European call option pricing formula
Valuation of European Options. The former Chapter, as was summarized in its Concluding Remarks, provided the bridge between the european call option pricing formula discrete model and a one period model with a continuum of states of nature. This latter type of model is sufficient to allow the valuation of European-type options. Utilizing the reader's intuition and extending the results of Chapter 1 to the continuous-time setting enables the valuation of European-type contingent claims.
Specifically if is the european call option pricing formula of the contingent claim, where is the price of the underlying asset at maturity, the value of the derivative security is given by equation 5. Chapter 5 was also, at least partially see footnote 6 on pageintent on supporting the choice of the SDF and the risk-neutral probability as the lognormal distribution. We are therefore now at the point where calculating the value of a European-type option amounts to calculating the value of a certain integral.
The current Chapter is dedicated to the valuation of European puts and calls, i. Having established the pricing formulae, the investigation of combinations of options with different maturity dates follows. Prior to the derivation of this formula we lacked the ''tool'' to compare options with different maturity times and, of course, to value options prior to maturity.
Consequently, we were limited to investigating only the payoffs of combinations at maturity. In this Chapter we have the tools and we are able to explore combinations across time. The Chapter also looks into the european call option pricing formula of dividends on the price of European-type european call option pricing formula. The Chapter concludes european call option pricing formula the issue of volatility vs.
We can now calculate the Black-Scholes formula see  for the price of a call option. As we have seen in equations 5. In either approach, the formula is obtained by the evaluation of essentially the same integral. If we approach the call option pricing formula from the perspective of the discounted expected value we must use the risk-neutral probability which we defined as the MAPLE function given below. European call option pricing formula price of the call option based on equations 5.
We have seen in equation 5. The function is defined as below. The price of the call option can also be calculated based on equation 5. Clearly these two expressions are the same.
We shall proceed now with the calculation of CallPrice. Some readers may wish to skip these calculations. The result is reported in equation 6. Similarly, the cumulative normal probability function is defined by. To demonstrate the relationship between erf and the normal probability function we evaluate Normalcdf at for and.
Conversely, we can express the function erf in terms of the function Normalcdf. Mutliplying Normalcdf by two, evaluating it at the point is equivalent to the erf function plus one. This is demonstrated as follows: To european call option pricing formula the MAPLE calculation the integral defining CallPrice is rewritten in terms of the density function of -- the continuously compounded rate of return.
As you recall, we assumed it to be the normal distribution with expected value of and a standard deviation of. This could be achieved by simply replacing the variable of integration in the above integral using the relation in equation, 5.
Alternatively, one can apply the fact that given a random variable the expected value of may be calculated based on the density of as. Thus in our case plays the rule of and is the normal density function. Consequently we can equivalently define CallPrice as below. Since when the value of vanishes we can change the lower limit of the integral to be and replace in the integrand with.
Thus, the integral to be calculated is. Now the resemblance between the CallPrice and the erf functions and consequently the Normalcdf function is more apparent. Given the relationship between Normalcdf and erf we can convert this expression in terms of Normalcdf. Indeed, the classical way of stating this expression is not european call option pricing formula terms of erf.
We shall return to this point soon, but first we define a function in MAPLE that calculates the price of a call option. We name this function Eqbs. The reader can now calculate the price of a call option given specific parameter values. For example, an option that expires in one month of a year where and are given per annum will have the following value: Given the relation between erf and the normal distribution it is european call option pricing formula to write the pricing formula in terms of the normal distribution.
We can also ask MAPLE to express the pricing formula in terms of the cumulative standard normal distribution, i. For simplicity let us call this function. Let us define a function called newerf ; this function is simply the erf function in terms of the function. As discussed earlier, this relationship is.
We can now substitute newerf for erf in the CallPrice expression. The pricing formula is now expressed in terms of the function: Recognizing thatwe restate equation 6. The above is the Black-Scholes option pricing formula.
The formula is classically restated as. Using the above example, the parameters for this procedure are as follows: Let us value an option with the above specifications. As discussed earlier it is common practice to use 1 year as the basic unit of time. Hence a one month time period is inputted as. We assume the interest rates and standard deviation are given in per annum values. We can now value such a call option as follows: Footnote 1 We see that the result from the procedure BstCall is the same as the calculation based on our function Eqbs.
European call option pricing formula we investigate some properties of the Black-Scholes formula, we would like to compare it to the bounds we obtained by relative pricing arguments. The bounds in equation 4. The value of a call option is graphed as a function of the current price of the stock and time to maturity. In the same plane we also graph the value of the stock the upper bound and the payoff at expiration the lower bound.
Looking at this picture the reader may gain insight as to how the value of the option approaches its value at expiration as time progresses and as the stock price changes. Recall that you may look at this graph from different perspectives by dragging the picture with the mouse.
The insights gained by close examination of the picture will be revisited very shortly. Pay attention to the way the middle graph the value of the call gets closer to the european call option pricing formula of the call at maturity as the time to maturity approaches zero. The on-line book shows an animation of the bounds. Run the animation and observe how as time approaches maturity the value of the call moves within the bounds toward its value at maturity.
The reader may have skipped this discussion as suggested.
Codes related to Option Pricing m file Description simdtree1. Checks european call option pricing formula of computation Also investigates how long it takes to evaluate tree. Displays bi-dimensional trajectories of Brownian motions Download BrownianMotion.
May be extended for timing european call option pricing formula. Download Simulation of geometric Brownian motion and implementation of dynamic hedging strategy.
Horizon over which option is simulated is NbDays. Horizon over which option is simulated is NbD days. Simulates price trajectory at 5 minute level. Once trajectory is constructed extract data for time where one wants to hedge creates a module that constructs for a given price series a dynamic hedging strategy.
Download Download all Files. Returns the option price European call or putthe option value matrix and the underling price matrix of a binomial tree. A function that tests binomial tree model for call and put evaluation. Estimates binomial tree model for a set of N. Calls to binomial tree procedure to perform a comparative static exercise i. Various programs that a brilliant student european call option pricing formula to me.
Displays bi-dimensional trajectories of Brownian motions. Computes the option price using Heston's model. Returns the hedging ratio delta of a European call option using B-S formula. Uses a simple integration rule to compute numerically the price of a call option.
Simulates trajectories of geometric Brownian motions in a risk neutral world and returns the European call option price. Simulation of geometric Brownian motion and implementation of dynamic hedging strategy.