The Peculiarities of Derivative Securities - Coursework Example

Securities price behavior depends much on economics climate (positive or negative growth), and on existing deposit and credit risk-free rates, which can also change and depend in turn on economic processes. It's familiar that deposit and credit risk-free rates can increase or reduce economic growth. It explains an existing interrelation between economic development and assets value. If we analyze any specific asset, for example stocks of a joint-stock company, we can surely emphasize, that economic successes of this company, its marketing divisions influence the price behavior. Particularly, stock price depends on 3 factors: annual dividends, stock sale price and interest rate for discounting. All these factors are undefined. If our security is a bond, only one factor is undefined, - interest rate. In the primary market, state emits securities to cover state expenses. Dealers create a secondary market, offering to sell or buy these securities between the dates of issue and maturity. Apart of marketable securities state on behalf of government sells several types of non-marketable securities, which can't be sold or given any other person. Gonchar, M. (2002, p. 28) pointed out that the holder (for instance, in the USA) has a right to repay them in a commercial bank, in Federal Reserve System banks, in a public treasury. Private investors own a significant part (75% in the USA) of government securities. Public institutions and Federal Reserve banks own about 25% of these securities. Securities price drop, generated with interest rates increasing, can cause problems for those, who bought them during the period of high price. Financial institution, which bought securities in the time of low interest rate, will experience losses, if it sells them after the rate uprising. If the rate is falling, financial institutions have an opportunity to attract necessary finances with securities selling. During the interest rate uprising we observe converse effect - financial institutions try not to sell government securities, avoiding capital losses. As security buyers don't know the way, in which interest rates will change, they can't avoid the risk of their securities price drop. This is interest-rate risk. Government securities with short repayment period have a little interest-rate risk, as their prices don't fall much during interest rate increasing. The situation with long-term securities is quite another. Their prices plummet during interest rate increasing. It testifies about a considerable risk, due to holding securities with low annual interest bearing over a long period. Another kind of risk aligned with dividend policy of a company, which emitted securities, for instance, stocks. There are some other accidental causes, which influence on security price, which are called psychological by analysts. So, as we've noticed, the price behavior such an asset as stock is a random value. Finances theory and financial mathematics has a task of building veridical models of stock value evolution and calculating on their base fair option price and investor strategy (investment portfolio) in the security market with the help of statistical data. Simple models are important. Let's analyze here discrete models of asset value evolution and related questions of fair option price calculating and hedging strategies. The development of calculating fair option price researches development started with famous Black-Sholes theory. The Black Scholes Model The seminal work of Fischer Black and Myron Scholes in 1973 produced an elegant closed form solution for pricing European style call options on stock. The standard Black-Scholes equation and its derivatives have dominated the derivatives markets for 25 years. According to F. Black and M. Scholes (1973, pp. 637-659), firstly, it is useful to examine the assumptions underlying the mathematical alchemy used to derive the Black-Scholes equation for the pricing of options: The price of the underlying asset follows a Markov process with an average m and volatility s. The short selling (selling of securities you do not own) with the full use of the proceeds is allowed. This ensures that we can invest or borrow money at the same interest rate. There are no transaction costs or taxes. Securities are fungible and infinitely divisible. There are no dividends during the life of the option. There are no riskless arbitrage opportunities, i.e. we cannot make money for free. Arbitrageurs are traders who exploit the mis-pricing of assets to lock in risk-free money. Security trading is continuous. The risk-free rate of interest, r, is constant and the same for all maturities. To make the understanding of the model simpler, let's use the special case of the Black-Scholes equation called the Black model used to price options on forward assets. This abstracts the pricing of the forward, which is implicit in the Black-Scholes equation. Let's divide the model describing into next phases. Option definition Simple options can be either call options or put options. Call options give the purchaser the right (as compared to the obligation) to 'call' the underlying asset from the writer of the option (the writer is the party that sells the right) and pay the strike price (K) for the asset. The purchaser of the option will only exercise this right if the ruling price of the asset (S) is above the strike price of the option (K) (you would not pay more for something than you need to!). Put options give the purchaser the right to 'put' or sell the underlying asset to the writer of the option and receive the strike price (K) for the asset. The purchaser of the option will only exercise this right if the ruling price of the asset (S) is below the strike price of the option (K) (it is irrationally to sell an asset at a lower price than is possible in the market!). Calculation of the forward price The forward price is a function of: the spot price today - S the cost of money - r cash flows generated by the asset (dividends or coupons) - d (this is a relaxation of the starting assumptions for the Black-Scholes equation). time - T. The forward price is determined as: F = S+Interest - (Dividends + Interest on Dividends) Where interest is the cash flow cost (at the risk-free rate r) of borrowing money to pay for the asset with a price of S for the period T. Any cash flows received are reinvested from the date of receipt to the end of the period T, at the same risk free rate r. This is the arbitrage-free forward price F. The academic texts calculate it as The important results of this equation are that if the cost of money is above the income generated by the asset the forward price F will be above the spot price S. The converse also applies. Example If the spot price is 100, the risk-free rate is 12% and dividends are 2%, paid annually, then the forward price is: 100 +((100*12%) - (100*2%)) = 110 The normal distribution This is the infamous 'bell' curve that seems to reasonably accurately describe the distribution of many real-life variables such as the heights and IQ of the population and, in the case of the financial markets, the distribution of daily returns of an asset (the percentage change in the price of an asset from day to day). Kogut B. and Kulatilaka N. (1994, p.56) indicated that he actual distribution is well represented by the normal distribution but generally in the real world there are more results in the centre of the distribution and at the tails (fat tails). This real distribution is termed leptokurtic. For the purposes of the development of my model we will assume that the normal distribution applies. The Black Option Model This phase will put the previous phases together: The forward price The future distribution of the forward price The fixed forward price will have a distribution of uncertainty, represented below by the normal distribution. The width of the distribution is dependent on the expected volatility of the forward price, s. The average of this distribution is the forward price F. The option strike relative to the forward price F If the dealer writes a put option to a client at a strike price of K, which is the forward price F, the client will 'own' the portion of the distribution below the strike. Remember the holder of the option will exercise the put option when the price of the asset falls below the strike price. Hedging costs and option premium R.C. Merton (1973, p.149) confirms Having written the option the trader is now exposed if the asset price collapses. In order to neutralize this downside the writer of the option must hedge the option. As he is exposed to the price of the asset dropping he will need to execute a transaction that will benefit from dropping asset prices. This is achieved by selling the asset short. Profits on the hedge will offset losses on the option. The trader will hedge in the underlying market represented by the forward and its distribution with an average F. However, the distribution, the option writer is trying to hedge, is represented by the un-shaded portion on the right hand side of the strike, K (he sold the other portion to the purchaser of the put option). The average of this portion of the distribution is different from F and is represented by F*, which is higher than F (the option chopped off the low-priced values). The difference between F and F* is the expected hedging cost of the option. Expressed another way in trading terms, the option writer is 'long' or has bought a distribution with an average of F* (he sold the rest to the purchaser of the option). To hedge he will sell in the underlying forward market with an average of F. Since F* is higher than F this represents a loss and for the option writer to break even he must charge the client. This loss is in future value terms. Using the risk-free rate, this value can be present-valued to today. This PV is the option premium that the client must be charged. Now lets' develop a discrete model, offered by J. Cox, S. Ross and M. Rubinstein, which describes two assets market model - bank account and stock . According to this model capital evolution on bank account during time period [1,N] develops according the next law: , where , (1) r - deposit interest rate, - opening capital cost on bank account. We also make an assumption that stock value changes "chaotically" during the same time period: , where , (2) - first stock value, - random variable , which takes on two values a and b from inequaion -1 Read More