Sergei Esipov Centre Solutions

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Portfolio Based Pricing of Residual Basis Risk

• Sergei Esipov - Centre Solutions
• Don Mango - American Re-Insurance

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Introductions

• This is based on a paper in the 2000 CAS
  Discussion Paper Program Portfolio-Based
  Pricing of Residual Basis Risk
• Winner of the 2000 Michelbacher Prize

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Introductions

• Authors Sergei Esipov and Dajiang GuoCentre
  SolutionsFormer Capital Market Quantitative
  Analysts. Backgrounds in finance, economics and
  natural sciences

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Introductions

• Providing a CAS Translation.Don Mango,
  FCASAmerican Re-Insurance(formerly of Centre
  Solutions)Casualty Actuary interested in finance

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The Converging Worlds of Capital Markets and
Reinsurance
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Common Ground

• Insurers are levered financial trusts
• Life Insurers are selling investments
• Financial derivatives have insurance-like
  characteristics
• Time value of money
• Volatility and uncertainty
• Risk

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Significant Differences

• Probability Measures
• Financial Risk Neutral Probability
• Actuaries Objective Probability
• Prices
• Financial Market Prices
• Actuaries Indicated Prices
• Time Frames
• Days/Weeks versus Years

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Significant Differences

• Tradability and Liquid Secondary Markets
• Foundations of financial market theory
• But State Farm cant sell off an Auto policy it
  just wrote !!
• Hedging
• Banks and Securities firms are always looking for
  zero net risk
• Insurers are looking to retain the right risks

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Hedging
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Option Pricing and Hedging

• Black-Scholes theory
• Short Rate known and constant
• Price follows continuous random walk with known
  and constant volatility
• No dividends
• European option
• No transaction costs
• Can short-sell and subdivide without penalty

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Option Pricing and Hedging

• If all those assumptions hold true, a PERFECT
  HEDGE is possible
• Perfect Hedge means the Profit Loss or PL
  on the Option is KNOWN
• The Price of the Option The Cost of the Hedge
  Portfolio

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Option Pricing and Hedging

• The Reality
• Transactions have costs
• Short rate and volatility vary over time
• The Results
• Dealers cannot achieve perfect hedges...
• so they retain Basis Risk...
• and Black-Scholes formula prices do not match
  market prices

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Option Pricing and Hedging

• In particular, two Stylized Facts cause
  concern
• Implied Volatility gt Realized Volatility
• (index options)
• The Volatility Smile
• What do they mean?

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Implied and Realized Volatility

• Implied Volatility
• Black-Scholes formula reduces the Option Price to
  a function of Volatility
• Therefore, for a given Market Price, one can back
  into the Implied Volatility
• Realized Volatility
• That measured historically for the underlying
  asset

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Implied Vol gt Realized Vol

• Implied Volatility is greater than Historical
  (Realized) Volatility (index options)
• Market is pricing options as if they were riskier
  than history would indicate
• Perhaps there is an insurance element to the
  price - a Risk Premium?

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Volatility Smile
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Volatility Smile

• Black-Scholes theory makes no provision for
  varying Option Price with Strike Price
• Option Price f(Volatility)
• In addition to Strike Price dependence there is a
  maturity dependence. Together they form
  volatility surface.
• What exactly do we learn from translating Option
  Price into Vol by means of a smooth function ?

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Esipov Guo Approach

• Dealers employ an average hedging strategy
• Their Residual Basis Risk gets priced ACTUARIALLY
  (similar to Kreps), resulting in a Risk Premium
• Option price Average Hedging Cost Risk
  Premium

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How Did They Test It?

• Simulation Modeling of the SP 500 Index (SPX) -
  see Section 3 of paper
• Average Hedging Strategy for Options on the SPX
• Based on an average observed volatility
• Use Black-Scholes delta hedging based on the
  volatility
• Discrete in time (not continuous) or imperfect

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What Was The Result?

• The hybrid pricing approach produced prices much
  more similar to actual market prices than
  Black-Scholes using historical volatility
• ...and in many cases generated the implied
  volatility smile for index options

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What Was The Result?

• Significant for the Finance community
• Actuarial techniques providing a possible answer
  to serious problem
• More significant for the CAS !!!
• Reciprocal adoption of actuarial techniques by
  Finance quantitative analysts

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Dr. Sergei Esipov
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Why Do We Talk about Options?

• Actuaries are actively studying financial
  literature. How to combine new things with the
  existing knowledge?
• Options can be explained simply. What happens at
  the option trading desk?
• Options can be translated to NPV distribution
  (PL). How to convert this to price?

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How to Trade a Call or a Put in Practice?

• Set up an econometric process for the underlying
  security S. How?
• Sell (Buy) an option
• Establish a dynamic hedging position f. How?
• Each time f changes significantly - rebalance
• Accumulate hedging cost and use it to offset the
  option payoff

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Underlying Process

• Standard Poor 500 Index

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Underlying Process

• Standard Poor 500 Index

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Econometric Process

• A process for the underlying security S with
  little memory

• m - drift rate per time step 0.030
• s - volatility per time step 0.88 in 90 of
  cases
• qt - jump per time step in 10 of cases

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Simulation of the SP500 Index

• Which one is the original index?

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Sell 1 Year European Put Option

• This is just one of many liquid options for
  SP500

Strike K 1400 Maturity T 1
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Establish a Hedging Position

• Sell short units of the underlying index (in
  reality - futures)

Strike K 1400
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Dynamic Hedging Rebalancing

• In theory the option payoff and hedging cost
  together offset each other
• In reality, as mentioned before by Don Mango

• Difficulties in maintaining correct
• Problems with parametrization
• Transaction costs

Net accumulated PL is volatile
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Net Accumulated PL is Volatile

• This is the basis risk

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A Put with no Hedging

• What kind of PDF one can get? This depends on the
  hedging strategy,

Profit
Loss
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Perfect (Theoretical) Hedging
A put with perfect hedging in lognormal world
Only Profit at rate r
No Loss
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Real Hedging
A put with diligent hedging
at sunset (real world)
Profit
Loss
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From Hedging to PL Distribution

• In case the underlying index is lognormal (no
  jumps) the PL distribution density for arbitrary
  is described by the following backward PDE

http//papers.ssrn.com/paper.taf?ABSTRACT_ID14517
2 IJTAF, 2, 2, 131-152 (1999) Sergei Esipov
Igor Vaysburd
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Risk Management

• How do we go from distribution to price?
• Option trading desks are required to pass through
  a set of risk management tests (regulations)
• E.g.Value-at-Risk test demonstrate the capital
  sufficient for solvency of BB rating, i.e. in all
  but 1 of the cases.

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Porfolio Considerations

• What happens when we add PL distribution of the
  considered put option position to our portfolio?
• Percentiles of change a little after
  addition. How much?

correlation
Standard deviation
Standard deviation
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Change of the Percentile

• Expand in Taylor series assuming that scales of x
  are much smaller than scales of X
• To leave unchanged shift x by
  and by

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Change of the Percentile

• One has to come up with additional capital in the
  amount of

• to satisfy the VaR requirements
• What is the return on this risky investment that
  the firm should expect?

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Return on Allocated Reserve

• The return is defined as

• Solving this for the Price or Premium one finds

• This is a quick formula for translating PDF into
  premium

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Reverse Engineering

• What is the corresponding implied volatility?

• Solving this for volatility gives

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Market vs Modeled Implied Vol
There are no adjustable parameters
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Begin Conclusions

• We have presented a method (entirely based on the
  analysis of fundamentals) to evaluate options and
  reproduce the volatility skew
• Institutions (and capital market analysts) have
  to compute PL distributions of their (option)
  positions plus hedge positions as a keystone of
  pricing

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Conclusions

• New Role of Risk Management. Pricing and Risk
  Management are explicitly connected. One cannot
  do them separately
• Actuaries have to adapt to short time scales and
  seriously discriminate between prices based on
  fundamentals and actual market prices.
• It is imperative to have up-to-date econometric
  analysis

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Conclusions

• It is profitable to have direct access to trading
  desks to be able to monitor positions and perform
  dynamic hedging.
• The firms portfolio can be considered as a big
  option with uncertainty if the index goes up,
  the firm will have PDF_1, if index goes down, the
  firm will have PDF_2. If the index is tradable,
  one has to hedge! New questions.

Index 1100, Firm PDF is
Index 1000 Should one hedge? How many shares?
Index 900, Firm PDF is
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Conclusions End

• Answers depend on the firm business strategy and
  heavily depend on regulations/risk management
  rules. We have answers for a number of common
  cases. They require a separate technical
  presentation
• VaR analysis is forward-looking/ NPV PL analysis
  is backward-from-the-future looking. How to
  reconcile the difference?
• Actuarial approach to ruin probability
  (credit-related), reserving, return on reserve,
  portfolio-based pricing is at work

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Relation to Static CAPM

• The pricing formula generalizes the static
  Sharpe-Treynor CAPM formula consider static
  investment into log-normal equity-like asset

NPV of Expected Value
NPV of Standard deviation
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Relation to Static CAPM

• Change of VaR (Allocated Capital)

• Returns on this risky asset and on the market
  portfolio are to be equal

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Relation to Static CAPM

• Substitution results in

• For short time horizons
• (both the asset and market) one gets the static
  CAPM

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Dynamic Version

• How to invest dynamically to achieve a given
  expected NPV and minimal corresponding standard
  deviation?
• The solution of this problem can be found in
• http//papers.ssrn.com/paper.taf?ABSTRACT_ID1705
  74

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Special Thanks to

• Dajiang Guo and Igor Vaysburd for help and
  numerous discussions
• Karl Borch for motivation of this study
• Richard Timbrell for discussions and support
• To Caffé Dante staff for patience and
  understanding

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Recommended Reading

• Options basics J. Hulls book. 4th edition
  Options, Futures and Other Derivative
  Securities, Prentice Hall, 1999 (based on
  expectations).
• Early actuarial PDF pricing K. Borchs
  book Economics of Insurance, North-Holland,
  1992 (no explicit dynamics of the underlying
  asset).
• Up-to-date bookshelf of books on financial
  mathematics is maintained by Alex Adamchuk at
  http//finmath.com