Derivative Securities in Finance - Assignment Example

Derivative Securities-Finance Name of Student Institution Question 1 a) A one-step binomial tree Current ten month gold futures price (St) = 1734 Volatility= 21% per annum Risk-free interest rate= 5% per annum with continuous compounding Strike= 1700 Price of 10 month European call option on the gold futures: Where, e =2.718282, U=e σ √change in t and d=e -σ √change in t U= 1.1912, d= 0.8395 and t= 0.8333 Price of the stock in up move= Current futures price (1 + u) Price of the stock in down move= Current futures price (1 + d) Price of the stock in up move= 1734 (1+1.1912) = 3,799.54 Price of the stock in down move=1734 (1+0.8395) = 3,189.69 Since the stock price is 1700 the payoff in case of an up move= 3,799.54-1700= 2,099.54 and because this is a call option, the payoff in case of a down move is zero. Risk neutral probability of an up move (P) = (e r*t– d)/ (u-d) =(e0.05* 0.8333-0.8395)/ (1.1912-0.8395) = (1.04254517-0.8395)/0.3517 p= 0.20304517/0.3517= 0.5773 The value of the option = 2,099.54 * 0.5773* e-0.05 x (10/12)= 1212.0644 * e-0.04167 =1,162.60 b) A four-step binomial tree U= 1.0447, d= 0.9572 and t= (10/4)/12= 0.2083 Price of the stock in case of four up moves= Current futures price (1 + u)t =1734 (1+1.0447)1/4 = 2,073.51 Price of the stock in case of four down moves=1734 (1+0.9572)1/4 = 2,050.96 Since the stock price is 1700 the payoff in case of an up move= Price in four up moves- strike price =2,073.51-1700= 373.51 The payoff in case of a down move is zero because this is a call option. Risk neutral probability of an up move (P) = (e r*t– d)/ (u-d) =(e0.05* (2.5/12) - 0.9572)/ (1.0447-0.9572) P=0.053321627/0.0875=0.6094 The value of the option= Payoff * Probability * e r*t =373.51 * 0.6094 * e0.05* (2.5/12) = 227.616994* 1.010471108 =230.00 u d c) Comparison of the answers The value obtained by using Black’s formula is 186.68. This is different from both answers above. However, it is closer to the answer obtained using four step binomial tree. d) I) Implied volatilities of the gold futures price K 1500 1600 1700 1800 1900 T = 2 months 23.730 23.480 23.150 22.460 22.080 T = 6 months 13.700 13.820 13.470 13.150 13.270 ii) Plot of implied volatility as a function of strike price ii) The option prices are consistent with assumptions underlying Black Scholes model. The implied volatility depends on the maturity and value of the option. If maturity is for a short duration, volatility is higher than for options that have longer maturity period. If the value of the option is high, implied volatility is high. On the other hand, if it is low, the implied volatility is low. e) Difference between implied and historical volatility Implied volatility is estimated potential movement of an option’s price. It focuses on future stock changes. On the other hand, historical volatility is standard deviation of annual stock movements. It is the level at which stock price changed on daily basis in a previous one year. It is based on historical data changes. Implied volatility is also said to be higher than historical volatility. This can be observed from the volatility figures in the above table (Historical vs. Implied Volatility, 2011). Question 2 a) Portfolio insurance strategy The two portfolio strategies explained below are naïve hedge and risk hedge. S&P 500 futures multiplier= $250 P: Current portfolio value= $55,000,000 Beta= 1.5 Dividend yield of the portfolio= 2.4% simple compounding S&P dividend yield= 1.8% p.a. simple compounding Risk free interest rate= 4.8% p.a. continuous compounding A: current value of futures contract on index= 1327*250= 331,750 F: Current price of S&P 500 index= 1327 Naïve hedge (N): Hedging $55m portfolio with S&P 500 futures will mean that the investor will sell 55,000,000/ 331750= 166 contracts. If N is less than zero the investor should sell and if it is greater than zero, he should buy. N= (desired beta-portfolio beta) (Current portfolio value/ Current value of futures contract on index) N= (0-1.5) * (55,000,000)/ 331,750=-249 contracts (sell). Therefore, the investor should sell 249 contracts. Position necessary to reduce beta to 0.75: (0.75-1.5) * (55,000,000)/ 331,750= -124 contracts (sell) Position necessary to increase beta to .2.0: (2.0-1.5) * (55,000,000)/ 331,750= 83 contracts (buy) Risk hedge = (Dollar value of the portfolio/ Dollar value of the S&P futures contract) * beta = [$55,000,000/ (1327*250)] * 1.5= 248.68 = 249 contracts. Given beta, the investor should sell 249 contracts. If the S & P500 index falls by 5% over the next four months, from 1327 to 1260.65; then portfolio should fall by 5.0% * 1.5= 7.5%. This is equal to $7,125,000. However, if there are dividends of 2.4%* 0.333* $55,000,000= $183,150 Additionally, if after four months, the index receives dividends, it will rise by (331,750* 1.8%* 0.333)=1,988.51 If the investor sells 249 contracts, at 1,988.51 his account will benefit by (1,988.51-1327)* 250*249)= $41,147,250. b) Insurance premium Volatility: 20% p.a. Risk free interest rate= 4.8% p.a. continuous compounding Dividend yield of the portfolio= 2.4% simple compounding Premium= Portfolio return- Risk-free interest rate Portfolio return= [portfolio gain (loss) + dividend yield + interest]/ (Initial investment+ Added months/12) =$(41,147,250 + 2,640,000)/ (55,000,000+0.333) =79.61%- 4.8%= 74.81% c) Gain or loss if the level of market in four months is 1100 Initial futures price= 1327 Final futures price= 1100 Percentage change= (Final futures price- initial futures price)/Initial futures price * 100 The portfolio will fall by 17%* 1.5=25.5% * 55,000,000= 14,025,000 Dividends received = Rate * Time * Current portfolio value =2.4%* 0.333* 55,000,000= $439,560 If 249 stocks were sold, the gain will be: (Initial futures price-final futures price)*250* Number of contracts =1327- 1100= 227*250*249 = $14,130.750 The resulting gain= (439,560+ 14,130.750)- 14,025,000= $545,310 Gain or loss if the level of market in four months is 1500 Initial futures price= 1327 Final futures price= 1500 The portfolio will rise by 13%* 1.5*=19.5%= 18,525,000 Dividends received = 2.4%* 0.333* 55,000,000= $439,560 If 249 stocks were sold, the loss will be: 1327- 1500= -173*250*249 = -$10,769.250 The resulting gain= (18,525,000+ 439,560)- 10,769.250= $8,195,310 Question 3 a) Use of the Black Scholes’ model to estimate the premium Premium obtained from tradingshort 25 call options on S&P 500= $2,091.25 Premium obtained from tradingshort 50 put options on S&P 500= $4,977.57 My friend receives premiums of the call option from the buyer and pays put option premium. In general, he did not receive any premium. Actually, he incurred a loss of $(4,977.57-2,091.25) $2,886.32. b) Strategy that the investor uses The friend uses delta, rho, gamma and theta hedging strategies. Delta (hedge ratio) is a measure of how the option price will change in relation to $1 change in the stock price. Rho is a measure of how the option price changes given 1% change in the risk-free interest rate. Theta is a measure of how the option price changes given 1 day decline in option duration. Gamma hedging is a measure of how fast delta changes when stock price changes by $1. Profit/loss is the difference between the option’s premium and the cost of acquiring it.The cost deducted in the table below is interest.           European style       Call Put     Option premium $ 2,091.253   $ 4,977.566 Expected profit/loss 584.1238021 1963.308818       Option delta 0.4842   -0.4690                   Option rho 13891.61   -35794.26       Option theta -986.03   -2241.82     For Call or Put     Gamma 0.0001       The profits are different for the two options. This is because the number of put options is double the call options traded. Therefore, put option has higher premiums and subsequent higher profits than call option. c) Profit and loss potential associated with this strategy. The profits are different for the two options. This is because the number of put options is double the number of call options traded. Therefore, put option has higher premiums and subsequent higher profits than call option. The investor’s expectations were that the market volatility would be higher than 18%. This way, premiums and profits would be high (bear). d) Losses the investor would be making if he was exercising on 3 October, 2011. If the investor was trading on October 3rd, he would make a loss of 1234.81 from call option but a profit of 10,019.70 from put option. e) Delta and gamma of the investment Delta and gamma options for 1 July 2011 are unknown. This is because, the expiry period is zero. Additionally, at time zero, there is no change in price. As a result, because delta is zero or unknown, gamma will also be zero or unknown. The delta of the call option by 03 October 2011 is 0.4714 while delta for put option is -0.5203. Gamma for both call and put options by 03 October 2011 is 0.0001. This means that for every one cent change in stock price, calloption value will change by 0.47 while put option value will change by -0.52. Gamma measures the speed at which delta changes. It should always be as small as possible so that any delta changes will not require consistent gamma adjustments. f) Recommendation to hedging the portfolios: i) Delta neutral: The net change of the portfolio= Number of put options (delta) – Number of call options (delta) =50(0.32) - 25(0.32)=8. This is the neutral position for the portfolio. Therefore, for every 1$ rise in stock price, the put and call options fall by $8. Buying 8 units of the stocks would hedge the investor from small changes in stock prices. If the value changes from $1339.67 to $2000, then the 8 unit position would fall by (2000-1339.67). This is a (660.33-80)=$580.33 loss. ii) Delta – gamma neutral: Net change in portfolio= Number of put options (gamma) – Number of call options (gamma) =50(0.0021) - (25*0.0021)= 0.0525/8= 0.00656 Delta- gamma neutral position is 0.006656 units for both call and put options. A 660.33 rise in value would lead to no loss or gain. Cost of delta hedge=Current option value * rate * neutral position =$2000 * 0.045^0.2= 1,075.65* 8= $8605.20 Cost of delta-gamma hedge= 1,075.65 *0.00656= $7.06 g) Benefits and drawbacks ofdelta and gamma hedging Delta hedging immunizes a portfolio against small stock price movements. However, it changes with price changes. Therefore, it has to be rebalanced over time in order to maintain neutrality. Delta-gamma hedge helps to retain neutrality by ensuring that portfolio gamma is at zero. However, in case of volatility, it may not be able to retain volatility. h) Profit/loss of portfolios By December 19, 2011 the 25 call option’s premium is $2,928.04 while put option’s premium is $6642.25. Current call option’s trading value= Option price * number of units =1205.35*25 units= 30,133.75. Loss=30,133.75- 33,491.75=-$3,358. Current put option’s trading value= 1205.35*50 units= 60,267.5. Loss=60,267.5-66,983.50=-$6,716 1st July index price= $1322 19th September index price= $1210 Call option delta= Final index price- initial index price/ strike price- strike price =[(1210-1322)/25]/(1350-1339.67)=-10.84 Put option delta= [(1210-1322)/50] /(1350-1339.67)= -0.22 For every 1$ rise in stock price, the call and put options will fall by –10.84 and -0.22 units respectively. If the value changes from $1322to $1210, then the 10.84 unit position would rise by (2000-1339.67). This is a 1,196 gain. Additionally, the 0.22 unit position would rise by 682.33. Therefore, if delta hedge is used, both options will gain. Reference Historical vs. Implied Volatility. (2011). Retrieved May 24. 2012, from http://community.tradeking.com/members/optionsguy/blogs/83304-historical-vs-implied-volatility-what%E2%80%99s-the-difference Read More