Risk Neutral Pricing
Thursday, May 12, 2011
2:03 PM
We discussed last time how the Girsanov theorem allows us to reweight probability
measures to change the drift in an SDE.
This is used to construct a reweighted "equivalent risk-neutral" or "equivalent
martingale" measure in financial market models in the following way:
For a single asset, generalized GBM price model:
The Girsanov theorem tells us that there exists an equivalent probability
measure
under which the price trajectories follow:
where R(t) is the interest rate (also allowed to be a
general random process adapted to the filtration
generated by the Wiener process) that describes how
cash appreciates in value:
The Wiener process under the new risk-neutral
probability measure is related to the original Wiener
process by the transformation:
(and so really we need the volatility to be bounded from
below for this to work):
Under the risk-neutral probability measure, every asset, including cash, is a martingale when
discounted by the interest rate.
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This in particular means that no matter how we manage a portfolio of assets (including cash),
the portfolio will behave as a martingale.
If we hold
shares of the underlying asset
then portfolio value:
is always automatically a martingale no matter how we manage it.
In particular, if T is the expiry time of the financial option, then:
Now suppose that I have a perfect hedging strategy using such a portfolio so that I can perfectly
replicate the payoff of some financial option that I'm considering:
For a European option, this looks like:
but one can contemplate more general financial options.
Under this assumption:
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And from this equation, it would seem that the fair price to charge for the financial option,
at time t , is the value of the portfolio of assets and cash at that time that would perfectly
replicate the option. (Otherwise there's arbitrage for the smarter person).
The the value of the option at time t, using the fact that D(t) is measurable with respect to
Risk-neutral pricing formula
Recall that this risk-neutral pricing formula can only be expected to hold when:
a. one can find an equivalent probability measure that makes the total market look risk neutral
b. an active hedging strategy exists (a portfolio of cash and assets can replicate the financial option
payoff) (market completeness)
How does one know whether such a hedging strategy exists in general? The abstract way to
approach this is through the Martingale Representation Theorem which says the following
(Shreve Sec. 5.3)
If M(t) is a martingale with respect to the filtration generated by Wiener process
and M(t) is adapted to this filtration, then there exists a stochastic process
which is also adapted to this same filtration such that
This theorem is used in the following way: If we define V(t) as above, and if we specify that the
filtration in our market model is just the filtration generated by the Wiener process under the riskneutral probability measure (note this excludes allowing other sources of uncertainty), then we
can apply the Martingale Representation theorem to the discounted option price:
Active hedge:
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Active hedge:
How does one actually use the risk-neutral pricing formula?
To apply it for the European call option with geometric Brownian motion model with deterministic timedependent coefficients:
One could in principle evaluate this risk-neutral expectation by inserting the Radon-Nikodym
derivative (reweighting factor Z) and then taking averages with respect to the original probability
measure.
But an easier alternative is actually to express S(T) in terms of the Wiener process under the riskneutral probability measure, and complete the calculation in risk-neutral world.
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Now the calculation is straightforward. One recovers
the Black-Scholes-Merton formula but with:
Note the Black-Scholes-Merton PDE is indeed the Kolmogorov backward equation for the risk-neutral
probability measure.
Even though the geometric Brownian motion model is used as the basis for option pricing, people
don't really believe it.
Volatility smile:
The reason for the discrepancy is generally attributed to prices having larger probabilities for
large fluctuations than the geometric Brownian motion model would give.
Several modeling frameworks try to take this into account:
○ stochastic volatility models:
• jump-diffusion models for price shocks
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The simplest version is framed in terms of a compound
Poisson process.
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c.f. Paul and Bauschnagel, Stochastic Processes: From Physics to Finance, Ch. 5
These jump-diffusion models are still Markovian, so can still use the Markov process theory with
infinitesmal generator
How is the risk-neutral option-pricing formula affected?
○ Girsanov theorem transformation to a risk-neutral probability measure still seems to be OK
under the usual conditions
○ Martingale representation theorem exists that might suggest market completeness, but be
careful: if one wants to conclude this, then one would need the option price and the trading
strategy to be adapted to the filtration generated jointly by the Brownian motion and the jump
process
But in practice, it would seem more reasonable to only
impose that:
For the case of jump-diffusions,
Then one cannot perfectly hedge the risk of a financial option. The general
method for finding prices then involves measuring the unhedged risk (say through
a variance of the amount of money at risk) and charge a "risk premium" for that.
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