Derivative Portfolio Optimisation

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Derivative Portfolio Optimisation

• Raoul Davie
• Karen Lau Thierry Mbimi
• Quantitative Applications Division
• Macquarie Bank

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Outline

• Challenges of managing a Derivative Trading Book
• Typical approach
• Example hedging interest rate risk
• Risk-return trade-off ?
• Benefits of portfolio approach
• Simple examples

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Typical Approach

• 2-Step Process
• Trading decisions
• Motivated by profit
• Hedging decisions
• Neutralise risk to moves in market rates
• (asset price, asset volatility, interest rates)
• Trading decisions are independent of hedging
  decisions

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Example hedging interest rate risk

• AUD interest rate curve
• RBA Cash rate
• Physical Bills -gt 90 days
• Bank Bill futures -gt 3 years
• Swaps -gt 15 years
• Derivative trading book
• Maturities at monthly -gt 15 years

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Hedging interest rate risk cont

• Approach 1 (naive)
• Compute sensitivity to 1 bp move
• Hedge sensitivity with 1 hedge
• e.g. 1-year swap
• Problem
• hedges parallel shifts only

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Hedging interest rate risk cont

• Approach 2 (brute force)
• Compute sensitivity to each input rate
• RBA Cash rate
• Physical Bills -gt 90 days
• Bank Bill futures -gt 3 years
• Swaps -gt 15 years
• Hedge sensitivity to each instrument
• Problem
• Need to trade 20 instruments

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Hedging interest rate risk cont

• What if you want to hedge with a subset of
  instruments?
• Portfolio choice problem
• Correlated assets
• Derivative Trading Book
• 20 possible Hedge Instruments

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More Generally risk

• Trading Book encapsulates a range of risks
• Vanilla and exotic derivatives
• Each derivative has a unique exposure
• underlying asset(s)
• volatility (S,T, K)
• interest rates
• Many hedge instruments, including ETOs

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More Generally return

• a range of return opportunities
• Exotic options margin to fair value
• ETOs margin to value
• and constraints
• Limited liquidity in hedge instruments
• Management limits

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Decision Time

• How can a trader
• Take advantage of derivative spreadsfor exotics
  and ETOs ?
• Choose a short dated hedge for a long dated
  position?
• Choose which strikes to use for hedge?
• while accommodating limits constraints?
• this is a Portfolio Optimisation Problem !

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How do we describe Portfolio Risk?

• Derivatives?
• risk characteristics of derivatives are
  continually changing
• Instead, base analysis on markets rates
• underlying correlated market rate variables
• Derivative price PModel(Si , ?i,k , rT .)
• links prices with market rates

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Risk cont (exposure vector)
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Risk cont portfolio variance

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Arent returns on derivative portfolios skewed?
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Risk summary

• Portfolio risk is driven by correlated market
  rates
• Si , ?i,k , (rT .)
• Market rates common across many derivatives
• Pricing model
• Risk Transformation rates -gt derivative
  holdings
• Benefit
• Rates provide unifying stationary framework
• Analogous to factor models used in funds
  management

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Returns, Utility and Constraints

• Expected Return, for example
• per derivative
• or drive from underlying rates, as margin on
  ?i,k
• Utility k(expected profit) (variance of
  PL)
• profit is good
• VaR is bad
• Constraints are straight-forward to add

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•  Examples
• hedge a 2-month atm european option with
• underlying asset
• 1-month 3-month atm options
• Rates
• Implied volatilities (IV) 25
• Correlation between IV 0.8
• Volatility of IV 20
• (unless otherwise stated)

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Is zero vega the target?

• Even with perfect correlation and sufficient
  hedge instrumentsthe minimum risk portfolio may
  not have zero vega
• If the IVs, or volatility of the IVs, are
  differentthen optimal portfolio will not have
  zero vega
• Since, volatility exposure depends upon the
  product
• Vega IV (volatility of IV)

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Summary

• How can a trader
• Take advantage of derivative spreadsfor exotics
  and ETOs ?
• Choose a short-dated hedge for a long-dated
  position?
• Choose which strikes to use for hedge?
• Choose to hedge with a related asset?
• while accommodating limits constraints?

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Conclusion

• portfolio optimisation techniques provide a
• robust, scalable and disciplined framework
• for addressing all of these issues