Options and Bubble

1. Options and Bubble Written bySteven L. Heston Mark Loewenstein GregoryA. Willard Present byFeifei Yao

2. Definition • Option Pricing Bubble: An asset with a nonnegative price has a "bubble” if there is a self-financing portfolio with pathwisenonnegative wealth that costs less than the asset and replicates the asset's price at a fixed future date.”

3. Article Structure • New solutions for CIR, CEV and Heston Stochastic Volatility model • 3 Conditions to prevent the underlying assets from being dominated in diffusion models. • Findings & Consequences

4. CIR Model • With linear risk premium ϕ0+ϕ1r, where ϕ0 ϕ1 are constants • Riskless interest rate under P measure by • Assume • Given: A unit discount bond has a payout equal to one at maturity T.

5. CIR Model • Bond’s value G(r,t) satisfies the valuation PDE • Define: • One solution is using where

6. CIR Model • If inequality holds, but • Then a cheapest solution is • Note : G2 is nonnegative and less than G1 prior to maturity

7. CIR Model • There is no equivalence (local martingale measure ) Given Under measure P Under measure Q

8. CIR Model • G2 − G1 is negative, implying that arbitrage which bounded (>-1) temporary losses prior to closure • The original CIR bond price has a bonded asset pricing bubble since G1 exceeds the replicating cost of G2

9. CEV Model ZQ : Local stock return equal to r under a given equivalent change of measure Q • Stock-Price process • A European call option pays max(ST- K,0) atmaturityT. PDE • Boundary conditions

10. CEV Model • Solution where • The p1 satisfy • Subject to

11. CEV Model • Using the probability density produce a new formula for CEV model • Cheapest nonnegative solution subject to the boundary condition

12. CEV Model • There is an arbitrage even though an equivalent local martingale measure exists. • There are assets pricing bubbles on options values, as well as on the stock price. • Put-Call Parity or Risk-Neutral Option are mutually exclusive. Option bubble: G1- G2 Stock bubble: Set K= 0 in G1 formula so that G1=S

13. Stochastic Volatility Model • Stock price • Stochastic variance • Denote the time T payout of a European derivative by F(ST, VT) , PDE • Subject to

14. Stochastic Volatility Model • Bubble: G2(S, V, t) = G1( S, V, t) + Π(V, t) • Stock bubbles are not (mathematically) necessary for option bubbles.

15. Condition 1 to rule out bubbles • Absence of instantaneously profitable arbitrage • Ensures the price of risk is finite • Local price of risk (Sharpe ratio): • Example CIR

16. Condition 2 to rule out bubbles • Absence of money market bubble Under stock price is given by • The exponential local martingale has to be a strictly positive martingale

17. Condition 3 to rule out bubbles • Absence of stock bubbles • There exists an equivalent local martingale measure Q, and the Q-exponential local martingale is a Q-martingale Where

18. Findings & Consequences • A European-style derivative security pays F(ST) at time T. • The nonnegative solutions of G(S, Y, t) is Bubble for solution G The lowest cost of a replicating strategy with nonnegative value

19. Findings & Consequences • Risk-Neutral Pricing VS. Put-Call Parity • American Options • Lookback Call Option

20. Furthermore… • Personal Thoughts • Betting Against the Stock Market: Buying Bear Funds Placing Put Options Shorting Stocks