Black-Scholes Formula and Comparative Statics 4. The current price of the SPY is
ID: 2792280 • Letter: B
Question
Black-Scholes Formula and Comparative Statics 4. The current price of the SPY is $250 per share, the risk-free interest rate is 5%, and the annualized volatility is 15%. Use the Black-Scholes formula to calculate N(dl), N(d2), and the premium for a 3 month call with a strike of $275? Remember that 3 months is 0.25 of a year. This will be our baseline scenario for comparative statics. 5. What is the premium on a call if the current price is $0.50 higher? What is it if the current price is $0.50 lower? Hint: the difference in the premium is the delta of the call. 6. Return to the baseline scenario. What is the call premium call if the volatility increases to 16%? Hint: the difference from the baseline premiumis vega of the call. Black-Scholes Formula and Put-Call Parity 7. Return to the baseline scenario. What is the premium on a 3-month put with a strike of $275? Use the call premium from question 4 and the put premium from question 7 to determine the 3-month forward price. Assume the interest rate is 5% per year as in the baseline scenario. Hint: use the put-all parity relationship. 8. Option Structures, Payoffs and Breakeven Return to the baseline scenario. An investor purchases a call with a strike of $275 and purchases a put with a strike of $240. 9. What is the premium of the combined structure and how does it change if the price of the underlier increases to $255? Hint: this involves the delta of both the call and the put. 10. Draw the payoff diagram for the combined structure from question 9. Indicate on the figure when the investor will break even. Hint: consider the values of ST where the payoff from the option be equal to the combined premium.Explanation / Answer
a.) Given that,
Current Stock Price, S0=250,
Strike Price, K=275,
Risk Free Rate, r=0.05,
Volatility, =0.15,
Time to Maturity, T=3/12 years = 0.25 years
Continuously Compounded Dividend Yield, q = 0
Using relation, d1 = ln (S0/K) + (r+2/2) x T
T
= ln (250/275) + (0.05 + 0.152/2) x 0.25
0.150.25
= -0.0414 + 0.0153
0.0750
= -0.3477
Now, d2 = d1 - T
= -0.3477 - 0.150.25
= -0.3477 - 0.0750
= -0.4227
European Call Price as per Black-Scholes Merton,
= S0 x N(d1) - {Ke-rT x N(d2)}
= 250 x N(-0.3477) - {275e-0.05 x 0.25 x N(-0.4227)}
= 250 x 0.3640- {275e-0.0125 x 0.3363}
= 91.0075 - 91.3228
= 0.3153
European Put Price as per Black-Scholes Merton,
= Ke-rT x N(-d2) - S0e-qT x N(-d1)
= 275e-0.05 x 0.25 x N(0.4227) - {250 x e-0 x N(0.3477)}
= 275e-0.0125 x 0.6637 - {250 x 1 x 0.6359}
= 271.5839 x 0.6637 - {250 x 0.6359}
= 180.2502 - 158.975
= 21.2752
Call Delta = e-qT x N(d1)
= e-0.00 x 0.25 x N(-0.3477)
= 1 x 0.3640
= 0.3640
Call Price for $0.50 higher price of stock = 0.3153 + 0.50x0.3640 = 0.4973
Call Price for $0.50 lower price of stock = 0.3153 - 0.50x0.3640 = 0.1333
Call Vega = 1/100 S0e-qT T x 1/(2Pi) e-d1*d1/2
= 1/100 x 250 x 1 0.25 x 1/(2x3.14) e-0.3477*0.3477/2
= 1.25 x 0.3755
= 0.4693
Call premium if volatility increases to 16% = 0.3153 + 0.4693 =0.7846
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