Portfolio-Based Pricing of Residual Basis Risk: CAS Discussion Paper Summary

1. Portfolio Based Pricing of Residual Basis Risk • Sergei Esipov - Centre Solutions • Don Mango - American Re-Insurance

2. 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

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

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

5. The Converging Worlds of Capital Markets and Reinsurance

6. 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”

7. Significant Differences • Probability Measures • Financial = Risk Neutral Probability • Actuaries = Objective Probability • Prices • Financial = Market Prices • Actuaries = Indicated Prices • Time Frames • Days/Weeks versus Years

8. Significant Differences • Tradability and Liquid Secondary Markets • Foundations of financial market theory • But State Farm can’t 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

9. Hedging

10. 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

11. Option Pricing and Hedging • If all those assumptions hold true, a PERFECT HEDGE is possible • Perfect Hedge means the “Profit & Loss” or “P&L” on the Option is KNOWN • The Price of the Option = The Cost of the Hedge Portfolio

12. 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

13. Option Pricing and Hedging • In particular, two “Stylized Facts” cause concern: • Implied Volatility > Realized Volatility (index options) • The Volatility Smile • What do they mean?

14. 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

15. Implied Vol > 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”?

16. Volatility Smile

17. 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 ?

18. 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

19. How Did They Test It? • Simulation Modeling of the S&P 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

20. 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

21. 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

22. Dr. Sergei Esipov

23. 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 (P&L). How to convert this to price?

24. 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 = φ. How? • Each time φ changes significantly - rebalance • Accumulate hedging cost and use it to offset the option payoff

25. Underlying Process • Standard & Poor 500 Index

26. Underlying Process • Standard & Poor 500 Index

27. Econometric Process • A process for the underlying security S with little memory • m - drift rate per time step 0.030% • σ - volatility per time step 0.88% in 90% of cases • qt - jump per time step in 10% of cases

28. Simulation of the S&P500 Index • Which one is the original index?

29. Sell 1 Year European Put Option • This is just one of many liquid options for S&P500 Strike = K = 1400 Maturity = T = 1

30. Establish a Hedging Position • Sell short units of the underlying index (in reality - futures) Strike = K = 1400

31. 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 P&L is volatile

32. Net Accumulated P&L is Volatile • This is the basis risk

33. A Put with no Hedging • What kind of PDF one can get? This depends on the hedging strategy, Loss Profit

34. Perfect (Theoretical) Hedging A put with perfect hedging in lognormal world No Loss Only Profit at rate r

35. Real Hedging A put with diligent hedging at sunset (real world) Loss Profit

36. From Hedging to P&L Distribution • In case the underlying index is lognormal (no jumps) the P&L distribution density for arbitrary is described by the following backward PDE http://papers.ssrn.com/paper.taf?ABSTRACT_ID=145172 IJTAF, 2, 2, 131-152 (1999) Sergei Esipov & Igor Vaysburd

37. 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.

38. Porfolio Considerations • What happens when we add P&L distribution of the considered put option position to our portfolio? • Percentiles of change a little after addition. How much? correlation Standard deviation Standard deviation

39. 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

40. 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?

41. 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

42. Reverse Engineering • What is the corresponding implied volatility? • Solving this for volatility gives

43. Market vs Modeled Implied Vol There are no adjustable parameters

44. 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 P&L distributions of their (option) positions plus hedge positions as a keystone of pricing

45. 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

46. Conclusions • It is profitable to have direct access to trading desks to be able to monitor positions and perform dynamic hedging. • The firm’s 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

47. 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 P&L 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

48. 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

49. Relation to Static CAPM • Change of VaR (Allocated Capital) • Returns on this risky asset and on the market portfolio are to be equal

50. Relation to Static CAPM • Substitution results in • For short time horizons • (both the asset and market) one gets the static CAPM