crash and hence starts to buy Dollar put options – the bigger the fear of the crash, the more out of the money will be the strike, and hence lower delta – then the implied volatility of the USD put options will go up as compared to USD call options. Hence the 25 delta RR will become negative.

In December 2008, the 25 delta RR for Dollar-Yen touched -10% (negative ten percent). This was indicative of the fact that the Dollar-Yen carry trade was finally coming to an end.

But Dollar-Yen was not the only market that was getting battered. Major equity markets around the world got hammered in the aftermath of the Lehman Brothers’ collapse. The VIX index, developed and quoted by CBOE, which measures the volatility of the S&P500 index using implied volatility of index options, hit a record high of 89.53 on October 24, 2008. Prior to this the high recorded for VIX was 38.00 on August 8, 2002. A high value of VIX indicates greater fear amongst the investors that the markets will crash. In general, the VIX and the underlying equity index – the S&P500 – are negatively correlated, so a high VIX means a low S&P500 and vice versa. The Lehman collapse had brought the bears out in droves and the equity markets around the world were in free fall.

How did all this affect the volatility models that were being used by the major banks?

Ever since 2001, stochastic volatility models had come to dominate the quant modeling space within the banks. For both equity and FX exotic options and products, the quants and traders were using some kind of stochastic volatility model to price exotic options, especially products that had a forward skew such as cliquets, reverse cliquets, Napoleons, etc. Even otherwise, for almost all exotic products in the FX and equity domains, stochastic volatility models had become some kind of a benchmark for pricing. Within the equity derivatives space, even though traders continued to use local volatility models, Heston’s stochastic volatility model had come to be regarded as a serious contender for pricing exotic options. Many large banks were using Heston’s stochastic volatility model to price all kinds of exotic products. Within the FX domain – and even in the interest rate space – stochastic volatility models, such as the SABR (stochastic alpha beta rho), etc. had become an important tool in a quant’s repertoire of valuation models. Even though stochastic volatility models had a lot of flaws, such as the problem with the estimation of greeks (option sensitivities), giving rise to incomplete markets, calibration, etc. and despite many traders’ dislike for these models due to their cumbersomeness these models had somehow become quite popular amongst traders and quants. One reason could be that under most circumstances stochastic volatility models, at least Heston’s model, consistently overprices certain kinds of exotic options that are popular with investors.

All this came crashing to the ground with the financial crisis of 2008.

A curious, and baffling, fact that was noted by many FX option traders during this period was that somehow volatility was moving independent of the underlying exchange rate. Two-factor stochastic volatility models were incapable of explaining such behavior. Even though volatility is stochastic in a Hestonlike model, the movement in the variance of the asset (and thereby the volatility) is intricately linked with the movement in the underlying asset. One is influenced by the other and almost always the correlation is non-zero. As noted above, in most circumstances the movement of one variable (say, the asset) causes the other variable (the variance or the volatility) to move in the opposite direction.

Even before the financial crisis many veteran traders had strong doubts that both the stochastic volatility models and the local volatility models – the other dominant volatility measurement technique used by exotic equity option traders – would not be able to stand the test of time should there be a full blown market panic. And that is exactly what happened in the aftermath of Lehman Brothers’ collapse. With the equity and the FX markets is full blown turbulence and crash mode, Heston like pure stochastic volatility models flew out of the window. And local volatility models used in the equity derivatives markets also appeared grossly inadequate to capture the true dynamics of option volatilities.

What dawned on a few astute finance theoreticians is that volatility needs to be both "local" and "stochastic" to address the new market dynamics. In fact, work was already underway to create hybrid – hybrid signifying, part "local volatility" and part "stochastic volatility" – volatility models even as the market turbulence reached its peak in end 2008. The aim was to add new stochastic properties to local volatility models and to modify the pure stochastic volatility model in such a way that would allow volatility of an underlying asset to move independently without any associated movement in the underlying exchange rate or the equity index.

These models have come to be known as the "stochastic local volatility" (SLV) models and an example of one such sophisticated model is one developed by Ren, Madan and Qian in 2007. SLV models are mathematically quite complex and there is a marriage of the essential features of local and stochastic volatility processes. Another popular SLV model, one that is significantly complex in terms of the underlying math, has recently been developed by Tartaru and Fisher at Bloomberg. Tartaru and Fisher’s model developed around 2010 is now being carried on Bloomberg’s system and many traders and quants are getting to use them. Both these models are outlined in *The Book of Greeks* by Rahul Bhattacharya and published by CFE School

In its most basic form, an SLV model can be understood as a mixture of local volatility and stochastic volatility models whereby these are implemented separately. However, once the calibration is done, these two processes are combined using some sort of a "mixing rule". But this view is highly simplistic and to some extent naïve. In practice, the SLV models far more complex and involves rigorous mathematics and calibration algorithms before they can be used to price options. However, they seem to capture the market dynamics of the post-financial crisis well.

**Reference:** *Options on Foreign Exchange, *3rd Edition, David F. DeRosa, John Wiley & Sons (2011).

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