Modeling European Options

Option contracts can be valued by using the Black-Scholes equation, a partial differential equation with initial conditions. An exact solution for European style options is known. The computation time and the error need to be minimized simultaneously. In this paper, the authors have solved the Black-Scholes equation by employing a reasonably accurate implicit method. Options with known analytic solutions have been evaluated. Furthermore, an overall second order accurate space and time discretization has been accomplished in this paper.

price in response to external effects. It is represented by a random sample drawn from a normal distribution with mean zero denoted as dX  , Putting the above two factors together we obtain a stochastic differential equation, which is the mathematical representation for generating asset prices [1,4].
The equation (1) gives interesting and important information concerning the behavior of S in a probabilistic sense. At time ' t t  , suppose the price is ' S , and then ' S will be distributed about 0 S with a probability density function. The future asset price ' S is thus most likely to be close to 0 S and less likely to be far away. Thus, the equation generates time serieseach time the series is restarted a different path results. Each path is called a realization of the random walk.
The Black-Scholes Partial Differential Equation, the derivation of which is well-known, is given below If E S  at expiry, then in financial sense, for call option, there will be profit E S  , handing over an amount E , to obtain an asset worth, while, if E S  , then there will be loss of S E  . Thus the value of the call option at expiry can be written as As the time tends to expiry date the value of call option approaches (6), it is known as pay-off function for European Call Option. This is known as the final condition of PDE (12).
which means pay-off is also zero. Thus, the call option is worthless on 0  S even if there is long time to expiry. Hence, we have i.e. the asset price increases without bound it becomes ever more likely that the option will be exercised and the magnitude of the exercise price becomes less and less important.
And the initial condition   t P , 0 is determined by calculating the present value of an amount E received at timeT . For time-dependent interest rate, the boundary condition at And, Thus, the Black Scholes equation and boundary conditions for a European Put Option are given by equations (2), (6)-(8).
The analytical solution to the BSDE is not the focal point of the paper but given below, nonetheless as [2] The corresponding calculation for a European Call Option follows similar lines.

Analytical Solution:
for some constants  and  , subsequent differentiation yields , so that the PDE assumes the standard parabolic form The argument follows from the fact that Next, evaluate the inner integral, The corresponding calculation for a European Put Option follows similar lines. Its transformed pay-off is and can be computed as above. However, having evaluated the Call, a simpler way is to use the put-call parity formula for the value P of a Put, given the value of the Call. This yields

Definitions/Terminologies:
Writer: The person selling the asset.
Holder: The person buying the asset.
Expiry Date: At a prescribed time in the future, the holder/writer of the option may purchase/sell a prescribed asset. Underlying Asset: The holder/writer may purchase/sell a prescribed asset at a prescribed time.
Exercise Amount/Strike Price: The holder/writer purchases a prescribed asset at a prescribed time at a prescribed amount.
Arbitrage: In financial terms, there are never any opportunities to make an instantaneous risk-free profit.
Risk: Risk is commonly of two typesspecific and non-specific, specific risk is the component of risk associated with a single asset, whereas nonspecific risk is associated with factors affecting the whole market.

Notation table:
t -Time

Software System Attributes
Dept. of CSE, PES Institute of technology, Bangalore South Campus  Reliability: This implementation is based on the Black-Scholes and Crank Nicolson's Equations; it is mathematically sound with no error included during coding. This does not slow down the system or does not cause any harm to the environment.
Availability: This software is built with a simple user interface and easily available resources.
Portability: Since this product uses platform independent HTML and C codes, it is portable. Assign to y maximum value between z-K and 0, which symbolizes the call option else Assign to y maximum value between K-z and 0, which symbolizes the put option Step3: stop