An option is a contract between two parties, the writer (seller) of the contract and the buyer. The buyer has the right but not the obligation to exercise the option at some time $T>0$ or even at any time at the time interval $[0,T]$. The payoff of the option that the buyer will take is an amount $H(S_1,S_2,\cdots ,S_d)$ where $H:{\mathbb{R}}_{+}^d\to {\mathbb{R}}_{+}$ and $S_1,\cdots ,S_d$ are the underlying assets. Some examples are $P_T=max\{S_T-K,0\}$ (call option), $P_T=max\{K-S_T,0\}$(put option), $P_T=max\{{max}_{t\in [0,T]}S\left(t\right)-K,0\}$ (call on maximum), $P_T=max\{S_1\left(T\right),S_2\left(T\right),K\}$ and $P_T=max\{S_1\left(T\right)-S_2\left(T\right),0\}$ (spread option) and many others.

For some options (contracts) there is the possibility of bankruptcy for the writer. For example the writer of a spread option may go bankrupt if the price of the asset $S_1$ reach high levels. The writer can buy a call option written on $S_1$ in order to eliminate that risk. However this is not the case for the call on maximum option for which the writer can buy a series of call options in order to shrink the possibility of bankruptcy but not to eliminate it.

It is clear that the writer of an option will accept to undertake the risk of selling it only for speculative reasons. So, in what price should the writer sell the option? The option pricing problem is equivalent to the hedging problem. The writer first designs a hedging strategy and then prices the option based on the amount that she need to follow that strategy. The purpose of this hedging strategy is to minimize the probability of bankruptcy and also to increase the probability of profit. Of course the competition will play a decisive role in the final sale price.

What do we mean by "has a good chance of profit"? The exact meaning will be given by the writer herself, so in general, a different writer will price a contract differently. At [5] we have given some practical ways of doing this and will refer to them below.

One way to hedge the risk of selling an option is to built a replicating portfolio. We have two main methods to construct replicating portfolios: the first one is in discrete time and the second is in continuous time. The main representative from the first category is the binomial option pricing model, while from the second category is the Black-Scholes model. We will discuss next these two main methods and see their disadvantages.

Let us recall here the binomial option pricing model written on one asset S for the time period $[0,T]$.

We call $u,d$ the upward and downward rate of the future price of the asset. There are several choices for the rates in the bibliography, see for example [3,4] and [9] and the references therein.

For a specific contract, the binomial option pricing model computes the initial value of a replicating portfolio. The basic problem, however, of the binomial model is that the assumption 1 is not realistic, i.e. $\mathbb{P}(S_T=u{S}_0)=\mathbb{P}(S_T=d{S}_0)=0$ in the real world or very small! Therefore the binomial option pricing model does not produce a price with a physical meaning as we define it at definition 1. That is we are almost sure that the writer’s guess will not come true so the question is: what happens in this case concerning the value of the replicating portfolio and the payoff?

Now let us recall the Black - Scholes option pricing model, see [2] (see also [1]).

In [1] the author, in fact, proposes the price of a call or a put option to be with $\mathbb{P}$ is the real world probability. Here $g\left(x\right)={(x-K)}^{+}$ for the call option and $g\left(x\right)={(K-x)}^{+}$ for put option. This approach have a clear physical meaning and this price can be considered as a fair price for the put option since the payoff is bounded. On the other hand the Black - Scholes model proposes the price to be where $\mathbb{Q}$ is not the real world probability but an equivalent probability measure to $\mathbb{P}$. Note that ${\mathbb{E}}_{\mathbb{Q}}\left(g\left(S_T\right)\right|{\mathcal{F}}_0)=V(0,S_0)$ where $V(t,x)$ is the solution of the following famous Black - Scholes partial differential equation At the first sight the above price is not intuitive at all. However, if someone go deeper to this theory understands that this price is in fact the initial value of a replicating portfolio and this explains the proposed price. Therefore, the Black - Scholes model and the binomial model have the same basic pricing principle which is based on the construction of a hedging portfolio.

So the Black - Scholes model prices a contract by the initial value of a replicating portfolio that has to be reconstructed continuously in time. Assumption 2 maybe is not the best for a specific asset but is realistic. However, the problem of the Black - Scholes model is the reconstruction of the replicating portfolio which is not realistic because nobody can reconstruct such a replicating portfolio continuously in time in order to hedge the risk. To be more precise, suppose that at time t the replicating portfolio contain $a(t,S(t\left)\right)$ assets and $b(t,S(t\left)\right)$ at a bank account. At the next time that the price of the asset changes to $S(t+h)$ from $S\left(t\right)$ the number of the assets should also change instantaneously to $a(t+h,S(t+h\left)\right)$ which is impossible. Of course every pricing model via replication with reconstruction in continuous time has the same serious problem and therefore does not produce a price with a physical meaning (see definition 1).

In the spirit of [1], for general options with payoff $P_T$ (see Assumption 2.2 of [5] for example), we can define the amount ${\mathbb{E}}_{\mathbb{P}}\left(P_T\right)$ and suggest it to the writer as the possible value of the option, i.e. the average value of the payoff which has an obvious physical meaning for the writer. In [5] we have used for simplicity the Brownian motion to the assumptions concerning the payoff of a contract but is obvious that someone can use any other suitable random variable or stochastic process.

Note that the no arbitrage arguments (see for example [6]) does not give us a specific price for the call and put options but only a relation between them and some bounds for their prices. The situation is worse for more complicated contracts because even if we can prove some bounds for their prices the competition is maybe too low for them in order these bounds come true in the real world.

Therefore, in order to propose a pricing model via replication we should combine the realistic assumption 2 with a replication in discrete time. In [5] we have modified the binomial model in this direction.

As we have seen in the previous section, the writer’s guess concerning the rates $u,d$ will never come true in the real world or the probability is very low. So what happens when the true upward rate becomes bigger or less than she expected?

In [5] we have show that for the usual call and put options the writer will have a profit if the future upward rate becomes smaller of what the writer predicted or if the downward rate becomes bigger than she expected. Denote the profit of the writer of a call or put option by $\Pi$ when she is pricing by the binomial model. Let us denote the rates that she used to price the contract by $u,d$ and by $u^{*},d^{*}$ the future rates. Then the profit of these contracts is a function of $u-u^{*}$ and $d^{*}-d$, that is $\Pi :=\Pi (u-u^{*},d^{*}-d)$, and moreover this function is increasing in both variables.

Let us recall it here for the reader convenience.

In [5] we have proposed a way of choosing $d,u$ so as to have for a given probability p under assumption 2. We call this as the realistic binomial model because we assume a realistic assumption for the future prices of the underlying asset and we choose the rates $d,u$ so as to fix the probability of the event $\{\frac{S_T}{S_0}\in (d,u)\}$. We have already see that this is not enough to have a practically useful pricing model via replication so we should also define a useful property of the replicating portfolio regarding the cases in which the writer has a profit.

Let a contract written on d assets and let the $u_i,d_i$ are the rates guessed by the writer while by $u_i^{*},d_i^{*}$ we denote the future real rates where $i=1,\cdots ,d$. Let a replicating portfolio constructed by the (realistic) binomial model, that is let some $a_1,\cdots ,a_d,b$ such that assuming for simplicity the risk free rate equal to zero.

That is the profit property is a property of a replicating portfolio that has to be reconstructed in discrete time and not a property of the realistic binomial model.

So we have combined the realistic assumption 2 for the price of the asset with the realistic reconstruction of the replicating portfolio in discrete time.

What about the well known arbitrage theorem (see for example [8]) in the binomial model setting? Let us suppose that the writer chooses $u,d$ such as $e^{rT}>u$. In this case the notion of arbitrage has no meaning because the probability of the event $\{u^{*}>e^{rT}\}$ is strictly positive!

In this section we will describe a generalization of the safe price notion by combining it with the notion of static replication. Let an option written on d assets with payoff $P_T$ at time T. The problem can be stated as follows: given some $p\in (0,1)$ and $\sigma >0$, construct a portfolio with price process $V_t$ by buying $a_i$ number of asset $S_i$, with $i=1,\cdots ,k$ at time 0 and by putting the amount b at a bank account in the following way, or any combination of the above. This point of view has a clear physical meaning for the writer and is in fact a generalization of the safe price that we have proposed before because one can choose $a_i=0$ at the above construction. Here $k\ne d$ in general because a very important advantage of the (generalized) safe price is that we can built a replicating portfolio as above by buying assets which are, in general, different from the underlying assets! This is very useful in the case where the underlying assets can not be traded at the market.

Of course we can built a portfolio by only one asset but a diversified portfolio is more stable. An interesting problem is the following.

One way of course is to built a portfolio that will contain all the n assets and then simply try to choose appropriately the corresponding $a_i$, i.e. the weight of each asset.

After the construction of this replicating portfolio the writer can rebuilt it accordingly as soon as she has new information from the market.

Note that the model parameters are dependent also on the number of the particular N past years which have used to get the corresponding data. That is, an investor can use any N past years, not necessarily consecutive, in order to estimate the parameters of the model.

Assumption 3 is obviously not that realistic, as most investors rely on their intuition about future market movement, so the reality is even more complicated. Anyway, the calculations we propose here using the historical data are certainly a good first estimate to which one can then add her/his intuition about the future evolution of the phenomenon. For example, a recent event has not left its mark on historical data, yet it is going to affect future ones. An investor should take this into account by increasing or decreasing the model parameters appropriately which has been calculated based on historical data.

We gave the concept of a price with a physical (or realistic) meaning and gave ways to produce such prices. It will be very interesting if we can find other, different from the above, realistic ways to price an option! Unfortunately, the binomial option pricing model and the Black - Scholes model do not produce prices with some physical meaning!

In [5] we have classified the options into two main classes. At the first one belong all the options with unbounded payoffs (like the call options) and at the second one belong the options with bounded payoffs (like the put options). The first class can be divided into two subcategories. At the first belong the options with unbounded payoffs but in which we can bound the payoff buying some call options (like in a spread option) and at the second belong the options with unbounded payoff in which we can not bound it buying call options (like in a call on maximum option). This classification is important because the notion of the fair price has a meaning only at the class of options with bounded payoff and this price does not come necessarily by a replicating portfolio as we have seen at the Section 4.1.

We have also recalled the realistic binomial option pricing model as we had proposed it in [5] and we have point out four open problems. We have given some examples of option pricing using the realistic binomial model and the notion of the safe price. We compare them and give the advantages and their disadvantages in every situation so as the writer can decide how to hedge the option and consequently how to price the option. The notion of the safe price can be used also by the buyer in order to estimate the probability of profit. The Black - Scholes model and all the models that assumes replication continuously in time is not practical since no one can construct such a portfolio in practice. In the hypothetical case that one could construct such a portfolio it is not certain that it would be preferred by writers over the safe or acceptable price as for example we have seen comparing the realistic binomial price with that of the safe price at the Section 4.1.

In the case where someone can find a super - hedging strategy which hedge the option without any risk then the same maybe can be done by the buyer of this option. Moreover, if we assume that there is some competition, there will be someone else which will sell this option for a lower price undertaking some risk.

Summing up, in order the writer to price an option she first have to design a hedging strategy. For example, for a spread option she should buy some call options concerning the asset $S_1$ in order to bound the possible payoff. She estimate the amount of money that she need for building a replicating portfolio with the one period realistic binomial method. If this replicating portfolio assumes borrowing assets she should also buy call options for each such an asset! Next she estimate the safe price given a probability of profit and finally the average value of the payoff. All the above hedging strategies have a clear physical meaning and moreover they are applicable in practice. The writer need to know the smallest price, but with a physical meaning, for the contract in order to be competitive. The competition, if any, will drive the price of this option to the minimum assuming A3. The buyer has also a way to estimate the probability of profit buying at a specific price, assuming that she need this contract for speculation, by using the safe price notion. For example a writer of a spread option will buy a call option of the asset $S_1$ for security reasons and not for speculation. Finally, one way to eliminate the risk of bankruptcy selling a call option is by owning one stock per call option.

Assuming the d assets follows the following stochastic differential equation we can estimate by means of the Monte Carlo method the expectations where $\mathbb{P}$ is the real world probability and $\mathbb{Q}$ is an equivalent probability to $\mathbb{P}$ so that the stochastic processes $S_i$ are martingales under $\mathbb{Q}$. Here $f_i,g_i$ are suitable functions chosen for each asset and H is the payoff function of the contract written on d assets. The physical meaning of the first expectation is clear for both the writer and the buyer. On the other hand the physical meaning of the second is clear only if the writer construct the corresponding replicating portfolio. If she don’t construct the replicating portfolio then, in fact, she sell at the fair price with appropriately chosen probability p. That is, if the price Y is such that $\mathbb{P}(P_T\le Y)=1/2$ with $Y>{\mathbb{E}}_{\mathbb{Q}}\left(H(S_1,\cdots ,S_d)\right)$ and the writer sell at the price ${\mathbb{E}}_{\mathbb{Q}}\left(H(S_1,\cdots ,S_d)\right)$, i.e. with probability of profit less than $1/2$, then she should construct the replicating portfolio in order to hedge the risk which is impossible in practice. If $Y<{\mathbb{E}}_{\mathbb{Q}}\left(H(S_1,\cdots ,S_d)\right)$ and the competition is high then there will be someone that will sell this contract at the price Y.

Every investor needs to know the future beliefs of other investors as well. This can be done by computing the implied probability of profit (fixing the parameters $m,\sigma$) using the above models, i.e. first compute the implied $z_p$ and then p. The implied probability of profit is a criterion of how other investors see the future market concerning the underlying asset and is equivalent with the implied volatility using the Black - Scholes model.

As we have seen, the concept of fair price is not well defined because of the different perspective on the future of the writer and the buyer. However, the notion of the unfair price for the writer or for the buyer can be well understood. Consider for example a put option with strike price K. In this case, any price $Y\ge K$ is an unfair price for the buyer because the payoff will be less than K. Consider now an option with payoff $P_T=max\{S_1\left(T\right),S_2\left(T\right),K\}$. Any price $Y\le K$ is an unfair price for the writer because the payoff will be more than K. Therefore we can have the following definition of the unfair price.

Consider now the case of a spread option with payoff $P_T=max\{S_1\left(T\right)-S_2\left(T\right),0\}$ and consider the price $Y=C(S_1,K)+K$ where $C(S_1,K)$ is a call option written on $S_1$ with strike price K. This price will drive the writer to certain profit without risk (arbitrage) but there is also the possibility for profit also for the buyer. So this price is not an unfair price for the buyer according to the above definition. However, if the competition is strong there will be another writer which will sell this contract for less money undertaking some risk.