There is no method of predicting the price of an option other than hedging strategies such as the binomial hedging strategy, the Black-Scholes hedging strategy and others. The price of an option is defined to be equal to the amount the seller needs to implement a hedging strategy. We will study these two basic hedging strategies in terms of their feasibility and we will see that the Black-Scholes hedging strategy is not feasible because this strategy demands instantaneously rebuilding the replicating portfolio. Consequently, the real world prices of the options are not relevant at all with the Black-Scholes hedging strategy! We will suitably redefine the binomial hedging strategy so that it will be practically useful and present other feasible and generally more effective hedging strategies with some of them are practically useful for options with no tradable underlying assets. Finally, we will mention some open questions related to the above.

In this note we describe a new approach to the option pricing problem by introducing the notion of the safe (and acceptable) for the writer price of an option, in contrast to the fair price used in the Black-Scholes model. Our starting point is that the option pricing problem is closely related with the hedging problem by practical techniques. Recalling that the Black - Scholes model does not give us the price of the option but the initial value of a replicating portfolio we observe easily that has a serious disadvantage because assumes the building of this replicating portfolio continuously in time and this is a disadvantage of any model that assumes such a construction. Here we study the problem from the practical point of view concerning mainly the over the counter market. This approach is not affected by the number of the underlying assets and is particularly useful for incomplete markets. In the usual Black-Scholes or binomial or some other approaches one assumes that one can invest or borrow at the same risk free rate $r>0$ which is not true in general. Even if this is the case one can immediately observes that this risk free rate is not a universal constant but is different among different people or institutions. So, the fair price of an option is not so much fair! Moreover, the two sides are not, in general, equivalent against the risk therefore the notion of a fair price has no meaning at all. We also define a variant of the usual binomial model, by estimating safe upward and downward rates $u,d$, trying to give a cheaper safe or acceptable price for the option.

In this note we study the computation of the minimum polynomial of a matrix $A$ and how we can use it for the computation of the matrix $A^n$. We also describe the form of the elements of the matrix $A^{-n}$ and we will see that it is closely related with the computation of the Drazin generalized inverse of $A$. Next we study the computation of the exponential matrix and finally we give a simple proof of the Leverrier - Faddeev algorithm for the computation of the characteristic polynomial.

In this note we study the binomial model applied to European, American and Bermudan type of derivatives. Our aim is to give the necessary and sufficient conditions under which we can define a fair value via replicating portfolios for any derivative using simple mathematical arguments and without using no arbitrage techniques. Giving suitable definitions we are able to define rigorously the fair value of any derivative without using concepts from probability theory or stochastic analysis therefore is suitable for students or young researchers. It will be clear in our analysis that if $e^{r \delta} \notin [d,u]$ then we can not define a fair value by any means for any derivative while if $d \leq e^{r \delta} \leq u$ we can. Therefore the definition of the fair value of a derivative is not so closely related with the absence of arbitrage. In the usual probabilistic point of view we assume that $d < e^{r \delta} < u$ in order to define the fair value but it is not clear what we can (or we can not) do in the cases where $e^{r \delta} \leq d$ or $e^{r \delta} \geq u$.