Chrif YOUSSFI

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Convexity adjustment for volatility swaps

• Chrif YOUSSFI
• Global Equity Linked Products

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Outline

• Generalities about volatility/variance swaps.
• Intuition and motivation
• The framework of stochastic volatility.
• Convexity adjustment under stochastic volatility.
• Convexity adjustment and current smile.
• Numerical results.
• Conclusions.

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Volatility and Variance Swaps

• A volatility swap is a forward contract on the
  annualized volatility that delivers at maturity
• A variance contract pays at maturity
• The annualized volatility is defined as the
  square root of the variance
• where is the closing price of the
  underlying at the ith business day and (n1) is
  the total number of trade days.

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Hedge and Valuation

• When there no jumps, the variance swaps valuation
  and hedging are model independent.
• The vega-hedge portfolio for variance swaps is
  static and the value is directly calculated from
  the current smile.
• The valuation of a volatility swap is model
  dependent and the pricing requires model
  calibration and simulations.
• The vega hedge portfolio is not static.

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Motivation

• 
  
  

Smile of volatility generated by a stochastic
volatility model wherespot is 100,maturity 1y
and correlation is estimated at -70

• What is the price of ATM option?

By considering the linearity of the option price
w.r.t. volatility, the price is approximately
14.11

• What is the value of the variance swap? 16.17.

• What is the value of the volatility Swap? More
  difficult.

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Intuition

• is the spot density at maturity T and
  the diffusion factor which be stochastic
• A rough estimation of the volatility swap
• Question What can the moments of the implied
  volatility teach us about the value of volatility
  swap?

The weighting is not exact.
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MIV Moment of Implied Volatility

• We define by MIV (n) as the nth moment of the
  implied volatility weighted by the risk neutral
  density.
• We define the smile convexity by
• The convexity adjustment for the volatility
  swaps
• Question What is the relation between
  and ?

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Stochastic volatility assumptions

• The underlying dynamics are
• The volatility itself is log-normal
• with the initial condition
  and
• The dynamics correspond to the short time
  analysis and the factor can be considered
  proportional to the square root of time to
  maturity.
• (Patrick Hagan Model (1999))

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Forward and backward equations

• The backward equation for the call prices is
  
• The transition probability
  from the state
• to satisfies the
  forward equation (FPDE)
• When integrating the Forward PDE (Tanakas
  formula)

The curve
Smile effect
Intrinsic
Integral over calendar spreads
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Call Price and Implied Volatility

• Define by
• 
  and
• The solution of the system (S) is
• In the BS case we have a similar formula with
  

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Volatility swap convexity adjustment

• The expected variance under the model
  assumptions
• The value of the expected volatility
• It follows that the convexity adjustment is

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Smile convexity

• By considering the value of the log-contract and
  the square of the log-profile
• The smile convexity of the implied volatility is

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Convexity adjustment

• As long as the is large enough (which is
  satisfied in the equity markets), to the leading
  orders show that the relation between the two
  convexities is very simple
• There no dependencies on maturity and volatility
  of the volatility.
• The value of the volatility swap does not depend
  on the correlation, however the implied
  volatility depends on and therefore
  intuitively we need to strip off this dependency.

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Numerical Results (4)
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Numerical Results (5)
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Numerical Results(6) Heston
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Numerical Results (7) Heston
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Conclusion

• This analysis shows that option prices can be
  very insightful to estimate the convexity
  adjustment.
• Even though the results are derived in the case
  of Hagan model, they can be extended to other
  models of stochastic volatility (Heston) as long
  as the correlation is in an appropriate range.
• It sheds some light on the importance of the
  curve factors to decide the value of a volatility
  swap.