Distribution-Based Pricing Formulas are not Arbitrage-Free

The Risk Discount Function

The Casualty Actuarial Society

Spring 2003 Meeting

Marco Island, Florida

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Summary of Main Points

• Roulette-like binary derivatives

• Arbitrage-free pricing

Same probabilities different payoffs

• Distribution-based formulas cannot model this

• Risk Discount Function characterizes risk measurement

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Overview of Main Points

• Derivatives can be created from call options that are equivalent to roulette-like bets on the stock price.

• Probabilities and payoffs are calculated with the Black-Scholes pricing model, which is arbitrage-free.

• Unlike actual roulette, two bets with identical probabilities will generally have different payoff ratios.

• A distribution-based risk load formula would assign the same risk load to bets having identical probabilities, so it cannot reproduce Black-Scholes prices.

• The result holds in general: Risk load formulas that use only the outcome distribution do not produce arbitrage-free prices.

• The “Risk Discount Function” characterizes risk measurement, and distinguishes between investment and hedging derivative types.

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Derivative-as-Wager Concept

• The ray derivative is binary.

• Binary derivatives are like bets.

• The bet is on whether the stock price will be above 120 at expiration, or not.

• Similar to a roulette bet, with different odds and payoff.

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Since it’s a bet…

What are the odds, and what’s the payoff?

• Black-Scholes implies lognormal prices, so can use a normal table to get odds.

• Probability of winning = 33%

• Black-Scholes price = 0.2551

• You bet $25.51, you have a 33% chance of winning, you get $100 in a year if you win.

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Is this a good bet?

• NPV Expected Gain Analysis:– PV($100) at 4% = $96.15– Expected = 33% ($96.15) = $31.73– Expected Net @ PV = $31.73 - $25.51 = $6.22

• Return Analysis– Exp’d Return = $33.00 / $25.51 – 1 = 29% – Choice: 4% risk-free or 29% exp’d + risk

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Risk Discount Concept

• Bond prices are discounted based on risk.• More risk higher yield more discount

relative to the price of a risk-free bond.• Example: 5% 1-year, when risk-free = 4%:

– Price = $1,000 / 1.05 = $952.38

– Risk-free price = $1,000 / 1.04 = $961.54

– Discount Factor = $952.38 / $961.54 = 99.05%

– Discount Factor = Price / Discounted Face Value

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Risk Discount Concept

• Same reasoning applies to any instrument:– Expected yield > risk-free discount in price– Risk Discount Factor = Price / PV[Expd Value]– Risk Discount Factor = $25.51 / $31.73 = 80%– Can also ratio risk-free / expected yield:– Risk Discount Factor = 1.04 / 1.29 = 80%

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Risk Discount Factor for Binary

• Rays are binary derivatives

• Payoff = $1 if win, $0 if loss

• Expected Value = Probability of win

• Risk Discount = Price / PV[Probability]

• Risk Discount = 0.2551 / (.33/1.04) = 80%

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Segment Derivative

• Bet on: Expiration price between 120 and 150

• Probability = 33.00% - 11.82% = 21.17%

• Price = 0.2551 – 0.0819 = 0.1732

• Lower price, lower odds than the ray

• Is this a better or worse bet than the ray?

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Segment Derivative

• Analysis: Better or worse than ray?– Exp’d Return = 0.2117 / 0.1732 – 1 = 22% – Win probability lower than ray more risk– Less expected return than the A*(120) ray– The ray would be a better bet– Risk Discount = 0.1732 / (0.2117 / 1.04) = 85%– Not as much discount in price as ray has

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Roulette

• Wheel with 38 equally-likely spaces, numbered “00” and “0” through “36”

• Probability of win = 1/38• $1 bet pays $36 (including return of $1 bet)• Same as binary derivative• Negative expected return:

– Expected return = (1/38)($36) – 1 = -5% – Risk Surcharge Factor = $1 / ($36/38) = 106%

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Map derivatives to roulette wheel

• You can choose boundary prices for segments and rays for any win probability.

• Split up the entire price range into 38 segments and one ray, so that each of them has the same 1/38 probability.

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Map derivatives to roulette wheel

• Each segment / ray has the same odds as a roulette wheel space: p(win) = 1/38.

• All have the same outcome distribution:– P(value = $1) = 1/38– P(value = $0) = 37/38

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If the odds are just like roulette, how are the payoffs?

• Surprising fact: The spaces on this wheel all have different payoffs.

• Space “00” pays $25 (worst space)

• Space “12” pays $36 (like a normal wheel)

• Space “16” pays $38 (breakeven bet)

• Space “36” pays $60 (best space)

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Key points from example

• All spaces have the same probability distribution.

• All outcomes are determined by the same event, the stock price at expiration (like the wheel’s spin).

• All have different arbitrage-free prices.

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Why doesn’t everyone bet on space 36?

• One reason: People need hedges against economic risks that naturally arise in the course of business and living.– Risk of loss of equity in a business.

– Risk of loss of house in a hurricane.

• Same reason a business founder sells stock in an IPO for cash: less expected return, but hedge against potential loss

• Lower-numbered spaces are hedges, like put options.

• Higher-numbered spaces are speculative investments.

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Arbitrage

• Roulette: Only one side available

• Players, not casino, set the bets

• If you could bet from the casino’s side, you’d bet on every number certain win

• This is arbitrage

• Securities markets: Either side available (short or long), and player sets the bets

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Arbitrage

• If all spaces have same payoff, it has to be $38, or else arbitrage is possible

• $38 payoff zero risk load

• Only two possibilities:– All spaces pay $38 for $1 bet– Spaces have varying payoffs

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Distribution-based risk load formulas are not arbitrage-free

• A distribution-based risk load formula gives the same risk load to risks that have the same distribution.

• Unless the risk load = 0, this will not produce arbitrage-free prices.

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The Risk Discount Function

• The central function that describes:– when risk is compensated by return, and how

much (investment or speculation)– when risk assumption is surcharged (hedging or

insurance)

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Further Reading: Ruhm-Mango

• Paper presented at Bowles Symposium, April 2003 by David Ruhm and Donald Mango

• Ruhm-Mango theorem: Any formula that produces additive prices has a risk discount function at its core, which completely describes it (up to a scale factor).

• The underlying risk discount function is like a pricing method’s DNA. It contains all of the method’s risk-pricing information.

• All additive pricing formulas can be condensed to one:Price = W (E[R] + Cov[Z,R])

with Z = underlying risk discount function.• This formula produces Black-Scholes, CAPM, etc. prices.