Options and Bubble Written by Steven L. Heston Mark Loewenstein Gregory A. Willard Present by Feifei Yao

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

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

CIR Model With linear risk premium ϕ 0 + ϕ 1 r, where ϕ 0 ϕ 1 are constants Riskless interest rate under P measure by Assume Given: A unit discount bond has a payout equal to one at maturity T.

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

If inequality holds, but Then a cheapest solution is Note : G 2 is nonnegative and less than G 1 prior to maturity

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

CIR Model G 2 − G 1 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 G 1 exceeds the replicating cost of G 2

CEV Model Stock-Price process A European call option pays max(S T - K,0) at maturity T. PDE Boundary conditions Z Q : Local stock return equal to r under a given equivalent change of measure Q

CEV Model Solution where The p 1 satisfy Subject to

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

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: G 1 - G 2 Stock bubble: Set K= 0 in G 1 formula so that G 1 =S

Stochastic Volatility Model Stock price Stochastic variance Denote the time T payout of a European derivative by F(S T, V T ), PDE Subject to

Stochastic Volatility Model Bubble: G 2 (S, V, t) = G 1 ( S, V, t) + Π (V, t) Stock bubbles are not (mathematically) necessary for option bubbles.

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

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

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

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

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

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