Options and BubbleWritten 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 ModelWith 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.

Bond’s value G(r,t) satisfies the valuation PDE

Define:

One solution is using

where

CIR Model

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 ModelStock-Price process

A European call option pays max(ST - K,0) at maturity T. PDE

Boundary conditions

ZQ : Local stock return equal to r under a given equivalent change of measure Q

CEV ModelSolution

where

The p1 satisfy

Subject to

CEV ModelUsing the probability density produce a new formula for CEV model

Cheapest nonnegative solution subject to the boundary condition

CEV ModelThere 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

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