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

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Wacek Discussion of Ruhm Paper 2 Wacek Discussion Ruhm illustrates difficult concepts from financial economics in a refreshing way 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 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 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 We discuss distinction between price and cost and its significance, particularly for the seller

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Wacek Discussion of Ruhm Paper 3 Wacek Discussion Overview Ruhm also seeks to generalize about arb-free pricing of calls and puts 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 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? Why would anyone buy puts? He reasons that risks have a qualitative nature as either insurance or investment – puts reflect insurance nature 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 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 Explanation for investor behavior does not require “qualitative nature of risk” concept

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Wacek Discussion of Ruhm Paper 4 Wacek Discussion Overview Discussion divided into three sections: 1. Distinction Between Price and Cost – Seller’s Perspective 2. Value for Money – Buyer’s Perspective 3. w(s) function (Ruhm’s “risk discount function”) Discussion divided into three sections: 1. Distinction Between Price and Cost – Seller’s Perspective 2. Value for Money – Buyer’s Perspective 3. 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 This presentation will include a recap of key aspects of Ruhm’s original paper in order to put the discussion in context

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Wacek Discussion of Ruhm Paper 5 Summary Of Ruhm Paper

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Wacek Discussion 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.”

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Wacek Discussion of Ruhm Paper 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.”

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Wacek Discussion of Ruhm Paper 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.”

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Wacek Discussion of Ruhm Paper 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.”

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Wacek Discussion of Ruhm Paper 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.”

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Wacek Discussion of Ruhm Paper 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%

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Wacek Discussion of Ruhm Paper 12 “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% Ruhm’s Main Derivative Example Pricing Assume

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Wacek Discussion of Ruhm Paper 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

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Wacek Discussion of Ruhm Paper 14 Ruhm’s Roulette Wheel – Excerpt (1) from Exhibit 1 Roulette Wheel Space # Derivative Type Lower Strike (s) Upper Strike (t) Payoff Prob 00Segment0.0058.800.0263 0Segment58.8064.680.0263 1Segment64.6868.840.0263 2Segment68.8472.230.0263.…..........…..........…......... 33Segment153.10160.640.0263 34Segment160.64170.960.0263 35Segment170.96188.080.0263 36 Ray Ray188.08Infinity0.0263

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Wacek Discussion of Ruhm Paper 15 Ruhm’s Roulette Wheel - Pricing _____________ 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) Bet Amount

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Wacek Discussion of Ruhm Paper 16 Ruhm’s Roulette Wheel – Excerpt (2) from Exhibit 1 Roulette Wheel Space # Lower Strike (s) Upper Strike (t) Payoff ProbPrice 000.0058.800.02630.0384 058.8064.680.02630.0346 164.6868.840.02630.0330 268.8472.230.02630.0319..........…..........…..........… 33153.10160.640.02630.0194 34160.64170.960.02630.0187 35170.96188.080.02630.0179 36188.08Infinity0.02630.0162

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Wacek Discussion of Ruhm Paper 17 Ruhm’s Roulette Wheel – Risk Factor Risk Factor = _____________________ Discounted Probability Price

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Wacek Discussion of Ruhm Paper 18 Ruhm’s Roulette Wheel – Excerpt (3) from Exhibit 1 Roulette Wheel Space # Payoff ProbPrice Risk Factor 000.02630.0384151.91% 00.02630.0346136.68% 10.02630.0330130.35% 20.02630.0319126.00%....……....……....…… 330.02630.019476.65% 340.02630.018774.09% 350.02630.017970.67% 360.02630.016263.83%

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

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Wacek Discussion of Ruhm Paper 20 Ruhm’s Main Derivative Example Exhibit 2: Risk Discount as a Function of Strike

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Wacek Discussion of Ruhm Paper 21 Wacek Discussion

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

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

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

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Wacek Discussion of Ruhm Paper 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.”

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

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Wacek Discussion of Ruhm Paper 27 Wacek Discussion Overview Discussion divided into three sections: 1. Distinction Between Price and Cost – Seller’s Perspective 2. Value for Money – Buyer’s Perspective 3. w(s) function (Ruhm’s “risk discount function”) Discussion divided into three sections: 1. Distinction Between Price and Cost – Seller’s Perspective 2. Value for Money – Buyer’s Perspective 3. w(s) function (Ruhm’s “risk discount function”)

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Wacek Discussion of Ruhm Paper 28 Wacek Discussion Distinction Between Price and Cost Seller’s Perspective

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Wacek Discussion of Ruhm Paper 29 Wacek Discussion Ruhm Roulette Wheel Prices for $100 Payoff

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Wacek Discussion of Ruhm Paper 30 Wacek Discussion Ruhm Roulette Wheel Expected Costs for $100 Payoff

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Wacek Discussion of Ruhm Paper 31 Wacek Discussion Ruhm 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

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Wacek Discussion of Ruhm Paper 32 Wacek Discussion Excerpt from Wacek’s Exhibit 1

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Wacek Discussion of Ruhm Paper 33 Wacek Discussion Time to get out the hedging instructions! 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 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 Low numbers – hedging produces expected losses High numbers – hedging produces expected gains High numbers – hedging produces expected gains Hedge eliminates the adverse selection problem Hedge eliminates the adverse selection problem

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Wacek Discussion of Ruhm Paper 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 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 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) The payoff costs of a physical roulette wheel cannot be hedged in this way (another example of where outcome-based pricing will apply)

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Wacek Discussion of Ruhm Paper 35 Wacek Discussion Value for Money Buyer’s Perspective

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

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

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Wacek Discussion of Ruhm Paper 38 Wacek Discussion Value for Money – Buyer’s Perspective Excerpt (2) from Wacek’s Exhibit 1* *Based on Ruhm’s assumptions

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

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

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

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Wacek Discussion of Ruhm Paper 42 Ruhm’s Roulette Wheel – Modified to E = 0% Roulette Wheel Space # Lower Strike (s) Upper Strike (t) Payoff ProbPrice 000.0058.800.05260.0384 058.8064.680.04380.0346 164.6868.840.04050.0330 268.8472.230.03820.0319..........…..........…..........… 33153.10160.640.01640.0194 34160.64170.960.01550.0187 35170.96188.080.01430.0179 36188.08Infinity0.01200.0162

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Wacek Discussion of Ruhm Paper 43 Wacek Discussion Risk Discount Function, w(s)

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Wacek Discussion of Ruhm Paper 44 Wacek Discussion Ruhm’s Risk Function w(s)

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

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Wacek Discussion of Ruhm Paper 46 Wacek Discussion Ruhm’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

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Wacek Discussion of Ruhm Paper 47 Wacek Discussion Ruhm’s Risk Function w(s) Excerpt from Wacek’s Exhibit 2

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

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Wacek Discussion of Ruhm Paper 49 Wacek Discussion Investor 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) 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 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 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) 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”. 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”.

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Wacek Discussion of Ruhm Paper 50 Wacek Discussions Conclusions Ruhm’s roulette wheel is excellent metaphor Ruhm’s roulette wheel is excellent metaphor Critical distinction between price and cost 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 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 Cannot use w to draw conclusions about motivation for investor behavior unless investor belief about E vs. r is known

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