Assume that a derivative product has been developed such as it\'s value is equal

Question

Assume that a derivative product has been developed such as it's value is equal to the square root of the underlying asset, as such:

The price of the underlying asset follows an Ito Process as per below:

The asset is today qupted at 20,00 EUR, and has the following characteristics:

a) What is the expected value and the standard deviation for alterations in the price of the derivative asset G?

b) What is the expected value and standard deviation for the rendibility of the derivative asset G?

c) Assuming that the process is applicable to the daily observations, what would be the price of the underlying asset be tmorroe, if the aleatory variable was equl to +0,750?

PS. I hava an exam this week and since I'm graduated in International Relations I never learner how to do derivatives, which apparently you need to do in exercises a) and b). Could you please explain to me step by step? I have the final solution for each question but now the 'how to get there' part. Many thanks.

Explanation / Answer

Underlying asset is a term used in derivatives trading. Options are an example of a derivative. A derivative is a financial instrument with a price that is based on a different asset. The underlying asset is the financial instrument on which a derivative's price is based.

Derivative assets analysis enjoys an unusual status; it is a recently developed,
relatively complex tool of economic analysis, faithful to the core of economic theory,
and widely used to make real-life decisions. For example, most traders on the floors of
options exchanges use arbitrage-based option values at least as benchmarks in
establishing market prices and in constructing replicating strategies to hedge their
option positions.

The payoff Junction of a derivative asset is a mathematical representation of the
relation between the payoffs of the derivative and the prices or payoffs of the
underlying assets. One of the simplest and oldest examples is a forward contract, an
arrangement whereby the seller agrees to deliver to the buyer a specified asset on a
specified future date at a fixed price, to be paid on the delivery date. Typically, no
money changes hands until the delivery date. For example, the payoff function for a
Standard & Poors 500 Index forward contract is
C* =S* – F
where C* is the value of the forward contract on the delivery date, S* is the level of
the S&P 500 Index at that time, and F is the previously agreed upon forward price.
In this case, the seller is obligated to deliver on the future delivery date the cash value on that future date of the S&P 500 Index (S*), while the buyer is obligated to pay the
currently agreed fixed price (F) on the delivery date.
Payoff functions of this type are particularly easy to analyze. The present value
of the derivative asset (C) is simply the present value of the future index value, which
is defined as V(S*), minus the present value of paying out the forward price for
certain on the delivery date, defined as V(F). Using V, the valuation operator
C = V(C*) = V(S* – F) = V(S*) – V(F).
The final equality follows from this: if the payoffs from the constituent variables can
be purchased and sold separately and no arbitrage opportunities exist, then the
present value of the sum of the variables equals the sum of their present
values—another version of the whole equaling the sum of its parts and an application
of the "value additivity theorem" discussed by Varian in this journal.
The next step is to realize that the present value of the index is its current value
(S) minus the amount of money which would need to be set aside currently to pay
any dividends (D) through the delivery date. In short, V(S*) = S – D. Moreover, if r
is one plus the annualized riskless rate of interest and t is the time to the delivery date,
then the present value of paying the forward price for certain on the delivery date is
Fr–t. Putting this together
V(C*) = V(S*) – V(F) = (S – D) – Fr–t.
Clearly, the higher F, the amount that must be paid at the termination of the
contract, the lower the value (to the buyer) of the contract. In most real-life situations,
the forward price F is set so that no money need change hands at the inception of the
contract; that is, F is chosen so that the present value of the forward contract is zero.
Therefore,
V(C*) = (S – D) – Fr–t = 0 F = (S – D)rt
and the forward price is determined.
The replicating strategy for the forward contract is clear from the payoff
function. The problem is to arrange to receive the index value at the delivery date and
simultaneously incur a debt requiring payment of F dollars at that time. To do this an
investor could buy the underlying index, sell off rights to any dividends prior to the
delivery date, and borrow the present value of the forward price.2
More generally, this simple analysis suggests that whenever the payoff function is
linear in the prices of the underlying assets, an investor can hope to replicate the
derivative asset by a buy-and-hold position in the underlying assets, possibly supplemented
with riskless borrowing or lending.

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