Individual HW 2 — Options & Futures Pricing
| Due: May 8 (Week 6) at the start of class | Submit via BruinLearn |
1. (a) Tesla’s stock price is $194 and the risk-free rate is 5.5%. Assuming Tesla’s volatility is 50%, compute the price of call and put options with exercise price $175 and maturity of 1 year using Black-Scholes.
(b) Compute the prices of call and put options when each of the following changes, with all other parameters remaining equal to the baseline. Fill out a table for: underlying asset price becomes $200, exercise price becomes $200, maturity becomes 2 years, volatility becomes 60%, risk-free rate becomes 6%.
IMPORTANT: Choose 3 of problems 2–7.
2. Bull Spread. Tesla’s current stock is $140. Two call options with 1-year maturity at strikes $140 and $150. Risk-free rate 5.5%, volatility 25%. (a) Price both calls. (b) Buy the $140 call, sell the $150 call — upfront cost? (c) Payoff if stock is $150 in one year? (d) Net payoff at $150? (e) Why a bull spread vs. buying a call or the stock outright?
3. Collar. Tesla at $140. Call at strike $150, put at strike $130, both 1-year. Risk-free rate 5.5%, volatility 25%. (a) Price the put. (b) Buy stock + put, sell call — upfront cost? (c) Payoff if stock is $160? (d) Payoff if stock is $110? (e) Expected net payoff if 50% chance of $160 and 50% chance of $120.
4. Straddle. Tesla at $140. Call and put both at strike $140, 1-year. Risk-free rate 5.5%, volatility 25%. (a) Price the put. (b) Buy both — upfront cost? (c) Payoff if stock is $180? (d) Payoff if stock is $90? (e) Expected net payoff if 50% chance of $180 and 50% chance of $90.
5. Strangle. Tesla at $140. Call at strike $150, put at strike $130, both 1-year. Risk-free rate 5.5%, volatility 25%. (a) Buy both — upfront cost? (b) Payoff if stock is $175? (c) Payoff if stock is $95? (d) Expected net payoff if 50% chance of $175 and 50% chance of $95.
6. Butterfly Spread. Tesla at $140. Three calls at strikes $130, $140, $150, all 1-year. Risk-free rate 5.5%, volatility 25%. (a) Price the $130 call. (b) Sell two $140 calls, buy one $130 call and one $150 call — upfront cost? (c) Payoff if stock is $135? (d) Payoff if stock is $175? (e) Net payoff at $135.
7. Ratio Spread. Tesla at $140. Two calls at strikes $140 and $150, both 1-year. Risk-free rate 5.5%, volatility 25%. (a) Buy one $140 call, sell five $150 calls — upfront cost? (b) Payoff if stock is $145? (c) Payoff if stock is $170? (d) Net payoff at $145.
8. The S&R index price is $1,000 and the effective 6-month interest rate is 2%. The premium on a 6-month call is $109.20 and the premium on a put with the same strike is $60.18. What is the strike price?
9. A call option on the S&R index with 6 months to expiration, strike $1,000. The future value of the option price is $95.68. Find the S&R index price at which the call option profit diagram intersects the x-axis.
10. (Extra Credit) First Union National Bank CD promises to repay in 5.5 years the initial invested amount plus 70% of the gain in the S&P index (principal-protected equity-linked CD). $10,000 invested when S&P is 1,300. (a) Final payoff if S&P is 1,500? (b) If S&P is 1,200? (c) Write the final payoff as a function of the S&P index.
Synthesis Question
11. Client Question — Hedging a Concentrated Stock Position Ahead of Uncertainty
This is a required question in addition to Q1 and your 3 chosen problems from Q2–7.
A client holds 1,000 shares of Nvidia (NVDA), currently trading at $120 per share ($120,000 total position). The client is concerned about near-term downside risk due to potential tariff impacts on semiconductor supply chains but does not want to sell the shares because of the large unrealized capital gain. The client asks: “Can you protect my downside without giving up all my upside?”
Use the following parameters: risk-free rate = 4.5%, Nvidia volatility = 45%, and options have 3-month (0.25 year) maturity. Three strike prices are available: $100, $120, and $140.
(a) Using Black-Scholes, compute the prices of puts at the $100 and $120 strikes, and the price of a call at the $140 strike. (Use the Excel sheet from class.)
(b) Strategy 1 — Protective Put: The client buys 1,000 put options at the $120 strike. What is the total cost of this insurance? What is the client’s minimum portfolio value at expiration?
(c) Strategy 2 — Collar: The client buys 1,000 puts at the $100 strike and sells 1,000 calls at the $140 strike. What is the net upfront cost (or credit) of this strategy? What are the client’s maximum and minimum portfolio values at expiration?
(d) Consider two scenarios at expiration:
- Scenario A (tariff shock): NVDA drops to $85
- Scenario B (AI demand surge): NVDA rises to $160
For each strategy, compute the total portfolio value (stock + option payoffs − option costs) in both scenarios.
(e) The protective put costs real money out of pocket. The collar is cheaper (possibly free), but caps upside. Which strategy would you recommend for this specific client, and why? Consider that the client is worried about tariffs (a downside scenario) but also doesn’t want to miss upside from AI demand.
(f) Write a 1-paragraph client memo (4–6 sentences) explaining how options allow the client to reshape their risk profile without selling shares. Reference the trade-off between the cost of protection and the upside they retain, and explain in plain language what the collar “gives up” in exchange for cheaper downside protection.