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 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
Stock-Price process
ZQ : Local stock return equal
to r under a given equivalent
change of measure Q
A European call option pays max(ST - K,0)
at maturity T. 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.
Option bubble: G1- G2
Stock bubble: Set K= 0 in G1 formula so that G1=S
Put-Call Parity or Risk-Neutral Option are
mutually exclusive.
Stochastic Volatility Model
Stock price
Stochastic variance
Denote the time T payout of a European
derivative by F(ST, V T) , 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