The Call Price as a Function of Stock Price - Essay Example

NAME OF THE STUDENT: STUDENT ID: COURSE UNIT: THE INSTITUTION NAME: Table of Contents Table of Contents 2 1.0 Introduction 3 2.0 FIGURES & ANALYSIS 4 2.1.1 The Call Price as a function of Stock Price 4 2.1.2 Put Price as a function of Stock Price 5 2.1.3 The Delta of a Call as a function of Stock Price 7 2.1.4 The Delta of a Put as a function of the Stock Price 7 2.1.4 The Call and Put Vega as a function of the 9 3.0 CONCLUSION 10 THE BLACK SCHOLES MERTON MODEL 1.0 Introduction The Black Scholes Merton Model was developed in 1973 by Fisher Black, Robert Merton as well as Myron Scholes as a mechanism to be used in estimating the fair prices of options traded in the market (Jensen, Fischer and Myron 239-251) The Black Scholes Merton Model is a mathematical model used in financial market and it relates changes in prices of financial instruments such as stocks over a given period of time. It is often used to estimate the price of a European Call option. This model in most cases assumes that the price of heavily traded financial assets takes a lognormal distribution pattern with a drift and volatility. In relation to the stock option, Black Scholes Model takes into consideration the constant changes in the prices of stock as well as the time value of money, that is, the option’s strike price in the market until the time of expiry of the option (Henderson 317). As stated by Chesney, Marc, and Louis (277-281), this model has a number of assumptions which includes the following:- The stocks traded in the financial markets do not pay dividends; the shareholders should not expect to gain any form of returns from their investments’ yet The prices of stock tend to behave in a random manner; the stock prices often follow a normal geometric pattern while having a constant drift and volatility The options in the financial market are always European The volatility and risk free rate of the traded stock assets tend to be constant throughout the lifespan of the option in the market. The Black Scholes Merton Formula is as stated below:- Where N(d1), N(d2) is the cumulative distribution of a standard normal distribution T is the time taken for an option to mature S0 is the stock price of the underlying traded asset X is the strike price of an asset r is the risk free rate compounded continuously δ is the volatility of the expected returns of the underlying stock traded 2.0 FIGURES & ANALYSIS 2.1.1 The Call Price as a function of Stock Price Analysis A call option being an agreement between a buyer and a seller, whereby the seller willingly sells a financial asset to a buyer based on their agreement, the buyer eventually pays a price known as premium for the right to own the asset in question (Grabber 244-247) However, the premium to be paid by the buyer is always influenced by the stock price that is in the market at a given time, that is, the price at which the call options are traded in the financial market often increase with increase in stock prices. For instance, as indicated in the graph above, when the stock price was trading at $75, the call price was being traded at $0.23 each; this however started to experience an upward trend with the increase in the stock prices. When the stock price was at $125 which was the price at which the assets were being exercised in the financial market, the call price had also risen to $13.02 as shown in the graph above. 2.1.2 Put Price as a function of Stock Price Analysis Ideally, the Put option buyer often believes that the price of the underlying asset will fall by the exercise date of the asset. The buyer often expects the price of the put option to go up with decline in the price of the underlying asset below the strike price. The seller of the put option on the contrary normally wants the price put option to be become worthless as the price of the underlying asset above the strike price increases. In most cases, the price of the put option that is purchased is normally considered as a long put while the put option that is sold is often considered as a short put. Therefore, as the financial market stock prices increases, the put option on the other declines as desired by the sellers’ of the put option. The graph above indicates the trend of the fall in put prices as the underlying stock prices increase. 2.1.3 The Delta of a Call as a function of Stock Price Analysis As indicated in the graph, the price of underlying asset tend to decline as from $125 which was the exerciseprice since the call in the in this case is regarded as moving out of the money, the delta as well falls quite faster untill it is surfaces off.Likewise, when the call moves into the money, the delta tend to rise slowly. This implies that, when holding an option that is currently out of the money call, the reaction of the delta call tend to be high in order to increase the price of the underlying asset as compared to when holdinf the same option in the money call. 2.1.4 The Delta of a Put as a function of the Stock Price Analysis As shown in the graph above, as the time remaining to expiration draws nearer, that is, the exercise date at which the strike price will be $125, the time value of the option seems to vanish as the delta in the money option increases whereas the delta out of the money option decreases. 2.1.4 The Call and Put Vega as a function of the Analysis The Option Vega normally indicates the impact of the variations in the underlying volatility of the price of the option.Ideally, the Vega of an Option often shows the change in price of the Option for every given percentage change in the underlying volatility. The option prices in the financial market increases with increase in volatility,that is, when the volatility increases, the price of the options also increases and likewise when the volatility reduces, the price of the options also goes down (Black, Emanuel and William 36-37) As indicated in the graph above, when the time remaining to the expiration of the option is more, the Vega will be high, this shows that time value is associated to the amount of premium payable and is often sensitive to the variations in volatility. Therefore, the Vega of the options increases with increase in the underlying prices until it reaches the expiry date after which it normally begins to drop. 3.0 CONCLUSION Based on the discussions above, a number of aspects should be taken into consideration when trading in the financial market:- The Time to Expiration; a trader should note that, the longer the time to expiration, the more the chance the buyer of the option has for the underlying price to move in the desired direction and the more costly the option will be (Longstaff and Eduardo 09-11). The Implied Volatility; the implied volatility should be monitored closely in that when they are high, options tend to be much expensive in comparison to when they are low (Longstaff and Eduardo 09-11). Moneyless; In most case, the out of the money options tend to be less expensive to purchase than in the money options .Therefore, for call options, the higher the higher the exercise price, the less costly the option and at the same time put options with lower exercise price will be cheaper to acquire (Longstaff and Eduardo 09-11) BIBLIOGRAPHY Black, Fischer, Emanuel Derman, and William Toy. "A one-factor model of interest rates and its application to treasury bond options." Financial analysts journal 46.1 (2010): 33-39. Chesney, Marc, and Louis Scott. "Pricing European currency options: A comparison of the modified Black-Scholes model and a random variance model." Journal of Financial and Quantitative Analysis 24.03 (2009): 267-284. Grabbe, J. Orlin. "The pricing of call and put options on foreign exchange." Journal of International Money and Finance 2.3 (2013): 239-253. Henderson, Vicky. "Black–Scholes Model." Encyclopedia of Actuarial Science (2003):311-321 Jensen, Michael C., Fischer Black, and Myron S. Scholes. "The capital asset pricing model: Some empirical tests." (2010):236-312 Longstaff, Francis A., and Eduardo S. Schwartz. "Valuing credit derivatives." The Journal of Fixed Income 5.1 (2005): 6-12. Read More