The Simulation of European Call Options ’ Sensitivity Based on Black-Scholes Option Formula

As the Stock index futures with Cri 300 index for the subject matter launch, the research to stock options done by China’s financial market is gradually in-depth, which has great significance to the improvement of the financial markets. With the Black-Scholes option formula, this paper attempts to study the sensitivity of single stock’ call option named Industrial and Commercial Bank of China Limited to stock price changes, time changes and the situation that both of them occur. The simulation results were achieved based SAS, which not only has a very important practical significance to the launch of this kind financial derivative and the establish of a perfect pricing model of financial derivatives, but also can help financial market further promote economic growth.


Introduction
As a landmark, Black-Scholes option formula laid a series of financial derivatives, especially stock options [1,2].With the fast developing of China's financial industry, the research to financial derivatives is gradually in-depth, which has been proved by the launch of Stock index futures with Cri 300 index for the subject matter.With the Black-Scholes option formula, this paper attempts to analyze the changes of call option with single stock, ICBC [3], for the subject matter.

The Analysis of Black-Scholes Option Formula
The Black-Scholes option formula [2] is often described as following: Then represents the present value of finan-

 ,
V S t cial derivatives, which is based on the subject matter's value and changes over time .Without considering the bonus and delivery of the block as subject matter, we can get European call options formula as following when

S t
the applicable conditions of Black-Scholes option formula are satisfied (derivation process referring to literature 4).Then is the present value of call options and is present time.
is the exercise rice of subject matter stock to the maturity date .T

 
N x is standard normal distribution function.The approximate graphic of x in the interval from -3 to +3 is shown as in Figure 1 [4].
According to the definition of European call option, its value to the maturity date meet the condition: . Meanwhile it can't be executed in advance, which is the great difference between European call option and American call option.
The pricing and risk research of European call option is more significant [5,6].Meanwhile the development and application of financial derivatives and playing its innovation role in the modern financial markets have profound realistic significance.rict, with the Black-Scholes option formula, we can st only establish approximative option pricing model.This paper only analyze the pricing model of European call option due to the limit of paper length.

The Select of Subject Matter Stock for Call Options
Acco e author's statistical studies, the stock rding to th with small capitalization is difficult to meet the application conditions of Black-Scholes option because of its overlarge fluctuation.So we select ICBC, which has great capitalization, as the subject matter of single call option to establish approximative analysis model.

Collecting Related Data
Select each trading day's closing price of ICBC during 1/1/2009 to 18/10/2011 as the subject matter stock's approximative price S , which is selected in NetEase and will not be given pa ticularly.Removing the date without r trade, we can get 423 effective data.

Calculating Option's Related Parameters
Calculate sample average rate of return as approximation to replace drift rate and volatility in the option formula.Calculate the 423 data of ICBC With software SPSS 16.0, we get the result as following: T t  selects 12 months as typical analysis stage.
It is selected as a typical analysis basis that the strike price of call option 0 X is 10% higher than its recent average les, price.Of course we can choose other princip w we establish hich do not affect the results of analysis.

The Related Simulation of Call Option
With Black-Scholes option pricing formula, the model of ICBC by software SAS when we have calculated related parameters, and we can get relevant ue Changing graphics describe.

The Affluence on Call Option Value Made by Asset Val
From formula B-S we can get: The explain of symbols and parameters refers to section I. Setting the closing price of ICBC in /10/2011 as 18 a benchmark, strike price rises 10% that 4.59 X 0  (Yuan).Then we can get the changes of European call options' value influenced by the price of the subject matter stock.We can get the expected graphic ware SAS as shown in Figure 2 in the condition that there is 12 months leaving to due data.From the Figure 2 we can get that there is a jump when the price is strike with soft price, and jump range increased if the price is too far away from the present price.Vertical axis presents Euro-pean call options' price and horizontal axis presents stock price.

The Sensitivity of Call Option to the Price of Subject Mattrr Stock
The relation between the change rate d d c S that option value c atter stock and stock price ompares with subject m S is as following.

   
is the function value when standard normal density function is at x .Data and parameters calculation accord to the result in III(A), then we get the simulation graph as shown in Figure 3.
From the figure we can get that there is intermittent phenomenon when the price is strike price, which echo is far away from strike price, the change rate function's changes.When the price of subject matter tock s change rate that option value compares with stock value gradually becomes a constant value.That is to say call option's price changes constantly comparing with the changes of subject matter stock.ce 3) We study the change relation among call option, the price of subject mat r stock and due time in section te IV(C).Of course we can further introduce the sensitivity of the time functions to make the research result more operational.

The Analysis and Prospect of All Kinds of Risk
In a word, we can further study the relation among call option, the change rate of interest rate and the change rate of subject matter stock's volatility, which has a very important practical significance to establish a perfect pricing model of financial derivatives and help financial 0.00 0.33 0.67 1.00 t 4.17  Copyright © 2012 SciRes.JMF market further promote economic growth.

Acknowledgements
The

Figure 1 .
Figure 1.The figure of standard normal distribution function in the interval from -3 to +3.

SFigure 2 .SFigure 3 .
Figure 2. The function graph between call option and stock value.
to the result given in the previous sec-d are still th tions, we and parameters.With software SAS, we can get the simulation graph of the can get all kinds of data relation among call option's value, the price of subject matter stock and time function (as shown in Figure 4).

Figure 4 .
Figure 4.The relation among call option, the price of subject matter stock and due time.
projects of Beijing municipal education commission: No. KM201010009011 & National Natural Science Foundation of China (11171002), Beijing Natural Science Foundation (the theory of mixed effects models of multivariate complex data and its applications; 1112008) & College Student Research and Career Creation Program of Beijing (2012).