# Time to maturity european option

The former Chapter, as was summarized in its Concluding Remarks, provided the bridge between the one-period 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 payoff 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 page , intent 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 effect of dividends on the price of European-type options.

The Chapter concludes with the issue of volatility vs. We can now calculate the Black-Scholes formula see [8] 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.

The 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 facilitate 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. In the case that the strike price is greater than the price of the underlying asset at the time of maturity, the call option is worthless - the holder would prefer to purchase the asset at the current market price and thus would not exercise the option.

The payoff of a plain-vanilla call option at maturity is,. The graph below shows the relationship between the payoff of a call option and the price of the underlying security at maturity. The holder of a put option has the right but not the obligation to sell shares of the underlying asset at the strike price upon maturity. As such, it is only profitable for the holder to do so if they can sell the shares when the strike price is greater than the market price at maturity. The value of a put option at maturity is,.

An option's value and payoff is directly related to the price and volatility of an underlying asset, as well as factors such as the proximity to the expiration date. Options can be valued using different valuation methods including the popular Black-Scholes Model which uses many variables to calculate the estimated value of an option.

When someone purchases 1 call option on a stock which expires in 1 year, the value of the option will increase as the underlying security rises in value. At the same time, the option will slowly lose time value as time progresses and the option gets closer to the expiration date. Most options expire worthless at expiration becuase they are "out of the money.

On the flip side, a put option is considered "out of the money" when the underlying stock price is trading above the strike price of the option. From the makers of. Unable to complete your request. Please refresh your browser.

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