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On the practical point of view of option pricing

Nikolaos Halidias

Volatility (finance)
Binomial options pricing model
Abstract In this note, we describe a new approach to the option pricing problem by introducing the notion of the safe (and acceptable) price for the writer 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 this has a serious disadvantage because it 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 approach or some other approaches, one assumes that one can invest or borrow at the same risk-free rate <m:math xmlns:m=""> <m:mrow> <m:mi>r</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> r&gt;0 , which is not true in general. Even if this is the case, one can immediately observe 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 <m:math xmlns:m=""> <m:mrow> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> </m:math> u,d , trying to give a cheaper safe or acceptable price for the option.

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