# A note on closed geodesics for a class of non-compact by Secchi S.

By Secchi S.

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Consider the following option. The option holder can choose to ‘exercise’ it at any time Moreover, he/she can choose to ‘exercise’ it for a second time Any exercise times are recorded, but the actual at any time payoff is delayed up to time T. At this time T, the option holder obtains the amount of money equal to Fτ,θ(S0, S1,…, ST) (or obtains some benefits with this market price). Does this option belong to any of the classes described above? 45 may have economical sense. For instance, take and consider an electricity market, where St is the price of an electricity unit at time t.

2 We say that the strategy is self-financing if Xt+1−Xt=γt(St+1−St), t=0, 1,…. 3) Here X0>0 is the initial wealth at time t=0. For example, for the trivial risk-free strategy, when γt≡0, the corresponding total wealth is Xt≡X0. , are not taken into account. 3 A discrete time bond-stock market model A more realistic model of the market with non-zero interest rate for borrowing can be described via the following bond—stock model. We introduce a model of a market, consisting of the risk-free bond or bank account with price Bt and the risky stock with the price St, t=0, 1, 2,….

Equivalent to the original measure P). A market model is and E*ψ2<+∞, is replicable with said to be complete if any claim ψ such that ψ is some initial wealth. 25). 34 If a market model is complete and there exists an equivalent riskneutral measure, then this measure is unique (as a measure on ). and let Proof. Let ( is the indicator function of A). By the assumption, the claim ψ is replicable with some initial wealth By is uniquely defined. 27, this risk-neutral measure P*. 35 Let a market model be such that there exists an equivalent risk-neutral probability measure P*.