Please show your work. The following information is given about options on the s
ID: 2780534 • Letter: P
Question
Please show your work. The following information is given about options on the stock of a certain company. S0 = 23 X = 20 rc = 0.09 T = 0.5 2 = 0.15 No dividends are expected. Use this information to answer questions 1 through 8. 1. What value does the Black-Scholes-Merton model predict for the call? (Due to differences in rounding your calculations may be slightly different. “none of the above” should be selected only if your answer is different by more than 10 cents.) 2. Suppose you feel that the call is overpriced. What strategy should you use to exploit the apparent misvaluation? (Due to differences in rounding your calculations may be slightly different. “none of the above” should be selected only if your answer is different by more than 10 shares.) 3. The price of a put on the stock is: (Due to differences in rounding your calculations may be slightly different. “none of the above” should be selected only if your answer is different by more than 10 cents.) 4. To construct a riskless hedge, the number of puts per 100 shares purchased is: (Due to differences in rounding your calculations may be slightly different. “none of the above” should be selected only if your answer is different by more than 0.01.) 5. The call's vega is: (Due to differences in rounding your calculations may be slightly different. “none of the above” should be selected only if your answer is different by more than 0.05.) 6. If the actual call price is 3.79, the implied standard deviation is 7. If we now assume that the stock pays a dividend at a known constant rate of 3.5 percent, what stock price should we use in the model? (Due to differences in rounding your calculations may be slightly different. “none of the above” should be selected only if your answer is different by more than 10 cents.) 8. If we now assume that the stock pays a single dividend of 2.25 in three months, what stock price should we use in the model? (Due to differences in rounding your calculations may be slightly different. “none of the above” should be selected only if your answer is different by more than 10 cents.) Show your work
Explanation / Answer
The Black Scholes model formula for a call option without dividend can be written as:
C=S0× N(d1)-Xe-rt×N(d2)…….(1)
Where N (d1) and N (d2) are the standard normal cumulative distribution function
The formula for d1 and d2:
d1= (ln(S0/X)+((r+(2/2))t)/(t)…..(2)
d2=d1-t……..(3)
Given information in the question:
S0=23
X=20
r=0.09
t=0.5
2=0.15 (Therefore = 0.15=0.3872)
First lets find out d1 and d2 using equation 2 and 3:
d1= (ln(23/20)+((0.09+(0.15/2))0.5)/(0.38720.5)
= (0.139762+0.0825)/ (0.273792)
= 0.811791= (rounding to 0.81)
Therefore, N(d1) in the Z table (look up 0.81)= 0.7910
d2=0.811791-0.38720.5
= 0.538 (rounding to 0.54)
Therefore, N(d2) in the Z table (look up 0.70)= 0.7054
Using equation 1 the value of the call option is:
C=23×0.7910-20e-0.09(0.5) × 0.7054
=18.193-13.48721
= 4.705788= (rounded to 4.71)
2) Since the call is overpriced we should be selling the call option and buying the stock to create a riskless trade. But to do this we need to figure out the delta of the call option. This is equivalent to N(d1) which we figured in question 1. Thus, N (d1) = 0.7910. The delta of call option says that for every 1$ change in stock price the value of the call option will change by $ 0.7910.
Therefore, if we sell 1,000 call option we have to buy 1000×0.7910=791 shares to create a delta neutral riskless trade.
Conclusion: Sell 1000 call options and buy 791 shares.
3) Using the black Scholes formula we have calculated the value of call option using equation 1 in question. Now we will calculate the value of put option. The value of put option (P) is:
P= Xe-rt× N(-d2)S0× N(-d1)……(4)
The formula for d1 and d2 stay the same but instead of using N (d1) and N(d2) we would be using N(-d1) and N(-d2).
Therefore,
N (-d1) = 1-N(d1)=1-0.7910(from question 1)=0.209
N (-d2) = 1- N(d2)=1-0.7054 (from question 2)= 0.2946
Now simply plugging in the values in equation 4:
20e-0.09(0.5) × 0.2946-23×0.209
=5.632737-4.807
=0.825737 (rounded to 0.83)
4) To answer this question we need to know the delta of put option. Remember that delta of call +put for the same strike should always sum to 1. Expressed other way delta of put option equals
N (d1)-1 i.e. 0.7910-1=-0.209
Therefore, the number of put options required to hedge long position in 100 shares equals:
100/0.209= 478.46(rounded to 478 puts).
Conclusion: Thus, in order to hedge 100 shares you need to buy 478 put options.