Abstract

The textbook says that: given the price of the numerair at time t and T which is N(t) and N(T), the prices of a derivative financial instrument at the both time which is P(t) and P(T) has the following relationship where E is the probability measure induced by the Numeraire:

P(t)/N(t) = E[ P(T)/N(T) ]

To accomplish the above conclusion, usual textbooks have some rigorous probability and stochastic process theory first in some chapters. However, in this article, I will have a simple rational to accomplish the same goal. Further, we will claim that the probability and the stochastic process shall not be the central theme. Instead, the hedge plan is the key role.

My rational

By hedge plan, we mean a set of what to do at what state; i.e., a dynamic hedging deals strategy based on the filtration. Suppose that we have a hedge plan which will offset the value P(T) in such a way that the total value of the whole portfolio at time T is proportional to N(T), i.e.

There is a number k such that for any scenario s, P(T) + H(T) = k.N(T)

Since all the hedging deals are executed at market-fair condition, the p(t) must be k.N(t), otherwise by the arbitrage argument, say, p(t)> k.N(t), we can:

1. short 1 unit of the financial derivative and long k unit of Numeraire at time t
2. combined with the hedge plan
3. short k unit of Numeraire and buy 1 unit of financial derivative at time T

we get a riskless money: p(t)-k.N(t) at time t.

Note that the above argument has nothing to do with the probability measure; the central theme in the above argument is the hedge plan. However, if we impose a special probability measure q such that Eq[ H(T)/N(T) ] is zero, then we have:

There is a number k such that for every scenario, ( P(T) + H(T) )/ N(T) = k

Since k is irrelevant to the scenarios, we can take the expectation under any probability measure and get:

k = E[ ( P(T) + H(T) )/ N(T) ]

And under the specific probability measure q, k = Eq[ P(T)/N(T) ]. Since P(t)=k.N(t), we get:

P(t) = Eq[ P(T)/N(T) ] .N(t) , i.e., P(t)/N(t) = Eq[ P(T)/N(T) ]

In other words, P(.)/N(.) is martingale.

What if some action is allowed in the term?

Extend to the derivatives in which some choice of action is allowed. Let I(T) denote the filtration up to time T and g(I(T)) denote the choice of action based on the filtration. Suppose we can:

With the g(.) and a proper hedge plan,

P(T, g(I(T))) + H(T) is proportional to N(T) for all scenario
i.e. exist k such that for all scenario, P(T, g(I(T))) + H(T) = k.N(T)

To get a best of P(t), we shall get a best k because of P(t)=k.N(t). Because the measure q will induce Eq[ H(T)/N(T) ]=0 for all hedge deals' profit H, we have:

k = max( Eq[ P(T, g(I(T))) / N(T)] )
     g

and we get:

P(t) = max( Eq[ P(T, g(I(T))) / N(T)] ).N(t)
        g
i.e., P(t)/N(t) = max( Eq[ P(T, g(I(T))) / N(T) ] )
                   g

What if some target redemption is in the term?

Extend to the derivatives in which some target redemption is present. Denote the target redemption time by J; J must be a random variable based on the filtration.

Under a scenario where the derivative is expired at time J, we can postpone the value at time J by long the position of Numeraire at P(J)/N(J) unit and short the position back and get the value P(J)/N(J).N(T) at time T. Denote the value of the new derivative formed this way by L(). Therefore, the value of the original derivative shall be

Eq[ L(T)/N(T) ] . N(t)

which is Eq[ P(J)/N(J) ] . N(t) because L(T) = P(J)/N(J).N(T)

Therefore, this formula:

P(t)/N(t) = Eq[ P(J)/N(J) ]

What if some contingent cash flow before the maturity is in the term?

Follow the same logic in the above target redemption case.

Calibration

Calibration is in fact this formula Eq[ H(T)/N(T) ] = 0

Why? Suppose we evaluate a trade A and get its price or premium p, i.e.

p = Eq[ A(T)/N(T) ]

We denote by H the original hedge trade A with its price p, then of course

Eq[ H(T)/N(T) ] = 0

In other words, the calibration process is simply a process to calibrate the parameters of the model so that a specific measure q can be found. Ideally, if we can have a model with n degree of freedom (n parameters), then we can calibrate the model to n independent hedge trades.

On the other hand, if we have a model with less degree of freedom than the number of hedge trades, we will appeal to some calibration-error-minimization procedure, such as least of sum of square, to get a set of parameters of the model of best fit.

Practically, we shall calibrate the current available derivatives and simple underlying hedge trade relevant to the derivative in question, and assume those hedge trades triggered in the future shall be executed with zero profit condition in the model.