Option Pricing Issues - Essay Example

? Option Pricing Option Pricing An option is defined as a right to buy or sell a specific stock, debt, currency or even an index or a commodity, at a certain amount of money (Strike price) within a stipulated period of time. Stock options have a minimum amount of 100 shares to be delivered by the seller (writer) to the buyer (holder) in the contract. Securities on the other hand, are sold in cash thus settlement is reached. If an option is held longer than the specified time, it reaches the point of the expiration when the holder does not exercise his rights. The loss that amounts is in the loss of cash or amount paid for the option. Determinants of an option are stated as stock price, volatility, strike price, risk free (short term) interest rate and time to the expiration. The contract in this case, is called the option contract (Don, 2004, 142). Options are used by holders for leverage or for protection. The leverage function helps the holder to control the shares bought for a portion what they would have cost. On the other hand, protection measures are adopted when the holder wants to guard against price fluctuations. He enters in to a contract with the rights to acquire the stock for a fixed period and specific price. The contracts in either case should be highly observed and monitored for efficient outcomes. The methods used in pricing options have been applied for years and can only be effective if the worth of the option is achieved. This is determined by the probability that on the expiration, the option price will be on a substantial amount of money. Any holder of an option expects a gain on his underlying asset to attain the worth of holding for the time given. The Black Scholes and the Binomial method are the elaborated on below in determining the true worth of an option. The Black Scholes Model: This model dates back in the twentieth century in its application. It was developed by Fisher Black and Myles Scholes in 1973 hence the name Black Scholes (Marion, 2003, 16). It is still in use today. This model uses the theoretical call price where by the dividends amounting during the life of the option is not included in the computation.  Theoretically, the price of an option (OP) has been determined by the formulae given below: In this case: (Simon & Benjamin, 2000, 255; Brajendra, 2011, 372) The variables in the above formulae are expressed as shown below: S is the stock price X is the strike price t is the time remaining until the expiration, denoted as percent of a year r is the compounded risk-free interest rate predominant in the current market v is the annual volatility of stock price.  ln is the natural logarithm N(x) is the standard normal cumulative distribution function e is the exponential function Below are the necessary requirements for validating this model: Dividends are not paid during the stock period. Variance and interest rate does not change in the course of the option contract. There is no discontinuity in the stock price i.e. a shift from one price to another like the case of tenders. This model applies volatility and normal distribution to determine the movement of options. The Excel add-in format can be used to calculate the normal distribution. Volatility on the other hand, can be implied or historical. The implied volatility of an option allows market traders to observe the current prices of options to determine how volatile they are. This is done by calculating the standard deviation i.e. v2, and in this case all other variables have to be known. Nevertheless historical analysis is not left out. The traders have to observe the performance of the option over past years to assess volatility. This measure is, however, not reliable on its own but provides an insight of the volatility of the option. Volatility has been defined as the unknown change in price of an underling asset during the specified time period which gives the true worth of an option. Volatility of an underlying asset can be assessed and comparison made with a non-volatile asset. Consider the illustration below: Stock: Prices: Nonvolatile (N) 40 45 50 55 Volatile (V) 30 40 50 60 In this example, if the chance of the stocks N or V will be at a specific price as in the past is 1/5 at the expiration. The N stock has a range between $40 and $55, and that of a volatile nature has a range of $30 and $60. From this illustration, we deduce that a call for the volatile asset will call for more premiums at the $50 price on the expiration, while the nonvolatile price stands at the same price $50. This is because there is a 20% chance that V stock will be worth $20 of premiums per share, and 40% chance that N stock will be worth $10 per share. If the stock price was to be equal to the strike price on the expiration, then the option is worthless. Even though the trader knows that the least the price of the two stocks can fall is $30 for V and $40 for N, this is irrelevant should the case of strike price and stock price equate on the expirations. There is a 50% chance that both V and N will expire worthless. The V call has an opportunity of paying $20, and N has a chance of paying $10; with this in mind, then the premiums payable for the volatile assets will be higher. This is because the expected returns or the payoff from the volatile asset is higher. The strategy behind acquiring an underlying asset is to determine the volatility. Undoubtedly, a trader cannot enter an option contract if he presumes that the stock will have the same or lower prices and yet he doesn’t have the volatile computation in regards to the asset. The underlying concept is the unknown fluctuation value before the expiration that gives a particular asset its worth. The market traders view that the stock price will go up or down in the future are completely irreverent; unless the risk of generating the return is determined. The model assumes that the return on assets is independent on the expected growth. The black Scholes price is determined as the price of compensation for a trader for exercising a call and protecting him from the risk therefrom i.e. hedge. The risk free rate is often applied hence this model progresses as a risk neutral valuation i.e. risk is not considered in this case; the option price is independent of the risk thereto. Therefore, two investors can be indifference in risk e.g. risk neutral or risk averse, but on the exercising the call, the same value of the option is given (Jeffrey & Donna, 2005, 31). Advantages for using this model include ease of use to calculate premiums within a very short time. The model allows for a large amount of data to be analysed for options. Speed is of the essence when computing premiums to be paid for an option. Nonetheless, it has one major limitation i.e. it calculates the option price at one point in- the expiration date. The model doesn’t provide for options that are exercised prior to the expiration or in the course of the specified period (Kent, 2009, 309). This is specifically to American type of calls that are exercisable at any point of time of the options. This is unlike the European calls that are exercised only on the expiration (Eckhard & David, 2006, 290; Steven, 2005, 3). One factor, however, must be emphasized on, that irregardless of the exercise time, both the American and European models amount to the same price. This is because dividends paid are not accounted for in the computation of premiums. The Binomial Model: The Binomial Model differs from the Black Scholes model in that it assumes different prices for the option on the expiration. It discretises both time and price of an underling asset then maps them on a binary tree. In this model, an assumption is made that the underlying asset price can only increase or decrease with time, until the end of the contract when the option is worthless. This model is applied for American options which can be exercised before the expiration date. To be more precise, the model represents all possible prices on a binomial tree, for an underling asset, within the given time period of the contract. If two prices were considered u and d, and pseudo-probabilities applied for the likelihood of the occurrence, then we get pu and pd. The formula below is deduced: u = ev? dT , d = e?v? dT ;(u ? d = 1) . pu = 1-pd dT is the time period v is the volatility e is the worth of the option currently By applying the risk free rate the pu =(er.dT –d)/(u-d) (Yue, 2008, 315) The model allows the trader to iteratively the price of the option using the binomial tree. The value of the option can then be determined as vt = (pu .v u, t+1+ pd. vd, t+1) e-r.dT. This is worked out from backward from the binomial tree. The option price for each nodule in the tree is used to derive the option price for the next in the tree (Neil, 2005, 6). This module has been simplified through the use of softwares in excel add in to compute the option price and the value of the option before and at the expiration. The dividends are given as a single yield during the year; hence the American options incorporate dividends allowing for call exercise before the expiration. Illustration below allows traders understand a simple case scenario: Consider the share price S is $300 at time 0. t = ?t, the price can increase up to $350 or decrease up to $290. The strike price of the call is $320. The exercise price can be seen as either $350 ? - $30 (up) or $290 ? (down). The valuation expected can be denoted as ? then; 350 ?-30=290 ? ?=0.5 which is the risk neutral valuation (Michelle et al, 2008, 122). The underlying assumption made is that we do not know which price will be realized. In our example above, dividends are not incorporated. The risk free rate, r, is a constant over the period, and bids will not be exercised in this case (Shreve, 2004, 5). When dividends are present then, the value of an American call is given by the formula below: (Pcu +(1-p)cd)/(1+r) , s-k An early exercise of the call should be made if the dividends paid is less than k-pv(k) (David, 2008, 114; Don & Robert, 2009, 16). The intrinsic value should then be determined. The binomial model has an advantage over the Black Scholes in that it allows for American options to determine the worth of the options which are exercised before the expiring date is reached (Kent, 2009, 309). The binomial tree allows traders to assess the price of exercising the call at any point of the option contract. It is accurately used to price American options. This model has a limitation in that it is very slow to compute for large options. Speed is at the core in exercising calls thus time is taken before the traders can make a decision at the point of time to exercise the call (John & Robert, 2006, 15). The two models compare in that both use the underlying principle of stock price. The prices follow a stochastic process as described by Geometric Brownian. The two actual follow the same computation and binomial model is an extension of the Black Scholes. The binomial process has infinite steps in its calculation in order to obtain the different prices for the call to be exercised. The graphical interpretation from excel worksheets allows the trader to carry out different computations to obtain a convergence of the numerous steps for either of the two models. This enables the trader to assess the relation between the strike prices, stock prices, volatility, risk free rate and the time to the expiration, and the impact deduced on changes made to these variables. The Black Scholes model assumes volatility in the period of the contract, but the binomial model assumes a constant volatility and the risk free rate. The two methods use the intrinsic method of valuation of options. This is the price at which the option is said to be in-the- money (ITM). The international financial reporting board had attempted to change the valuation of options from intrinsic value to fair value, a move that was not practical. The best valuation is the intrinsic as was developed from the twentieth century. IFRB had to review this valuation requirement to the old method of valuation in 1993 (Thomas et al, 2008, 345). Various studies have been done to show how managers apply the valuation models. According to Eckhard & David, 2006, (25) firms have seen to use a considerable discretion in the determination of the input variables for the option under consideration. This has resulted in immaterial differences in the fair value of the options, on exercising the call. Other firms have been seen to underestimate the stock expenses when disclosure of this expense is not made in their financial statements. Firms that have been seen to use zero rate of volatility in their assumptions results to underrating the stock price. This has been compared to firms who do not zero the volatility of the options (Joe, 2005, 2). Management discretion and lack of audits in firms reduces reliance in the financial reporting of the compensation of these firms (Basu, 2011, 122) The Greeks influence on option pricing: The Greeks apply delta (?) to denote the rate of change of the underlying asset, either a decrease or an increase. When stock prices go up, the call option increases and that of put options decreases. Vice versa is also true. The strike price applies as in the two models. The option is said to be out of money (OTM) if the underlying price of the stock is lower than the strike value. The stock option is said to have no intrinsic value. Six indicators are used by Greeks i.e. delta, gamma, rho, zeta, theta and Vega. These are complicated and delta is mostly applied. Gamma shows the acceleration of delta. Theta measures the time value changes. Rho indicates the sensitivity of an option in regard to changes in the risk free rate. Vega measures the volatility in historical aspects. Zeta shows the implied volatile measures (Parames, 2008, 51; George, 2005, 176). The delta value has a range of -100 to 0 for a put option, and 0 to 100 for a call option. The put option has a negative since it does not have any relationship with the underlying asset price; the call option has positive value as there is a relationship with the price of underlying assets. If the call prices increase so does the premium for the call (see the summary below for influences) (Courtney, 2008, 46). If a trader has an option of selling, time is of the essence. Time value should be observed on aspects of theta. A further analysis of the volatility should be done. If an option is highly volatile, it offers better premiums and better profits (Joe, 2006, 58). If a trader has a buy option, he can opt to maximize delta hence maximal intrinsic value. Time value should be assessed here in order to allow ample time for the option to growth. One needs to understand that too much time may eat away the profits. Timing is the key hence theta should be critically observed. To determine the best option to buy at cheap and fair prices, Vega and zeta are observed. The two measures of volatility enable the trader to know the point at which to buy i.e. low prices, and the point at which to sell, when the option is expensive. All these indicators are computed through softwares that are widely available online (John, 2005, 118). The time decay is one of the vital aspects that are applied in the trading sector for options. The influence of volatility, time to decay and the underlying price of an asset are illustrated below for a call and put option: Options Increase in volatility Decrease in volatility Increase in the underlying asset price Decrease in the underlying asset price Increase in time to expiration Decrease in time to expiration Call + - + - + - put + - - + + - An example is illustrated below: Consider an at-the-money call option from oil that has a delta of 0.8; if oil makes a 20-cent move higher (an assumption) the premium on the option will increase by approximately 16 cents (0.8 x 10 =16), i.e. $900. Note that a cent is worth $250 for the premiums. Delta in the Greek model has been seen to increase towards the end of the specified period. It is not a constant and moves as the volatility changes. Gamma is usually at its highest nearing the end of the option contract. The Vega indicator falls nearing the expiration period (John, 2008, 61). Taking another example, if a trader owns a put of 60 in November, and the delta is -23.5 then a movement of the price by $2 will substantiate to a loss of $47. Delta is the most important factor to consider during the beginning of the contract. It is such a sensitive measure that needs to be monitored carefully since it can have different effects if not monitored keenly (Shani, 2004, 86). In conclusion, the option pricing models are used by traders to assess the valuation and worth of an option upon exercise. The use of Greeks allows traders to understand the changes effected by the sensitive indicators to the intrinsic value. It generally enables a trader to make profits if he monitors his trading with the six indicators. Delta and theta are the two most important to consider, in the beginning and towards the end of the period consecutively. Calls should be exercised when they at the cheapest price, and sold when they are expensive considering volatility. Highly volatile options calls for higher premiums and high profits are earned. Bibliography Basu, A., (2011), Studies in Accounting and Finance, America, Pearson Education India Brajendra, C. (2011), Dynamic Mixed Models for Familial Longitudinal Data, New Mexico, Springer. Courtney, S., (2008), Option Strategies: Profit-Making Techniques for Stock, Stock Index, and Commodity Options, Chichester, John Wiley & Sons. Dan, P., (2008), Trading option Greeks: how time, volatility, and other pricing factors drive profit, Chichester, John Wiley & Sons. Don, M. & Robert, B., (2009), Introduction to Derivatives and Risk Management (8e), Columbia, Cengage Learning. Don, M., (2004), An introduction to options and futures (5e), Pennsylvania, Dryden Press. Eckhard P. & David H., (2006), A benchmark approach to quantitative finance, New York, Springer. George, F., (2005), The options course: high profit & low stress trading methods (2e), Chichester, John Wiley and Sons. Jeffrey, O. & Donna L., (2005), Advanced option pricing models: an empirical approach to valuing options, New York, McGraw-Hill Professional. Joe, D., (2006), Futures & options for dummies, New York, For Dummies. Joe, D., (2011), Trading Futures For Dummies, Chichester, John Wiley & Sons. John, F., (2005), Excel models for business and operations management (2e), Chichester, John Wiley & Sons. John, H., (2008), Topics in Derivatives, London, BoD – Books on Demand. John, V. & Robert, J. (2006), Binomial models in finance, New Mexico, Springer Science & Business. Kent, B., (2011), Capital Budgeting Valuation: Financial Analysis for Today's Investment Projects, Chichester, John Wiley and Sons. Marion, A., (2003), Real options in practice, Chichester, John Wiley and Sons. Michelle, R., Martin, S. George, H., (2008), Corporate Finance: A Practical Approach, Chichester, John Wiley & Sons. Neil, C. (2005) Black-Scholes and beyond: option pricing models, New York, McGraw-Hill Professional. Parames, W., (2008), Valuation Of Options, New York, McGraw-Hill Education. Shani, S., (2004), A currency options primer, Chichester, John Wiley & Sons. Shreve, E. (2004), The binomial asset pricing model, London, Springer. Simon, B. & Benjamin, C. (2000), Financial modeling (2e), New York, MIT Press. Steven, E. (2005), Stochastic Calculus for Finance: The binomial asset pricing model, London, Springer. Thomas, R., Hennie, V., Elaine H. & Michael, A., (2008), International Financial Statement Analysis, Chichester, John Wiley & Sons. Yue, K., (2008), Mathematical Models of Financial Derivatives (2e), New Mexico, Springer. Read More