elle
(Elle)
#1
The beauty of this approach is that it carries over to even the most complex economies
with, for example, continuous time modeling (i.e., a continuum of points in time to
consider), large numbers of risky assets, complex derivative payoffs, etc.
Therefore, consider a general market model in discrete time:
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A general market model ℳ in discrete time is a collection of:
A finite state space
A filtration
A strictly positive probability measure P defined on ℘()
A terminal date T ∈ ℕ, T < ∞
A set of K + 1 strictly positive security price processes
We write ℳ = {(, ℘(), ೨,P),T,}.
Based on such a general market model, we can formulate the Fundamental Theorem of
Asset Pricing as follows:
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Consider the general market model ℳ. According to the Fundamental Theorem of Asset Pricing, the following
three statements are equivalent:
There are no arbitrage opportunities in the market model ℳ.
The set ℚ of P-equivalent martingale measures is nonempty.
The set ℙ of consistent linear price systems is nonempty.
When it comes to valuation and pricing of contingent claims (i.e., options, derivatives,
futures, forwards, swaps, etc.), the importance of the theorem is illustrated by the
following corollary:
If the market model ℳ is arbitrage-free, then there exists a unique price associated with any attainable (i.e.,
replicable) contingent claim (option, derivative, etc.) VT. It satisfies , where e
–rT
is
the relevant risk-neutral discount factor for a constant short rate r.
This result illustrates the importance of the theorem, and shows that our simple reasoning
from the introductory above indeed carries over to the general market model.
Due to the role of the martingale measure, this approach to valuation is also often called
the martingale approach, or — since under the martingale measure all risky assets drift
with the riskless short rate — the risk-neutral valuation approach. The second term might,
for our purposes, be the better one because in numerical applications, we “simply” let the
risk factors (stochastic processes) drift by the risk-neutral short rate. One does not have to
deal with the probability measures directly for our applications — they are, however, what
theoretically justifies the central theoretical results we apply and the technical approach
we implement.
Finally, consider market completeness in the general market model:
The market model ℳ is complete if it is arbitrage-free and if every contingent claim (option, derivative, etc.) is
attainable (i.e., replicable).
Suppose that the market model ℳ is arbitrage-free. The market model is complete if and only if ℚ is a singleton;
i.e., if there is a unique P-equivalent martingale measure.
