elle
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Fundamental Theorem of Asset Pricing
The Fundamental Theorem of Asset Pricing is one of the cornerstones and success stories
of modern financial theory and mathematics.
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The central notion underlying the
Fundamental Theorem of Asset Pricing is the concept of a martingale measure; i.e., a
probability measure that removes the drift from a discounted risk factor (stochastic
process). In other words, under a martingale measure, all risk factors drift with the risk-
free short rate — and not with any other market rate involving some kind of risk premium
over the risk-free short rate.
A Simple Example
Consider a simple economy at the dates today and tomorrow with a risky asset, a “stock,”
and a riskless asset, a “bond.” The bond costs 10 USD today and pays off 10 USD
tomorrow (zero interest rates). The stock costs 10 USD today and, with a probability of
60% and 40%, respectively, pays off 20 USD and 0 USD tomorrow. The riskless return of
the bond is 0. The expected return of the stock is , or 20%. This is
the risk premium the stock pays for its riskiness.
Consider now a call option with strike price of 15 USD. What is the fair value of such a
contingent claim that pays 5 USD with 60% probability and 0 USD otherwise? We can
take the expectation, for example, and discount the resulting value back (here with zero
interest rates). This approach yields a value of 0.6 · 5 = 3 USD, since the option pays 5
USD in the case where the stock price moves up to 20 USD and 0 USD otherwise.
However, there is another approach that has been successfully applied to option pricing
problems like this: replication of the option’s payoff through a portfolio of traded
securities. It is easily verified that buying 0.25 of the stock perfectly replicates the option’s
payoff (in the 60% case we then have 0.25 · 20 = 5 USD). A quarter of the stock only
costs 2.5 USD and not 3 USD. Taking expectations under the real-world probability
measure overvalues the option.
Why is this case? The real-world measure implies a risk premium of 20% for the stock
since the risk involved in the stock (gaining 100% or losing 100%) is “real” in the sense
that it cannot be diversified or hedged away. On the other hand, there is a portfolio
available that replicates the option’s payoff without any risk. This also implies that
someone writing (selling) such an option can completely hedge away any risk.
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Such a
perfectly hedged portfolio of an option and a hedge position must yield the riskless rate in
order to avoid arbitrage opportunities (i.e., the opportunity to make some money out of no
money with a positive probability).
Can we save the approach of taking expectations to value the call option? Yes, we can. We
“only” have to change the probability in such a way that the risky asset, the stock, drifts
with the riskless short rate of zero. Obviously, a (martingale) measure giving equal mass
of 50% to both scenarios accomplishes this; the calculation is . Now,
taking expectations of the option’s payoff under the new martingale measure yields the
correct (arbitrage-free) fair value: 0.5 · 5 + 0.5 · 0 = 2.5 USD.
The General Results
