CAS 2004 Spring Meeting Presentation of Proceedings Paper

1. CAS 2004 Spring MeetingPresentation of Proceedings Paper DISTRIBUTION-BASED PRICING FORMULAS ARE NOT ARBITRAGE-FREE DAVID RUHM DISCUSSION BY MICHAEL G. WACEK

2. Wacek Discussion • Ruhm illustrates difficult concepts from financial economics in a refreshing way • He emphasizes that probability distribution of outcomes does not always contain enough info to produce arb-free prices for a risk • We point out, however, that distribution of outcomes cannot be ignored in determining expected cost of risk • We discuss distinction between price and cost and its significance, particularly for the seller

3. Wacek DiscussionOverview • Ruhm also seeks to generalize about arb-free pricing of calls and puts • Using a “risk discount function”, w(s), he concludes that calls are priced at a discount, puts at a premium to expected payoff • Why would anyone buy puts? • He reasons that risks have a qualitative nature as either insurance or investment – puts reflect insurance nature • We show that Ruhm’s generalization is undermined by a key assumption he overlooked • Explanation for investor behavior does not require “qualitative nature of risk” concept

4. Wacek DiscussionOverview • Discussion divided into three sections:1. Distinction Between Price and Cost – Seller’s Perspective2. Value for Money – Buyer’s Perspective3. w(s) function (Ruhm’s “risk discount function”) • This presentation will include a recap of key aspects of Ruhm’s original paper in order to put the discussion in context

5. Summary Of Ruhm Paper

6. Summary Of Ruhm Paper – Excerpt (1) from Abstract “A number of actuarial risk-pricing methods calculate risk-adjusted price from the probability distribution of future outcomes.”“Such methods implicitly assume that the probability distribution of outcomes contains enough information to determine an economically accurate risk adjustment.”

7. Summary Of Ruhm Paper – Excerpt (2) from Abstract “In this paper it will be shown that distinct risks having identical distributions of outcomes generally have different arbitrage–free prices.”

8. Summary Of Ruhm Paper – Excerpt (3) from Abstract “Risk load formulas that only use the risk’s outcome distribution cannot produce arbitrage–free prices, and in that sense are not economically accurate for risks traded in markets where arbitrage is possible.” “In practice, most insurance underwriting risks are not traded in such markets.”

9. Summary Of Ruhm Paper – Excerpt (4) from Abstract “A ratio is used to measure the implicit discount or surcharge for risk that is present in a price: the ratio of price density to discounted probability density.” “This ratio can be used to identify the qualitative nature of a risk as investment or insurance: a risk discount factor less than unity indicates investment, whereas a risk surcharge factor above unity indicates insurance.”

10. Summary Of Ruhm Paper – Excerpt from Section 1 “The result is shown to be true in general: arbitrage-free pricing cannot be produced by any formula that uses only the distribution of economic outcomes.”

11. Ruhm’s Main Derivative Example • Define a “segment derivative” paying a fixed amount if expiry stock price is between s and t, else $0 • Assume stock price at expiry lognormally distributed • Stock price parameters • Market price $100 • Expected expiry price $110 • Expiry in 1 yr • σ = 30%

12. Ruhm’s Main Derivative ExamplePricing Assume • “B-S” conditions are met “Risk neutral” pricing framework applies • $100 grows risk-free to $104 in 1 yr • Then derivative priced as though expiry stock price parameters are: • Market price $100 • Expected expiry price $104 • Expiry in 1 year • σ = 30%

13. Ruhm’s Roulette Wheel Metaphor • Segment expiry stock price range into 38 intervals • Each interval equally likely (based on expected stock price distribution) • Price a “segment derivative” with $1 payoff for each interval • Map each interval and its related derivative to a number on roulette wheel

14. Ruhm’s Roulette Wheel – Excerpt (1) from Exhibit 1

15. Ruhm’s Roulette Wheel - Pricing Bet Amount • Price =_____________ Payoff Amount • In conventional roulette price is proportional to outcome probability • In Ruhm’s roulette price is governed by risk neutral framework (i.e. not proportional to outcome probability)

16. Ruhm’s Roulette Wheel – Excerpt (2) from Exhibit 1

17. Price Risk Factor = _____________________ Discounted Probability Ruhm’s Roulette Wheel – Risk Factor

18. Ruhm’s Roulette Wheel – Excerpt (3) from Exhibit 1

19. Ruhm’s Generalized Risk Factor “w” w = Risk Neutral p.d.f. Real World p.d.f.

20. Ruhm’s Main Derivative ExampleExhibit 2: Risk Discount as a Function of Strike

21. Wacek Discussion

22. Wacek Discussion • Ruhm’s paper is a fascinating attempt to make implications of B-S “risk-neutral” pricing framework clear • Mapping to roulette wheel makes clear how strange “risk-neutral” prices are compared to expected value prices

23. Wacek Discussion • Quibble with title – too categorical • Should be “Distribution-Based Pricing Formulas Are Not Always Arbitrage-Free” • (Arbitrage depends on the market not on the formula)

24. Wacek Discussion • Ruhm focuses on buyer’s perspective • Risk-neutral pricing poses challenge to seller • Discussion will address seller’s perspective

25. Wacek Discussion “Ruhm’s conclusions about his ‘risk discount’ function and the buyer’s motivation, which have the appearance of generality, depend on certain of his assumptions.”

26. Wacek DiscussionPrice vs. Cost – Seller’s Perspective • Price does not always depend on probability distribution of outcomes • (Unhedged) cost doesalways depend on distribution of outcomes

27. Wacek DiscussionOverview • Discussion divided into three sections:1. Distinction Between Price and Cost – Seller’s Perspective2. Value for Money – Buyer’s Perspective3. w(s) function (Ruhm’s “risk discount function”)

28. Wacek DiscussionDistinction Between Price and CostSeller’s Perspective

29. Wacek DiscussionRuhm Roulette Wheel Prices for $100 Payoff

30. Wacek DiscussionRuhm Roulette Wheel Expected Costs for $100 Payoff

31. Wacek DiscussionRuhm Roulette Wheel – Number 30 • Premium for $100 payoff (per Ruhm) is $2.08 • Average payoff cost is $2.63 • Even with interest (unusual feature of this wheel!) expected casino result is a loss • Number 30 not exceptional – premium ≠ cost for almost all numbers

32. Wacek DiscussionExcerpt from Wacek’s Exhibit 1

33. Wacek Discussion • Time to get out the hedging instructions! • By following hedging instructions, expected payoff cost for any given number together with the hedging gain/loss will match the arbitrage-free premiums for that number • Low numbers – hedging produces expected losses • High numbers – hedging produces expected gains • Hedge eliminates the adverse selection problem

34. Wacek Discussion • Remember the roulette wheel is not governed by the laws of physics, but by stock prices mapped to the numbers on the wheel • Because the wheel outcomes are the results of “Black-Scholes conditions”, the payoff costs can be hedged • The payoff costs of a physical roulette wheel cannot be hedged in this way (another example of where outcome-based pricing will apply)

35. Wacek DiscussionValue for MoneyBuyer’s Perspective

36. Wacek DiscussionValue for Money – Buyer’s Perspective • Why would anyone place bets on low numbers? • High number bets have discounted price • Low number bets have surcharged price • From this Ruhm concludes high number bets are motivated by “investment”, low number bets by “insurance” psychology • There is a simpler explanation

37. Wacek DiscussionValue for Money – Buyer’s Perspective • Ruhm overlooked importance of his assumptions about the expected return, E, on the stock • His E = 10% (> r = 4%) • If r < E, pattern reverses (higher number bets surcharged, low number bets discounted)

38. Wacek DiscussionValue for Money – Buyer’s Perspective Excerpt (2) from Wacek’s Exhibit 1* *Based on Ruhm’s assumptions

39. Wacek DiscussionValue for Money – Buyer’s Perspective • Suppose E = 0% (< r = 4%) • Now arbitrage-free price for 00-17 (“put”) is $43.08 • Arbitrage-free price for 18-36 (‘call”) is $53.08 • “Puts” are discounted, “calls” are surcharged

40. Wacek DiscussionValue for Money – Buyer’s Perspective • No one knows true value of E • Investors who believe E > r may find “calls” (but not “puts”)attractive to buy • Investors who believe E < r may find “puts” (but not “calls”)attractive to buy • Not necessary to appeal to differences in risk aversion or “qualitative nature of the risk” to account for buyer behavior

41. Wacek DiscussionValue for Money – Buyer’s Perspective Seller can be indifferent to true value of E, provided he hedges.

42. Ruhm’s Roulette Wheel – Modified to E = 0%

43. Wacek DiscussionRisk Discount Function, w(s)

44. Wacek DiscussionRuhm’s Risk Function w(s)

45. Wacek DiscussionRuhm’s Risk Function w(s) • w = Risk Neutral p.d.f. Real World p.d.f. • Ruhm describes w as function of s, but . . . • It also depends on E (and other parameters) • If E < r, w(s) has positive slope (vs. Ruhm’s negative slope)

46. Wacek DiscussionRuhm’s Risk Function w(s) Counterexample w($90, 10%) = 0.3776 = 1.083 (Surcharge) 0.3487 w($90, 0%) = 0.3776 = 0.966 (Discount) 0.3909

47. Wacek DiscussionRuhm’s Risk Function w(s)Excerpt from Wacek’s Exhibit 2

48. Wacek DiscussionRuhm’s Risk Function w(s) • w is a function of E as well as s • E is unknowable (hence not unique) • Therefore, w is not unique • Cannot use w to draw conclusions about motivation for investor behavior

49. Wacek DiscussionInvestor Motivation • If Ruhm thinks a stock will go up at more than the risk free rate [negatively sloped w(s)], he will see my purchase of a “surcharged” put as evidence of risk-averse behavior (suggesting “insurance” orientation) • However, if I expect the stock to decline or trade sideways [positively sloped w(s)], then I believe I am buying the put at a discount – a good investment • No unique w(s) function independent of E that can tell us whether a risk is viewed as investment or insurance • Ruhm overreaches in conclusion about usefulness of w(s) • Only if we know an investor’s w(s) (based on his view of E) could we correctly judge whether he is “investing” or “insuring”.

50. Wacek DiscussionsConclusions • Ruhm’s roulette wheel is excellent metaphor • Critical distinction between price and cost • Wrong to say “calls” priced at discount in risk neutral framework, etc., without being clear this depends on E > r • Cannot use w to draw conclusions about motivation for investor behavior unless investor belief about E vs. r is known