A new approach for option pricing under stochastic volatility |
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Authors: | Peter Carr Jian Sun |
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Institution: | (1) Bloomberg LP, 731 Lexington Avenue, New York, NY 10022, USA;(2) XE Capital Management, 24 West 40th Street, New York, NY 10018, USA |
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Abstract: | We develop a new approach for pricing European-style contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume
continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we
do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance
swap market, we directly assume continuous dynamics for the time T variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option
prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility
of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and
on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption
the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic
process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this,
we show that the payoff to a path-independent contingent claim can be perfectly replicated by dynamic trading in futures contracts
and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying’s
futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our
approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility.
As a consequence of our SVRH, the partial differential equation (PDE) governing the arbitrage-free value of the contingent
claim just depends on two state variables rather than the usual three. We then focus on the consistency of our SVRH with the
standard assumption that the risk-neutral process for the instantaneous variance is a diffusion whose coefficients are independent
of the variance swap maturity. We show that the combination of this maturity independent diffusion hypothesis (MIDH) and our
SVRH implies a very special form of the risk-neutral diffusion process for the instantaneous variance. Fortunately, this process
is tractable, well-behaved, and enjoys empirical support. Finally, we show that our model can also be used to robustly price
and hedge volatility derivatives. |
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Keywords: | Option pricing Stochastic volatility |
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