How Financial Models Shape Markets - Coursework Example

Option Valuation Report College: An employee stock option is a contract between two parties on company stocks which is granted by an employer to employee as compensation in exchange for the work and services performed. Employers who use stock option contracts as means of compensation, the call option often amounts to limited employers’ equity position. In today’s era many employers are using employee stock options to retain as well as to attract employees. Company’s give employees a certain limit of incentive in order to boost the employee morale in boosting a firm stock price. An employee would pay an exercise price then allocated company shares if they raise the a company’s stock market price abovethe normal call price. Employee cannot exercise the stock option when market prices drop below the stock exercise price. This may result to a stock option lapse ( Mackenzie Donald 2006) Employers may issue employee stock option to certain individuals in order to preserve and generate cash flow. This normally happens when a firm issues new shares while receiving a reduced tax rate similar to the value of the employee stock options. Most companies offer employee stock option to its managers or the top leaders of the company. Other non-executive staff may be offered stock option either through unprofitable business ventures. A company may also offer its employee stock to non-employees such as the suppliers, the consultants or any party that gives a hand in the company’s services needed.It is now becoming popular and accepted that companies should recognize employee work and compensate them in form of stock options as an income statement expense. Determining employee stock options forms is a difficult task in analyzing the fair value of the options in accounting. Different companies have developed standard methods that value trade on exchange and also over the counter market. However some employee stock option face difficult task in applying different standard methods. Different reports have analyzed evaluation of employee stock options. This report explains how different mathematical models of financial markets are used to estimate the price of stocks. The report also develops enhancement to the basic methodology suggested in financial standards. Financial engineering valuations and analyses are used to support financial reporting, risk management, income planning and investment decision making. Some spreadsheets base software are incorporated with the report and also the calculations needed to successfully implement various employee stock options (Amedeo Decesari 2010) In thegiven market trend, GlobeTechplcwith annualized stock volatility of 36% in the past 10 years has decided to offer an employee stock to Joseph amounting to 10,000. The company stock has a divided yield of 2% over the past 10years. Using the Black Scholes method which assumes that a market consists of one risky stock and one riskless stock called the money market. We made the assumptions that in the current market it is possible to borrow and lend any amount at riskless rate. It was possible to buy and sell any given amount of the given 10,000 stock; the other assumption was Mr. Joseph had no practice of taking advantage of different the stock option between the two competing companies, the WorldTech and the GlobeTech. Globtech is offering Joseph at 10,000 divided with 2% dividend yield over the two years. Using the Black Scholes formula Where t= time in years R= annualized risk free interest rate The Black Scholes equation which describes the price option over time would be Expanding the Black Scholes in order to calculate the price and call options, By solving the equation corresponding boundary conditions, the value of Josephs non dividend paying underlying stock would be Where n is the cumulative distribution function T-t is time to maturity S is underlying Spot price R risk free rate Alpha the volatility returns of the underlying asset. GlobTech Votality rate is 12%, risk free rate applying in the formula; P(S,t) = (25,000(25/100)-4*10) – ( 25,000*5/100 *5years* 10 years) 252,500-(62,500)=187,500 £15,625 a month After 5 years 937,500 stock value Decomposing the call option into Binary options will result asset nothing call – cash nothing call. The difference of the terms = binary call options. The formula will result to Where DN(d+) = none asset call, N(d+) F = probability of the expiring money and N(d+) underlying asset value. DN(d-)K = no cash present value D= discounting factor WorldTech will be = Monthly salary income plus stock options 15,400 a month After 5 years Total investment in stock value would be 924,000. This binomial model was first proposed by Cox, Ross and Rubinstein in 1979. It provides a basis for the numeric calculation of the valuation of options. It is also known as the kind of lattice model. This method is widely used as it can handle a variety of conditions which may not hold true for the other models. This is because the binomial tree method is based on the underlying financial instrument over a period of time rather than a single point. The steps involved in its application are; creating the binomial price tree, finding an option value at each final node and then finding the option value at the earlier nodes. This method traces the evolution of the options key underlying variables in discrete time. The tree gives the time steps between the valuation and expiration dates. The process of valuation is performed repetitively commencing at each of the final nodes and then working in reverse through the tree towards the first node (valuation date). The value calculated at each stage is the value of the option at that point in time. The option valuation follows the already stipulated process above which is price tree generation, calculation of option value at each final node and finally the sequential calculation of the option value at each preceding node. The price tee is generated by working forward from the valuation date to the expiry date. It is assumed that at each and every step, the underlying instrument will move up or down by a specific factor (u or d) per the step of the tree. So is S is the current price, then it will either be S*u or S*d where u stands for up and d stands for down. The up and down factors are calculated taking note of the market factors such as volatility and the time duration of the step. The CRR method ensures some sort of a balance such that if the underlying asset moves up then down(u,d) the price will be the same as if it had moved down then up(d,u) ensuring that the two paths merge or recombine. This in turn reduces the number of tree nodes and as a result fastening the computation of the option price. From this it can be seen that the value of the underlying asset at each node can be calculated directly via the formula and does not necessarily require that the tree be built first. The node value can be given by; Where Nu is the number of ups and Nd is the number of downs. The intrinsic, exercise or the option value is obtained at the expiration of the option at the final node of the tree. Max [(), 0], for a call option Max [( –), 0], for a put option: Where K is strike price and Sn is the spot price of the underlying asset at the nth period. The value of the option can then be found for each node staring at the time steps and then working back to the first node of the tree as at the valuation date which yields the calculated value as the ultimate value of the option. In case of use of the risk neutrality assumption at the nodes, the greater the binomial and exercise value at the respective nodes. Under this assumption, the current fair price is equal to the expected value of its future payoff discounted by the risk free rate. Therefore the following formula is used to compute the expectation value at each node: Expectation value= Binomial Value = [ p × Option up + (1-p) × Option down] × exp (- r × Δt), is the options value for the node at time , The binomial value gives the fair price of the derivative at each node, given the evolution in the price at that point and is the value of the option if it were to be held as opposed to its exercise at that point. At this point one can evaluate the possibility of early exercise at each point or node showing if the option can be exercised, if the exercise value exceeds the binomial value and finally if the value at the node is the exercise value. It takes into consideration similar assumptions with the Black- Scholes method. The binomial tree provides a discrete time approximation to the continuous process in the B-S model. With respect to the American options, the binomial model value converges on the Black- Scholes formula value as the number of steps increases. It assumes that movements have a binomial distribution which approaches the process of normal distribution assumed by the Black- Scholes method. If the upward and downward probabilities in the real world are considered it is difficult to find a proper discount rate to discount the expected payoff of options. Securities with more risk ought to apply higher discount rates to future expected payoffs. Options tend to be risky due to the high leverage characteristics for investing options. The advantages of this method are: there are no approximations involved, p maintains to be a positive number between 0 and 3, its convergence rate is better than the CRR model and even though it is slower compared to the Black- Scholes method, it is more accurate especially for longer dated options. The disadvantages include: it is less efficient to price the family of barrier options and it is rarely used for options with several sources of uncertainty such as real options and also in valuing options with complicated features as they tend to pose several difficulties.The discrete dividends with a continuous dividend yield q, often lead to a significant mis-pricing of the option near an ex-dividend date so most people model dividends as discrete payments on the anticipated future ex-dividend dates. Using the binomial model to discrete dividend payments we use the following rule; For all where is the present value of the -th dividend. Subtract this value from the value of the security price at each node (, ) at each step. Analyzing Joseph’s case; the dividends will be affected by the market volatility. Since they are earned annually or periodically, for the first year he will earn dividends given by 1*0.2*10000 pounds assuming a non volatile market. With the market volatility at 30% the dividends at WorldTech Plc for the 1st, 2nd , 3rd, 4th and 5th year will be given by; PV (0.7*0.02*10000) down= 140 PV (1.3*0.02*10000) = 260 upward increase. The dividends during the ten years will remain constant because they have a constant dividend rate and the market volatility is constant during this time period. Hence Joseph if he was to be at world tech he would be entitled to the above amounts depending on the market volatility. At GlobeTech assuming a volatility of 36%, the dividends would be; PV (1.36*0.2*10000) =2720 upwards PV (0.64*0.2*10000) = 1280 down In finding the value of Joseph’s options the expected values of the options is given by the following binomial tree taking into place the average market volatility of 25%; Taking into consideration 50 steps, the value of Joseph’s options is given by; 50steps*node value; where the node value is given by; Where Nu is the number of ups and Nd is the number of downs. Sn= 50000*1.25^ (25-25) =62500(assuming an equal number of ups and downs. This gives an option estimated value of 62500*50=3125000 to Joseph. This method proves to be more practically applicable to Joseph’s situation because it has not put into consideration any approximations and has taken note of the wider range of steps which are 50 hence providing a greater accuracy for the calculated expected value of the stock options. Since Joseph is highly uncertain on the market outcome, he would be at a better position if he relied on the outcome obtained by the binomial method. Depending on his origin he can make an appropriate decision since for European options, there is no option of early exercise and the binomial value applies at all nodes. For the American options their option may either be held or exercised prior to the expiry while for the Bermudan options the value at nodes and exercise is allowed. Trinomial method is more advanced than the trinomial tree model because it allows different stock prices ti appreciate, depreciate or stagnate with certain possibilities. Analysis of Joseph’s mathematical calculations to determine price barrier options and calculating price option will be an effective method of numerical calculation within the Black-Scholes sharing pricing model. The jump sizes areu and d and the probabilities Pu and Pd S(t)u with probability of Pu S(t + ∆t) =S(t) with probability 1- Pu-Pd S(t)d with probability Pd The assumptions are The underlying asset volatility σ = k Asset prices follow Brownian motion. R= risk free investment. Standard equation with no arbitrage is E{S(ti+i) S(ti)} = e r∆tS(ti) Average return from the asset should be equal to the risk return free return. Var[S(ti=1) S(ti)]= ∆tS(ti)σ2+0(∆) show an extra 4 parameter of the tree which comes from the requirement that the ascending jump is the reciprocal of the downward jump. Ud=1. Because the knowledge of jump size is already indicated u d and transitional probabilities of Pu and Pd the value of the underlying asset for nay sets of price movements; Assume middle jumps as Nu Nd and Nm the value of underlying share price at the different nodes is (j for time i) Sij= u Nu d Nd S(to)where Nu + Nd + Nm = n Trinomial tree The results show there exists a family of 3 models after imposing the 3 parameters on u, d, pu and pm Transitional probabilities will be given by Value Joseph’s option position GlobeTech K=25,000 σ = 25% r = 5% a share t = 10years * 1 Pu=0.235 Pm=0.125 Pd=0.64 Josephs options if he used WorldTech. K= 25,000 σ = 30 r= 2 T =10 Pu =0.325 Pm =0.234 Pd =0.909 The European option pricing will be for Globtech Cn.j=40,000 Expected stock value after 5 years = 200,000 Applying the concept of a trinomial tree using same methodology as binomial tree, the option of the interior nodes of the tree can be calculated considering it’s an option value at the future nodes. Option price at a time n, Cn puCn+1+ PmCn+1 the option price. The up move is puCn+1 the option price of the middle move is Pm Cn+1 + ascending pdCn+1 discounted by one step e-r∆t. The formula of induction formula is applied. The name of the algorithm can be easily accessible from this level. Cn,j = e-r When fewer times are modeled using the trinomial model the results are more accurate than the binomial results. It is more ideally used to analysis of resources and speed is a problem. On the other hand binomial is simpler to implement because vanilla options increase in steps results to rapidly coverage. Pricing for the WorldTech Cn,j= 25,000 stock value After 5 years 125,000 Comparing the option values obtained from the Black Scholes methods, Binomial method and the trinomial method it is evident that the binomial method yields a higher stock value compared to the other two methods. This is because binomial method does not take any approximations and the convergence rate is generally better compared to the rest. The binomial method is also more accurate compared to the Black Scholes method especially for longer dated options which is in this case is 5 years. Since the stock options are payable at the end of the vesting period Joseph can only earn his annual salary and dividends. Which are for WorldTech is 140 or 260 depending on the market volatility while in GlobTech the dividends is 2,720 or 1280. At GlobTech he has a salary of 25,000 and dividends of 2720 and he is also entitled to Stock Option value at the end of the 5 years. It is advisable for him to keep his current job The performance condition adds more uncertainty to the expected value of the stock options. He cannot be sure that by the end of the five years, the company stock price will be above 7.5pounds.if the stock price does not raise to that amount, Joseph stands to lose the stock options and this complicates Joseph’s position. If the stock price manages to raise to that amount, then he will exercise his stock options which is a rare chance due to the existing market volatility as it raised to 36% in the next year meaning they are highly volatile and unpredictable. Taking an average market volatility of 25%, the chances of the stock price reaching 7.5% are minimal hence would advise Joseph to quit his job because the existing performance condition puts his options at chance and he may never earn the 10000 stock options after the vesting period elapses despite his effort in the company for those years .the uncertainty is also supported by the number of employees who even left the company after ten years meaning that they were not able to exercise their stock options during that time period because the stock options are meant to keep and maintain the employees of the company. Depending on this trend and the market volatility and uncertainty, the performance condition gives joseph an only option of quitting his job. Bibliography Mackenzie Donald 2006 “How financial models shape markets” Cambridge ISBN 0 262 134 8 Black, Fisher Schole “The pricing of options and corporate liabilities.” Journal of political economy Sulvian Arthur 2005 “Economics Principles in actions” new jersey 07458 Michael Simkovic 2009The effect of enhanced Disclosure on open Market Stock Repurchases” Berkely Bus Amedeo Decesari 2010. “The effects of ownership and stock liquidity on timing of repurchase transaction” Read More