NON AUTOMATICALLY EXERCISED (NAE) EUROPEAN CAPPED CALL PRICING THEORY

The objective of this paper is to present a methodology for deriving Black Scholes formulae via a simple lognormal distribution approach and introduce European capped non automatically exercise (NAE) call option pricing theory.


INTRODUCTION
Option or option contract is a security which gives its holder the right to buy or sell the underlying asset under the contracting conditions. Option pricing theory has advanced along many fronts since the invention by Black and Scholes in [1]. The valuation standard option pricing theory based on distribution approach has been done by many researchers such as Brooks in [2] with normal and lognormal distribution, Corrado in [3] with generalized lambda distribution, and Markose and Alentorn in [4] with generalized gamma distribution. The objective of this paper is to present a methodology for deriving Black Scholes formulae via a simple lognormal distribution approach and introduce European capped non automatically exercise (NAE) call option pricing theory. In this option, if the stock price at time of expiration is greater than the cap value L, we deal that L as the price of stock and of course the payoff is capped at L−K, conversely if the cap is not crossed then the payoff becomes the standard call, max (0, S T − K). In this option we see that the payoff opportunities are more limited, so they are cheaper to buy than standard. The approach adopted here is based on the risk-adjusted discounting of expected future cash flows. In this model we have that stock price S T is distributed lognormal.

Lognormal and Brownian Motion
A log normal distribution is given by the following pdf function S T log n µ l , σ 2 l µ l and σ 2 l are the expected value and variance of ln S T respectively, and l denotes the underlying reference index having a lognormal distribution. Specifically ln S T has a normal distribution.
Here we take that stock price follows the Samuelson model in [5], that is stock price is a random process called geometric Brownian motion with where S 0 is stock price at time 0, r ≥ 0 is riskless interest rate, σ > 0 is the volatility, T is time of expiration, and W T is a standard Brownian process with mean 0, and variance T respectively. Then Mean and variance of ln S T are respectively. For standard European call options, the payoff function is assumed to depend on the last value S T and not on all the values S 0 , S 1 ..., S T . We define C BS (x) as the standard Black Scholes call option price with exercise price x. Thus for standard European call option with contract price K, the option price based on lognormal is given by Take a look and compute the first Integral in (7). By taking µ l and σ l in (5) and With a little bit algebraic manipulation the exponential part in (8) can be written and then So we have solution for the first integral in (7) is ,and N (x) is the cumulative standard normal distribution function. Next take a look at the second integral in (7) as a probability function of S T We know from (2) and (4) . Then the value of (9) becomes . From solution of two integration in (7), the European standard call option price based on lognormal distribution and Brownian motion is This result is exactly the same as the Black Scholes standard [1] .

NAE European Capped Option Pricing
Here, in this paper, we will introduce the NAE European Capped call option pricing. In this call option, if the underlying asset price at maturity time is greater than the cap value L, the payoff is capped at L − K. So we have the payoff function is [max (min(S T , L) − K, 0)] , and the price of this call option is given by the formulae : We calculate the NAE European Capped call option price formulae based on Black Scholes equation (10). Now The value of the integral in (12) can be found with algebraic manipulation and then using equation (10) leads to the analytical formulae for NAE Capped call option price . So we have this call option price formulae is the Standard European call option price with contract price K minus Standard European call option price with contract price L. In general we can see that this price is cheaper than standard option.

Properties
In this section we will present some properties and analytical results of this option pricing model compared to standard option.  figure  1(a). The solid curve represents the plot of standard option, while the dash-dot one represents NAE European Capped option price. Table 1 give an example of comparison between both options in different time of expiration. Notice that in standard option the price get more expensive tend to asset price as the time of expiration get longer. However in NAE European Capped option, the price go up and then go down tend to zero as the time of expiration get longer, see figure 2(a).
. This show that if the cap L → ∞, the option is exactly the same as that standard, see also figure 1(b). The dash-dot curve represents standard option price. 4. If L → K, then C BS (L) → C BS (K) and of course the option price C cap → 0. It means that if the value of the payoff function get smaller then the option price also get cheaper. figure 2(a) for more detail plot. We can search the volatility's value which maximizes the option price by differencing

CONCLUDING REMARKS
We have shown that Black Scholes formulae can be derived by lognormal distribution approach and more simple than the original Black Scholes. From the definition of NAE European capped, this option is cheaper than the real european capped option. In the real european capped option if the stock price reaches the cap value prior the expiration time, this option is automatically exercised. Both of them are cheaper than the standard.