Question

3. Each of the following market models consist of the savings account Be.1 and the stock St, 0 t 1. Decide which market model is free from arbitrage by using the First Fundamental Theorem of Asset Pricing. In cases when the model does not admit arbitrage explain how to obtain EMM. In cases when the model admits arbitrage, suggest an arbitrage strategy. (a) S Se 2B (b) St Soe02 sint. sin t (c) S Soe01(B-t)

Answer 1

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Answer:

Explanation:

As per question given

Part a)

In the market model S(t)=S(0)e^0.1(B(t)-t) the stock of current period S(t) is a function of the stock from the previous period S(0).Notice that S(t) increases at an exponential rate(e) with respect to the savings rate(B(t)) or in other words, as B(t) increases in the model S(0) also increases considerably and hence S(t) also goes up steadily.It indicates that either the consumption level in the market model given relatively less resulting in higher stock level in the previous period S(0) or this market model offers risk free or lower risk investment scheme which might possibly ensure higher returns.Hence, the model indicates arbitrage free investment scheme or low arbitrage in which investment is relatively safer compared to the model in part b).In this exponential function given in the model it is clear that the current stock S(t) will increase at a guranteed rate with no or very low chance of uncertain fluctuations in S(t).Therefore,any risk aversive individual would prefer investing in this model due to no arbitrage or very low arbitrage involved.

Part b)

In this market model,S(t)=S(0)e^0.2sint presents an exponential sin function implying that in this model,S(t) has relatively more fluctuations and does not guarantee a steady increase.As any sin function represents a parabolic structure,S(t) might increase at some stage but again may drop and then again increase reaching multiple maximum and minimum points indicating uncertain or unpredictable fluctuations.Therefore, the arbitrage level is higher in this model in comparison to the model in part a).

Now as this market model does not provide any guaranteed return on any investment,different arbitrage strategy can be employed to reduce the investment risk and ensure desired return.An investor can use spot interest rate by which he/she can agree on a fixed interest rate to safeguard against against any future interest rate fluctuations due to changes in inflation or any other economic factors.Alternatively,another risk free investment strategy can be contract trading by which the investment structure or the stock return will be free from any potential market fluctuations such as interest rate alterations,inflationary preasures,changes in financial policy and so forth.Furthermore,Strike Arbitrage can be an effective strategy under two or multiple market model scenario which have different risk levels but with same underlying security and expiration date.This strategy is basically effective when the difference in the expected return from the models is less than the difference between their extrinsic measures.

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