Financial Engineering Option Valuation Report (Coursework Sample)

Option Valuation Report
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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;
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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 option's 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 re...