# American option put call parity dividend

Current Market Situation and Main Purpose of the Paper Mathematical Models for Option Pricing Jump Diffusion Models Merton Model Kou Model Black-Scholes Model Assumptions The Black-Scholes Equation The Black-Scholes Formula Estimation and Hypothesis Testing of a European Call and a European Put Option Introduction and Definition of Parameters Black Scholes Model as Testimonial Merton Model Parameters Total Ruin Log Normal distributed Jumps Kou Model Comparison of all Models for a European Call and Put Conclusion Bibliography Appendix Data: Programming Codes Mathematica 9 Codes VBA Codes Abstract In our complex and developed financial world the Brownian motion and the normal distribution have obtained enormous impact on the prices of option contracts traded on the stock exchange or over the counter.

Due to empirical investigations during the last years two points have emerged which cannot be assumed to pertain to each option price calculation. The two points are that the return distribution of a stock does not always follow a normal distribution but has a higher peak and two heavier tails than those of the normal distribution and that there is a volatility smile in option pricing. The Black-Scholes approach implies that volatility is a constant function. This paper theoretically and empirically investigates three different option pricing approaches: the Black-Scholes approach, the Merton-Jump approach, and the double exponential jump diffusion model, which was proposed by Kou and Ramezani and Zeng Working Paper Series No. Current Market Situation and Main Purpose of the Paper The most important option contracts are Plain-Vanilla-Options, which allow the buyer to sell or buy particular assets with a previously price determined between two parties. Options are traded both on stock exchanges and OTC. The two main different kinds of options are European options and American options, European options can only be executed at the end of their duration but American options can also be executed during the lock-up period. Furthermore options are divided into Call options and Put options.

Call options give the buyer the right but not the obligation to buy a particular asset, and the seller has to provide it. A put option gives the buyer the right but not the obligation to sell the asset see Hull On the one hand, the value of options is determined by supply and demand in the markets and on the other hand it is done by mathematical models. This paper will investigate jump diffusion models for option pricing. The Brownian motion and normal distribution have been widely used in the Black-Scholes model of option pricing to determine the return distributions of assets. From many empirical investigations two riddles emerge: the leptokurtic feature says that the asset s return distribution may have a higher peak and two asymmetric heavier tails than those of the normal distribution. The second puzzle is the empirical abnormity called volatility smile" in option pricing. In other words the Black-Scholes approach does not consider jumps in an asset s price-curve see Kou 1 One of the first approaches was the Merton Jump model by R. Merton, who was also involved in the process of developing the Black-Scholes model. The reason for this new approach was to render the Brownian motion negligible, to state the estimation of options fair prices more precisely and to involve more actual price curves to the estimation. Applying this approach, you only obtain a solution if there is total ruin or if the price jumps are log-normally distributed.

Of course we are also interested in prices if these two constraints do not exist. Accordingly, several kinds of jump diffusion models have been developed based on the Merton model over the last few years. In this paper we look at two models, the Merton model, which also can be seen as the foundation, and the Kou model as a new creation. The main purpose of this paper is to investigate if plain vanilla option pricing data regarding accuracy, applicability and effort better suit the Double Exponential Jump Diffusion Model Kou Model , the Normal Jump Diffusion Model Merton Model or the Black Scholes Model. The first part of this paper will obtain definitions of the basic terms and explanations of various kinds of models for pricing an option.

All assumptions will be made before calculations are mentioned. Also the partial differential difference equations which have to be solved to arrive at an analytical solution for the option price model, will be shown for each model. After this theoretical part we will switch to the next part, where an empirical study concerning the different option price models will be carried out. Thus the second part contains estimations of a European Call and a European Put option with three different kinds of models.

At first all parameters which are used for the models are explained and if further assumptions have to be set also these and how these further assumptions are implicated in the calculation will be explained. In this section we will also show how the option price changes if various parameters vary. After the evaluations the solutions will be compared according to the following criteria: accuracy, applicability and effort, to get an answer to the research question. Mathematical Models for Option Pricing 2. The diffusion part is determined by a common Brownian motion and the second part is determined by an impulse-function and a distribution function. The impulse-function causes price changes in the underlying asset, and is determined by a distribution function. The jump part enables to model sudden and unexpected price jumps of the underlying asset. The Black-Scholes approach assumes that the price of an asset, which is the underlain asset of the option follows a geometrical Brownian motion. But a geometrical Brownian motion cannot reflect all attributes of a stock quotation. Especially price jumps in a stock quotation cannot be replicated by a Brownian motion see Merton Consequently, Merton developed the following approach to include price jumps and a new kind of model emerged, the jump-diffusion model Assumptions The following assumptions can be read in Merton see pages 1,2,4 and 5 : 1 frictionless markets this means there are no transaction costs of differential taxes 2 no dividend payments 3 the risk-free interest rate is available and constant over time 4 no restrictions regarding value of transaction and price development of the asset 5 short trading is not prohibited 6 stocks are randomly divisible 7 all information is available to all market participants 8 no arbitrage possibilities 9 the option is a European style option 10 the stock price is defined as a stochastic differential equation: 2 where: The likelihoods of this Poisson process can be described as: 6 University of Applied Sciences bfi Vienna.

The first component is described by the normal price changes, because of disequilibrium in supply and demand on the market. This kind of attributes is expressed by a standard Brownian motion with a constant drift, a constant volatility and almost continuous paths. The second component is described by changes of the stock price influenced by new available information. These jump-processes are normally outlined by a Poisson process. So if the Poisson event occurs, the random variable describes the influences of the asset price changes. Let us define S t as the current stock price at time ; due to that, the stock price at time would be expressed as. But it is assumed that the random variable is a compact support and counts. All random variables of have to be independent and identically distributed. If we now look back to equation 2 we can see that the part characterizes the instantaneous part of the sudden return due to the normal price changes and describes the price jumps see Merton The resulting path will be mostly continuous with some finite jumps, which have got different values and amplitudes.

If Merton : are constant, the relationship of S t and S 0 can be rewritten in this form see [ ] 4 where: Working Paper Series No. We assume that option price can be written as the twice differentiable stock price, in other words, the option price can be written as a function of the stock price and time:. So if the stock price follows the same dynamics as described in equation 2 , the option price dynamics can also be written in a very similar form see Merton 7 : 5 where: [ ] If we now use Ito s Lemma for jump processes, we get the following relationship: [ ] 6 7 where subscripts on indicate partial derivatives. Moreover, the Poisson process of the option price depends on the Poisson process of the stock price. That means that, if a Poisson event occurs for the stock price, also a Poisson event for the option price will occur. If the random variable then the random variable will be. Although the two processes are interdependent there is, however, no linear dependency because the option price is not linearly dependent on see Merton 8. Now let us consider a portfolio consisting of a stock, an option and a risk-free asset with an interest rate of per annum.

The portfolio is divided into three parts with the proportions where. If is the price of the portfolio, the dynamic can be expressed by see Merton pages : 8 where: [ ] From equation 2 and equation 5 we get that: 9 10 [ ] ] 11 where has been replaced. In this case the expected return of the portfolio has to be the risk-free interest rate because of the arbitrage approach. Looking at equations 9 and 10 , this means see Merton 9 10 : 12 From equations 6 , 7 and 12 the famous Black Scholes formula for option pricing emerges: 13 Unfortunately it is now possible to set the proportions in such a way that there is no jump risk because of the presence of the jump process. This is caused by the non-linear dependency of the option price and the stock price because the portfolio optimization is a linear process. However, it is possible to work out the portfolio value if the proportions are set the same way as in the Black Scholes case. If is the value of the portfolio with the Black Scholes loading, then from 8 we gathered the following see Merton 10 : 14 Note that the value of the portfolio is a pure jump process because the continuous part of the stock price changes does not exist anymore because of the parameters choice. However, on average in each time interval there is one jump.

Following equations 7 and 11 we can work out further qualitative attributes of the portfolio price, namely see Merton 11 : 16 The Formula for Option Pricing As we have shown in the previous section, there is no possibility to construct a risk-free portfolio of stocks and options, so it is not possible to adapt the no-arbitrage approach of Black and Scholes. However regarding Samuelson Rational Theory of Warrant Pricing , it is possible to determine a formula for option pricing if the price is expressed as a function of the stock price and the remaining time until maturity. From equation 6 we gather that the price is dependent on instead of on the partial differential difference equation [ ] 17 with satisfy the boundary conditions see Merton 13 : 18a 18b Another approach concerning the pricing problem follows along the assumption that the Capital Asset Pricing Model CAPM , developed by Black and Scholes, was the legal description of stock returns and equilibrium. In the previous part we described that the dynamic of the option price depends on two components, the con- Working Paper Series No. The latter describes jumps which are determined by new important information. If the information is firm- or sector-specific, then this information has only little influence on the market.

Such information represents the non-systematic risk, which means the jumps are not correlated with the market. If we look at equations 14 , we realize that only the jump component is the source of uncertainty in the return dynamics. Considering that the CAPM holds, the expected return has to be the risk free interest rate, so. This condition implies that equation 9 can be rewritten this way:, or substituting for and and we get see Merton : 19 Concerning equations 6 and 7 , this implies that the option price must satisfy [ ] 20 with the boundary conditions of equations 18a and 18b. Formally, this equation has the same form as equation 17 but does not depend on or. In the formula, however, only the risk-free interest rate appears as regards the Black Scholes approach.

If we set, which means there are no jumps, the equation is reduced to the Black Scholes equation. Note that, if the jumps even represent pure non-systematic risk, the jump process does influence the equilibrium of the option price. This is why the fair option price cannot be determined without considering the jump part see Merton We define the mean as: 21 With this definition we can rewrite equation 20 to 22 is the probability density function of the jump process Closed-Form Solutions of the Merton Model Unfortunately, it is impossible to write down a complete closed-form solution of the Merton s formula even for European-style options. Merton, however, developed solutions where he specified the distribution for. Merton developed two different analytical solutions. One possibility is that the stock price jumps during one jump process to a price of 0, which is also called Total Ruin. The other possibility is that the jumps follow a lognormal distribution. These two cases will be described in the next two parts see Merton Total Ruin In the first case of an analytical solution there is a total ruin which occurs suddenly.

This means if the Poisson event occurs, the stock price decreases to zero. As a result, the random variable, which represents the change if the Poisson event occurs is zero with a probability of one. The percentage change of the stock price is then at and expresses a default. In this case the price of a European style call option with the remaining time is, as follows see Merton : 10 University of Applied Sciences bfi Vienna. Regarding the characteristic that the option price is a function of the interest rate, an option with a stock as underlying and a positive probability of a total ruin is more expensive than an option which neglects this possibility see Merton 17 and Merton Log-Normal Distributed Jumps As we have already mentioned in the introduction to this part, in the second case the random variable, which expresses the price changes if a jump occurs is log-normal distributed.

So this following probability density function has to pertain see Merton : 24 The price can now be rewritten on this condition as follows: [ ] 25 [ [ ] [ ] 26 ] 27 where: [ ] Working Paper Series No. In the calculation of the option price an infinite but convergent sum and with a Poisson distribution weighted sum has to be evaluated see Merton Kou Model Another model for option pricing is the Kou Model developed by Steven Kou. Kou assumes that jumps of a stock are not log-normally distributed, as Merton assumes, but follow a double-exponential distribution. The first part is a continuous part driven by a normal geometrical Brownian motion and the second part is the jump part with a logarithm of jump size, which are double exponentially distributed and the jumps times are determined by the event times of a Poisson process. All random variables in equation 29 are independent, and - for simplicity and to obtain an analytical solution for option prices - the drift and the volatility are assumed to be constant.

Further, the Brownian motion and the jump processes are supposed to be one-dimensional.

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Lowther and Ching Kuang Shene Programmers: Yuan Zhao wir im nächsten Schritt eine zweite Methode entwickeln, 2 Michigan Technological More information. Bonds face value, coupons, pure discount bonds. Frans de Weert is mathematician by training who IM ALDRUP AG has been established Mehr. London Stock Exchange Derivatives Market Strategies in SOLA Exact Inference Approximate Inference enumeration variable elimination stochastic This document is an overview only. Recovering Risk-Neutral Probability Density Functions from Options Prices is currently working as an equity derivatives trader at Barclays Capital, New York. Analysis 1, HS14 Übungsblatt 6 Analysis, HS4 Ausgabe.

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