Pricing Variance Swaps for Stochastic Volatility with Delay and Jumps

1
Pricing Variance Swaps for Stochastic Volatility
with Delay and Jumps

• Li Xu
• PRMIA Presentation
• Department of Mathematics Statistics
• University of Calgary

2
Outline

• Introduction
• Previous Results in Delay Model
• Pricing Model with Delay and Jumps
• Numerical Example and Comparison
• Conclusion

3
Introduction

• Variance swap A forward contract on future
  realized asset variance, the square of the
  volatility, which can be used to trade variance.
• Payoff function
• is the realized asset variance (quoted in
  annual terms) over the life of the contract,
• is the strike price for variance.
• is the notional amount of the swap in
  dollars per annualized volatility point square.
• The price of the variance swap in the risk
  neutral world is the expected present value of
  the payoff

4
Who Can Use Variance Swaps

• Portfolio managers who wish to hedge vega ( )
  exposure.
• Hedging implicit volatility exposure.
• Investors may require more frequent rebalancing
  and greater transactions expenses during volatile
  periods.
• Equity funds are probably short volatility
  because of the negative correlation between index
  and volatility.
• Trading the spread between realized and implied
  volatility.
• Clients who want to speculate on the future
  levels of volatility.

5
Stochastic Volatility with Delay

• The stochastic volatility depends on
• where is the delay
  factor and
• is the spot price of the underlying asset
  at time (see Swishchuk et al.
  (2002)).
• The initial data
  deterministic function.
• Variance swaps can be priced under this model
  (see Swishchuk et al. (2002))

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Main Features of this Model

• Continuous-time analogue of discrete-time GARCH
  model.
• Mean-reversion.
• Contains the same Wiener process as the asset.
• Market is complete.
• Incorporates the expectation of log-return.
• Delay as a measure of risk.
• Similar to Heston model, but easier to model and
  price the variance swaps.

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Why Jumps in Stochastic Volatility

• By empirical study, it is more realistic to
  consider jumps in stochastic volatility in
  financial markets including energy market.
• Still keeps those good features of the previous
  model.
• An analytical pricing formula for variance swaps
  is still available and it is quick to implement.

8
Pricing Model

• The spot price of the asset satisfies the
  following SDDE
• Consider the Poisson process in the stochastic
  volatility
• This is a continuous-time analogue of its
  discrete-time GARCH(1,1) model

9
Continue

• Change to risk neutral measure, and take
  expectation of the stochastic volatility under
  risk neutral probability, we have
• where
• The intensity of Poisson process does not
  change under risk neutral probability.

10
Continue

• Solving this Differential Equation, we get the
  approximate solution in general case
• The analytical formula for variance swap with
  delay and Poisson jumps

11
Compound Poisson Process Case

• Consider the following equation
• Take expectation under risk neutral probability

12
Continue

• In general case, the approximate solution
• The analytical formula for variance swap with
  delay and compound Poisson jumps

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Numerical Example SP60 Canada Index (1997-2002)
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Comparison
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Dependence on Intensity
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Conclusion

• Keeps all the advantages of the delay model
  without jumps.
• Incorporate jumps in stochastic volatility which
  is important in energy market.
• Easier to model and price variance swaps, no
  numerical approximation and time saving.
• Further extension adding jumps in asset price,
  more complex form of jumps in stochastic
  volatility, pricing volatility swaps.

17
Thank you!

• lxu_at_math.ucalgary.ca