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Distribution-Based Pricing Formulas are not Arbitrage-Free

Distribution-Based Pricing Formulas are not Arbitrage-Free. The Risk Discount Function The Casualty Actuarial Society Spring 2003 Meeting Marco Island, Florida. Summary of Main Points. Roulette-like binary derivatives Arbitrage-free pricing Same probabilities  different payoffs

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Distribution-Based Pricing Formulas are not Arbitrage-Free

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  1. Distribution-Based Pricing Formulas are not Arbitrage-Free The Risk Discount Function The Casualty Actuarial Society Spring 2003 Meeting Marco Island, Florida

  2. 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

  3. 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.

  4. 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.

  5. 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.

  6. 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

  7. 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

  8. 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%

  9. 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%

  10. 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?

  11. 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

  12. 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%

  13. 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.

  14. 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

  15. 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)

  16. 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.

  17. 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.

  18. 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

  19. 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

  20. 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.

  21. 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)

  22. 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.

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