Understanding Option Pricing Bubbles: Models and Implications on Financial Markets
This article explores the phenomenon of option pricing bubbles, particularly through the lens of various stochastic volatility models, including CIR, CEV, and Heston models. It defines what constitutes a "bubble" in asset pricing and presents new solutions for modeling such phenomena. Key findings include conditions that prevent underlying assets from being dominated in diffusion models, the implications of linear risk premiums, and the relationship between option and stock price bubbles. The paper aims to clarify the mechanisms underlying these bubbles and provides insights for investors and financial theorists.
Understanding Option Pricing Bubbles: Models and Implications on Financial Markets
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Presentation Transcript
Options and Bubble Written bySteven L. Heston Mark Loewenstein GregoryA. Willard Present byFeifei Yao
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.”
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 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.
CIR Model • 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 : G2 is nonnegative and less than G1 prior to maturity
CIR Model • There is no equivalence (local martingale measure ) Given Under measure P Under measure Q
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
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
CEV Model • Solution where • The p1 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: G1- G2 Stock bubble: Set K= 0 in G1 formula so that G1=S
Stochastic Volatility Model • Stock price • Stochastic variance • Denote the time T payout of a European derivative by F(ST, VT) , PDE • Subject to
Stochastic Volatility Model • Bubble: G2(S, V, t) = G1( 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(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
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
