Risk Management: Principles and Applications - Assignment Example

?Risk Management: Principles and Applications Options on stocks Options on stocks give their owner the right to purchase or trade shares of a particular stock, at a specified price. It is vital to compare the market price of the stock, with the exercise price of the option (Hull, 1997). A short position in calls Selling a call on the stock offers some limited protection against modest price declines; however one would still incur losses if the prices were to fall by a large amount. This is because the potential profits for the call writer are opposite to those of the call holder. If the market price for the asset is greater than the strike price, it will be worth while for the holder to use the option. In this instance the option writer is forced to trade the asset to the option owner. The price at which a call is traded is called premium. In the case of calls, a higher stock price means that other things being equal; the option is more likely to be in the money on maturity. Therefore, a higher S is associated with a higher premium (Hull, 1997). Conversely, the higher the exercise price K, the lower is the premium. A higher exercise price makes it less likely that the option will be in the money on maturity. The time to maturity t has appositive effect on the option price: the greater t is, the more time the stock price has to climb above the exercise price. The volatility of the stock measured by the standard deviation of its returns ?, a;so has a positive influence on the call price. This due to the fact that call option holders are only concerned with the gain potential of their option, and this is greater the higher is the likelihood that the option will be beneficial on ripeness. Finally, the rate of interest, r has a positive influence on the call price. The greater the rate of interest r, the lower the strike value to be paid on maturity (e-rtK). The price of a call is given as C=S. N (d1) –K. e-rt. N (d2) N(.) is the cumulative distribution function of the standard normal random variable, and where the value d1 and d2 are defined as d1=ln (S/K) + (r+?2/2) t ?vt d2=d1-?vt Strike price=879000 t=91day (0.25) Volatility per annum=0.263 r=0.034 e-rtK= e0.034.0.25.$ 879000 = 227,349.9673 d1= ln (879000/736000) + (0.034+0.263/2) 0.25/v0.263 v0.25= 0.9532 d2= 0.9532-v0.263 v0.25=-0.06968 Nd1=0.3289 Nd2= 0.0239 Replacing them in C= (879000) (0.3289) - 227,349.9673 (0.0239)= $ 23,476.6458 Using put option Put options are the options which give the holder the right to sell the specified asset at the specified price. They are issued by sellers or their writers. The price of an option is known as premium. The value at which the underlying asset can be sold is called the strike price (Brigham, Gapenski & Ehrhardt, 1999). If you have a long position, one of the strategies to be utilized in order to hedge against a price decline is a long position in puts. One could buy put options to be fully hedged against price decline, but could still profit from price raise. The purchaser of put options can hedge their drawback price risk for a time and still gain from the potential price increases if the market should rise. The holder pays a certain premium to shield against a likely loss. Once the premium is remunerated, the holder has no further obligation. Depending on value progress, the manufacturer can either take or leave the assured price. If the price of the stock goes down, one has a price protection with the put option at the selected strike price. The same aspects that influence the cost of a call, also influence the price of put options on stocks. The premium on puts is now a decreasing function of the stock price and an increasing function of the exercise price. The conditions that should be fulfilled for a put to be in money: an increasing function of the time to maturity and the volatility of the stock, and a decreasing function of the rate of interest: this last result is owing to the truth that an elevated rate of interest decreases the present worth of the exercise cost to be received on prime of life. The premium on the put is therefore, P=C-S+e-rt K = 23,476.6458-879000+227,349.9673=162,926.6131 Using a combination of options It is possible to join the call and the put options to make a new security with the required payoff profile. This security that is achieved is known as a synthetic option. There is a particular grouping of puts, calls and the fundamental asset which allows us to obtain an entirely riskless speculation, this speculation is equivalent to the present worth of the exercise price. One can use a bear put spread This is a strategy of buying an amount of put options at a definite strike value and selling the equal amount of put options (on the same principal asset) at a lesser strike price. Given that the put options with elevated strike price is more costly, the investor incurs a definite cost whilst setting up the plan. Conversely, if the asset price remains lower than the strike price of the options he purchased, the dealer builds a profit from the variation between the strike price and the stock price at increase, less the cost of setting up the strategy. Lower strike price=736000. Higher strike price= 879000 Suppose the premium of the lower strike price put is at 182300 and the higher price one is valued at 222000, then the holder will under go a cost of 39700 of setting up the strategy. Initially we have: S+P-C=e-rt K where S represents the cost of the principal asset that is the market value of the stock, P is the value of the put option, C represents the price of the call, K represents the exercise price, r is the rate of interest, and t is the period to termination of the options. The expression e–rtK is the present value of the strike price; e–rt represents the discounting factor at rate r, consistent with continuous compounding. The left hand area of the equation can be perceived as a ‘synthetic option’, created of a long position in the stock (S), a long position into the put on (P), and a short position into the call (C). Equation (2) therefore implies that the synthetic option on the left-hand side is as important as the inexpensive value of the exercise value on the right-hand side. The second one can be taken as the current value of a sum K to be obtained at expiration. This is a rigid sum, and can be considered as a riskless speculation. Hence, the synthetic option on the left-hand side is as important to investors as a riskless speculation. The basis for this end result is that the payoffs on a long stock, a short call and a long put give rise to a steady payoff profile, as a function of the stock cost (Kolb, 1995). 1b) Using a call option By using a call option on the stock, it would be profitable when it is in-the-money, which occurs when the market price, S is greater than the exercise price of the option (S-K>0). The holder can purchase the stock at the strike value and subsequently sell it at the market price. When S-K 0, the put option is out-of-the-money. The intrinsic worth of a put option is the greater of K – S and 0. It is positive once the put is in-the money, and equivalent to zero if it is out-of-the-money. When S = K, the put is at the cash and its intrinsic price is also zero (Kolb, 1995). P=C-S+e-rt K=616331.8476-879000+227,349.9673= -35318.1851 Using combination of options A bear call spread can be used as a strategy. This is a strategy of purchasing a number of call options at a definite strike price and trading the same quantity of call options at a lesser strike price with the similar maturity and principal asset. Since the worth of the calls with the lesser strike price is elevated, the dealer earns a little profit if the stock price decreases below the strike price at which he traded the options. Conversely, he confines his downside. If the stock price rises significantly, he would lose cash on the options he traded but recover a few of it as stock price crosses the strike price of the options he purchased (Elton, Gruber, Brown & Goetzmann, 2011). A stock of the company is trading at $ 900,000 trading a call option with a strike of 879000 and purchasing a call option with a strike of $ 885000 would create a bear call spread. Assuming that the call at $879000 sells for $289000, and the call at $885000 sells at $ 360000, if the stock price at maturity falls below $879000, the investor would earn $71000 in profit since neither calls would be exercised. As the synthetic option and the riskless asset give the same payoff, they must be similarly important to investors: therefore, they must have the equal price. Equation (2) can also be written as: C=P+S-e-rt K (3)which means that buying a call is the same as buying a put option and the stock and borrowing the amount e-rt K. Alternatively; P=C-S+e-rt K (4) Equations 3 and 4 indicate that the put-call parity relationship. They can used to find the right price of a call, once we identify the price of the put P, the stock price S and the exercise price K. otherwise; they provide us with the value of a put P after we know the value of the call C, the stock worth S, and the exercise worth K (Cornell, 2009). Question 2 The price of the stock after one year is 1225000 while the strike price of the call used in question 1a is 879000. This means that it is worth while to the holder. The holder can use the call to purchase stock at the strike value or he can sell it to another person at a profit. The call is likely to sell at a higher premium that the seller bought it since it is worth. In the case of a put option the stock price being higher than the strike price indicates that the put option was not worth while. The holder is out of money since he can sell the stock at a higher price than the strike price. 12250000-879000=11,371,000 is the distinction between the strike cost and the market cost of the stock. This could have been better if the holder had bought a put option of a higher strike value (McKeon, 2011). In the case where the holder uses a bear put spread to hedge against price decline, he makes losses as the price of the stock has increased more than the strike price. The holder incurs a loss of 39700 plus the difference between the cost of the stock and the strike price of the put option implying that the option was not worth while. He would have made a worth investment if the price was below the put strike price. The call option used is worth while as it has a strike value of 879000 and the prices have increased to 1225000. The call is in the money and by exercising it the holder will benefit. This means that the holder can buy the stocks of the company at the strike value. The price of the call is 616,331.8476 which are lower than the strike value implying that the holder can still sell it the option at a profit (The Nobel Foundation, 2008). Since the stock value has increased, it means that the dealer of the call loses but he can recover a few of it as stock price of the options are purchased. References Brigham, E., Gapenski, C. & Ehrhardt, C. (1999). Financial Management: Theory and Practice. 9th ed. Fort Worth, TX: Dryden Press. Cornell, B. (2009). Warren Buffett, Black-Scholes and the Value of Longdated Options. The Journal of Portfolio Management, 36(40):107-111. Elton, E., Gruber, J., Brown, S. & Goetzmann, W. (2011). Modern Portfolio Theory and Investment Analysis, Eighth Edition. New York: John Wiley and Sons. Franklin, E. and Ma, C. (1992). Futures and Options. New York: McGraw-Hill, Inc. http://www.nobelprize.org/mediaplayer/index.php?id=1134 Hull, J. (1997). Options, Futures, and Other Derivatives. 3rd ed. Upper Saddle River, NJ: Prentice Hall. Hull, J. (2011). Fundamentals of Futures and Options Markets, Seventh Edition. New Jersey: Pearson. Kolb, R. (1995). Understanding Options. New York: Wiley. McKeon, R. (2011). Traders Reacting to Bad News: The SEC versus Goldman Sachs, Working Paper series. San Diego: University of San Diego. The Nobel Foundation. (2008). Interview with Myron S Scholes. Retrieved from Read More