Introduction
European options are often priced and hedged using Black's
model, or equivalently, the Black-Scholes model. In Black's model there is a
one-to-one relation between the price of a European option and the volatility
parameter óLN. Consequently, option prices are
often quoted by stating the implied volatility
óLN, the unique value of the volatility which
yields the option's dollar price when used in Black's model. In theory, the
volatility óLN in Black's model is a constant.
In practice, options with different strikes K require different
volatilities óLN to match their market prices.
Handling these market skews and smiles correctly is critical
to fixed income and foreign exchange desks, since these desks usually have
large exposure across a wide range of strikes. Yet the inherent contradiction
of using different volatilities for different options makes it difficult to
successfully manage these risks using Black's model.
The development of local volatility models by Dupire [11] and
Derman-Kani [10] was a major advance in handling smiles and skews. Local
volatility models are self-consistent, arbitrage-free, and can be calibrated to
precisely match observed market smiles ans skews. Currently these models are
the most popular way of managing smile and skew risk. However, the dynamic
behaviour of smiles ans skews predicted by local volatility models is exactly
the opposite of the behaviour observed in the marketplace: when the price of
the underlying asset decreases, local volatility models predict that the smile
shifts to higher prices. In reality, asset prices and market smile move in the
same direction. This contradiction between the model and the marketplace tends
to de-stabilize the delta and vega hedges derived from local volatility models,
and often these hedges perform worse than the naive Black-Scholes' hedges.
To resolve this problem, Hagan, Kumar, Lesniewski and Woodward
derived the SABR model, a stochastic volatility model in which the asset price
and the volatility are correlated. Singular perturbation techniques are used by
the former authors in order to obtain the prices of European options under the
SABR model, and from these prices they obtained a closed-form algebraic formula
for the implied volatility as a function of today's forward price and the
strike. This closed-form formula for the implied volatility allows the market
price and the market risks, including vanna and volga risks, to be obtained
immediately from Black's formula. It also provides good, and sometimes
spectacular, fits to the implied volatility curves observed in the marketplace.
More importantly, the formula shows that the SABR model captures the correct
dynamics of the smile, and thus yields stable hedges.
2 INTRODUCTION
INTRODUCTION
Why models ? Objectively, it is no good pricing a liquid
asset; getting its price directly from the market is largely sufficient. The
purpose of models is the pricing of illiquid or scarce assets, such as vanilla
options with extreme strikes. Thus, a usable model is one which doesn't break
down under extreme conditions. However, SABR model is rather used a reading
tool: market data is usually transformed into model parameters through
calibration to vanilla assets. Then, the obtained market data (stored as a
matrix of SABR parameters) is used for the calibration of more complicated
models designed for the pricing of exotic options.
However, since the financial crisis that began in 2007, the
american Federal Reserve conducts monetary policy to achieve maximum
employment, stable prices, and moderate long-term interest rates. In addition,
the Fed purchased large quantities of longer-term Treasury securities and
longer-term securities issued or guaranteed by government-sponsored agencies
such as Fannie, Mae or Freddie Mac. With such low rates, the SABR model,
endowed with Hagan approximation for implied volatility, yields arbitrage. This
arbitrage is observable through the negative density of the underlying
process.
Furthermore, Interest rate option desks typically need to
maintain very large amounts of interlinked volatility data. For each currency,
there might be 20 expiries and 20 tenors, that is, 400 volatility smiles.
Furthermore, the smiles might be linked across different currencies.
Interpolation of observed discrete quotes to a continuous curve is needed for
the pricing of general caps and swaptions. At the same time, extrapolation of
options quotes are needed for constant maturity swap (CMS) pricing. The SABR
model only has four parameters to handle the mentioned tasks, which is not
enough flexibility to exactly fit all option quotes.
In this document, we shall outline some problems encountered
with SABR model nowadays. We will first solve each problem, then highlight a
new model that solves both of our problems.
3
Chapter 1
Problems encountered with SABR
model
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