/40 R&D: Model calibration Source: Getty Images Source:

Exotic matters Careful selection and calibration of pricing models for exotic derivatives is essential, writes Łukasz Dobrek of Markit

n the aftermath of the 2008 finan- ments in interest rates, exchange rates, cial crisis, it was widely expected stock prices and commodity prices. A that complex financial instruments, similar approach is taken to support perceived as too risky and esoteric, services to the retail market, such as would no longer play a significant the issuance of structured notes, which role in the global financial markets. are designed to give the exact profile of IWhile many banks and trading houses risk and reward that investors demand. partially or fully unwound their exotic In addition, speculative trading in exotic books and the focus turned to derivatives offers the opportunity of simpler or “vanilla” instruments, exotics significant profits for those who prove to are still very much part of the overall have the correct view on more complex derivatives landscape. market factors such as volatilities and Banks use exotic derivatives as a correlations, as opposed to simple price hedg­ing tool for their corporate clients movements. Moreover, it may take years with highly bespoke exposure to move- to liquidate positions in long-dated

the markit magazine – Autumn 2011 R&D: Model calibration /41 and illiquid exotics, during which time 1Y Cliquet with Monthly Reset on S&P 500 Equity Index trading books still have to be priced and Price as % of Notional risk-managed appropriately. 35% Selecting and calibrating pricing models 30% Derivatives that are actively traded with observable market prices represent only 25% a limited subset of the entire derivative 20% universe, meaning that the remainder have to be priced using some type of 15% Heston Model Calibrated mathematical model. Over time, the Fully to Cliquet Price Data range of derivative pricing models has Heston Model Calibrated 10% Only to Vanilla Option Data grown steadily, both as new types of Dupire Local Model derivative have been introduced and as 5% Black-Scholes Model with weaknesses in models previously used At-the-money have been identified, usually arising 0% 94% Put 97% Put 100% Put 100% Call 103% Call 106% Call from a failure to take market behaviour Resetting Relative to Prevailing Index Level into account adequately. Before the market crash of October 1987, equity Source: Markit options on a given stock, with the same but different strike prices, were traded with the same Black- generates prices in line with where the an approach which is known to provide Scholes implied volatility parameter. product would trade, regardless of less accurate pricing. As well as being After the crash, they began trading with market conditions. In order to achieve important in pricing exotic derivatives, a now-familiar volatility smile, meaning this, some market data giving the price properly-calibrated model parameters different implied volatilities for different levels at which related exotic products facilitate the hedging of traded struc- strike prices. have actually been quoted or traded tures, thereby enabling exposures to Alongside the recognition that more are needed. In most cases, these data market movements to be quantified and sophisticated models may be needed can only be sourced through dealers either hedged out or retained, depending to price exotic derivatives accurately, who are actively making markets in on the desired risk profile. there has been a desire in some these products. quarters to create models which are particularly intricate, in order to take into account every theoretical feature “ It is important to select a model that is of market behaviour that might be exhibited, however rare. This approach sufficiently rich...but not so complex that the would appear to be attractive but, in practice, can be counterproductive, required parameters cannot be calibrated.” introducing unnecessary model param- eters that cannot be accurately esti- Assuming that data of this type can be Pricing of cliquets mated from information available in the obtained, the actual process of calibra- A cliquet option consists of a series of market. So, when choosing a model tion involves repeatedly repricing these individually exercisable, forward-starting to price a particular type of derivative, structures with the chosen model until options in which the strike prices peri- it is important to select a model that a combination of parameters is found odically reset to the prevailing level of is sufficiently rich to capture market which accurately reproduces the market the market. Although cliquets can be features which have a material effect prices for both these products and vanilla defined on any type of underlying asset, on the price of the derivative, but not so instruments. These parameters can then they are most commonly traded on an complex that the required parameters be used within the model to price the equity index such as the S&P 500 or cannot be calibrated. exotic product in question, in the knowl- Euro STOXX 50. So a standard cliquet Once a particular model has been edge that they reflect the current market might comprise a strip of 12 options chosen to price a derivative product, it is outlook and have not been selected on the S&P 500 index, such that at the essential to identify a way of calibrating arbitrarily. The alternative consists of end of each month one of the options its parameters so that it consistently analysing historical market behaviour, expires and the next one starts, with the

Autumn 2011 – the markit magazine /42 R&D: Model calibration

index level at expiry determining both However, Dupire does not model vola- realised volatility is above or below the the payout for the expired option and tility dynamically, i.e. does not consider fixed level determines which party pays the strike price for the following option. how the implied volatility of the under- the other. Cliquets can also be more structured, lying asset will behave as the price of Volatility swaps are an attractive way with caps or floors on the payoffs of the the asset itself changes. This is impor- of gaining direct exposure to the vola- individual options or the entire structure. tant when pricing a forward-starting tility of a chosen asset without having to buy and delta-hedge vanilla options. FX volatility swaps, in which the underlying “ Dupire...does not consider how the implied asset is an exchange rate, have been traded for quite some time, although in volatility of the underlying asset will behave the equity derivatives world, variance swaps have generally been preferred. as the price of the asset itself changes.” Variance swaps are similar to volatility swaps but pay out based on the real- A cliquet can be preferable to a series option since, when the strike set date ised variance, i.e. square of the realised of vanilla options with strike prices that is reached, the option will effectively volatility, and are much easier to price are fixed initially and do not reset. For become a vanilla option with a value and risk-manage since they can at least example, if an investor correctly judges which reflects the prevailing market in theory be replicated using a portfolio that the market will trend upwards or implied volatility relative to the asset of vanilla options (in practice this can be downwards, then cliquets can be used price at that time. hard to execute). However, equity vola- to benefit from this, even if the market tility swaps have recently been growing moves in the opposite direction initially. Pricing of volatility swaps in popularity, particularly in Asian Furthermore, cliquets are compara- A volatility involves exchanging markets where variance swaps may be tively cheap, since a forward-starting the volatility realised by the move- viewed as more risky, owing to the non- option is less valuable than a fixed strike ments in an asset’s price over a defined linear relationship between the pay-off equivalent as it does not benefit from period against a fixed level of volatility. and the volatility of the underlying asset. any market movements before the strike For example, one party pays the other The difficulty in pricing a vola- is set. an amount based on the difference tility swap arises from computing the In order to price a standard cliquet, between the volatility calculated by expected value of the future realised each of the forward-starting options in observing the daily levels of the Nikkei volatility which, unlike with a vari- the series needs to be valued individu- 225 index over a one-year period and a ance swap, depends not only on the ally. It would be tempting to value each fixed contractual volatility; whether the expected value of the future variance of these options as a vanilla option and to apply Black-Scholes with the usual market inputs for this model, in on AUD / JPY Currency Pair particular selecting a single volatility Fair Volatility input from the implied volatility surface 23% constructed from market quotes for

vanilla options. However, since the strike 22% price of a forward-starting option is not fixed in advance, it is not possible to 21% choose an appropriate implied vola- tility for the exact strike and maturity 20%

as would be the case when valuing a 19% vanilla option. A relatively simple extension to the 18% Heston Model Calibrated to Black-Scholes model is the Dupire Volatility Swap Price Data 17% model which, unlike Heston Model Calibrated Only to Vanilla Option Data Black-Scholes, allows multiple volatility 16% inputs to be applied simultaneously At-the-money Implied Volatility when pricing. So, the implied volatili- 15% 1 3 6 12 18 24 36 48 60 ties observed in the market for different Maturity (Months) strikes and maturities can all be used in combination to price exotic structures. Source: Markit

the markit magazine – Autumn 2011 R&D: Model calibration /43 but also on the variance of the future (iii) the volatility of variance, which prices are updated on a regular basis. variance. So even after pricing a vari- represents the amount by which Markit’s Portfolio Valuations service ance swap by replicating it using a the variance can fluctuate; and uses dealer consensus quotes for portfolio of vanilla options and applying (iv) the correlation between the asset cliquets to estimate the relationship Black-Scholes to value each of them, price and its variance, which between the spot implied volatility, one would still need to compute a governs the extent to which which is the volatility implied by the convexity adjustment in order to price these two key factors move in a vanilla option market observed today, the equivalent volatility swap. This similar way. and the forward implied volatility, which adjustment can be applied to the is the projected volatility implied by price to give the vola- Of these, the volatility of variance and the vanilla option market observed at tility swap price. As with cliquets, this correlation are the key parameters for specified future dates. Given that both requires a model which can correctly using the Heston model to price exotic the spot and forward implied volatili- represent the behaviour of the implied derivatives such as cliquets and volatility ties have their own term structure and volatility of the underlying asset as it swaps accurately. smile, i.e. variation with option matu- evolves through time. Once values for the Heston param- rity and strike price, this relationship is eters have been obtained, solving the not simple to determine or to express Heston model model to find the price of a cliquet numerically. However, once this has We have discussed how important it is option or volatility swap is relatively been done, it can be used to derive the when valuing both cliquets and volatility straightforward and can be achieved appropriate values for the volatility of swaps to model the implied volatility either through closed-form solutions variance and correlation parameters. appropriately. So, in order to capture the for simple cases, or numerical tech- These parameters can then be applied correct dynamics of the implied volatility niques such as Monte Carlo simulation. within Heston to price a variety of cliquets on the same underlying asset, with differing maturities, and strike reset “When pricing volatility swaps, it is desirable levels and frequencies. When pricing volatility swaps, it to calibrate the Heston model to observable is likewise desirable to calibrate the Heston model to observable volatility volatility swap price data.” swap price data, which is the approach adopted within Markit. This is particu- surface, it is standard practice to resort The substantive difficulty when using larly important in order to compute the to a stochastic volatility model such as Heston to price exotic derivatives of this volatility of variance parameter correctly, the Heston model. This models the future type is not in solving or implementing which largely determines the amount of evolution of not only the underlying asset the model itself, but in calibrating the convexity or curvature of the volatility price (as is the case with Black-Scholes), parameters so that the prices produced smile, and thereby the price differen- but also the implied volatility (or vari- by the model reflect the levels at which tial between the volatility swap and the ance), where these two processes have those derivatives would be traded. corresponding variance swap. random components which are corre- As demonstrated in the charts, the lated with one other. By doing this, Heston Calibrating Heston parameters importance of having a suitable model is able to take into account known market In order to calibrate Heston so that it with high-quality calibration of its effects such as a noticeable increase in can be used to price a range of cliquets parameters is underlined by the signifi- implied volatility associated with a sharp on a particular underlying asset, one cant mispricing that can result from decrease in asset price. must have access to market informa- naive calibration techniques or simplistic The Heston model contains the tion on the prices of cliquets of a similar model assumptions. following four parameters which cannot nature. More generally, when pricing be directly observed or backed out from any exotic using a model such as market quotes for vanilla instruments: Heston, we require price data for a set of calibration instruments covering the (i) the long-run variance, to which same or similar structures, underlying the variance of the asset price assets and maturities as the derivative Łukasz Dobrek, Markit converges over time; we intend to price. Furthermore, since it (ii) the speed at which the actual is known that parameters can fluctuate variance reverts to the long-run significantly over time, it is also impor- variance; tant that these calibration instrument

Autumn 2011 – the markit magazine