Why a Delta-Gamma Option Basket Cannot Fully Hedge Jumps in SVJ

Summary

The document asks why options used to hedge stochastic volatility in a Heston model cannot also create a riskless portfolio for pricing options under a stochastic volatility jump (SVJ) model. Delta and gamma matching can address local exposure to movements in the underlying, and additional options can help hedge volatility risk. However, SVJ includes Poisson-driven jumps that create discontinuous price changes, which local delta and gamma matching alone do not eliminate.

The text poses the question but provides no answer, derivation, or numerical evidence. It therefore does not establish whether a sufficiently broad set of traded instruments could hedge jump risk under particular assumptions. Its value is as a conceptual prompt about the limits of dynamic hedging and the risks that remain when the model includes jumps.

Key ideas

• Delta and gamma matching hedge local exposure to changes in the underlying price.
• Stochastic volatility introduces a separate source of risk that option positions may help hedge.
• Poisson jumps create discontinuous price moves that local delta and gamma matching do not remove.
• The document raises, but does not resolve, whether a traded option basket can hedge all SVJ risks.

Tags

# Why can't we create a "magic" basket of options to sell for no-arbitrage pricing in SVJ model?


# Why can't we create a "magic" basket of options to sell for no-arbitrage pricing in SVJ model?












I am learning how to price SVJ options and am reading some stuff on no-arbitrage pricing for SVJ model using the typical approach you would use (like in BSM option pricing) of creating a risk free portfolio. I understand the part on why you CAN'T use the typical approach and just create a portfolio that consists of a long option and short underlying as you still have the Poisson process that isn't hedged out.

However, why can't you create a portfolio of long option, short underlying, and short a "magic" basket of options that just happens to match the gamma and delta of your long option. When deriving the pricing equation for Heston's stochastic volatility model, we do the exact same thing by selling options to hedge the stochastic volatility process.

Why can't we do the same when trying to price options using SVJ model?

Thanks!

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