LOL Ltd has a current share price of $45. A put option written on LOL, which has a strike price has $38 and 9 months to expiry, is trading at $1.42. If the riskfree rate of interest is 3% pa (continuously compounded), what is the correct price for a call option on LOL with $38 strike price and 9 months to expiry? Do not enter the dollar sign "$" in your answer. Your answer should have at least two decimal places.

Question

BlackTom AI · Accepted Answer

Put–call parity for European options is the key relation: c + K e^{-rT} = p + S. Rearranging gives the call price c = p + S − K e^{-rT}.

1) Relevant concepts
- Put–call parity links the prices of European calls (c) and puts (p) on the same underlying with the same strike (K) and expiry (T), in a no-arbitrage market.
- Continuous discounting: the present value of the strike is K e^{-rT} when r is the continuously compounded risk-free rate.

2) Step-by-step substitution
- Given: S = 45.00, p = 1.42, K = 38.00, r = 0.03 (continuously compounded), T = 9 months = 0.75 years.
- Compute rT = 0.03 × 0.75 = 0.0225.
- Compute the discount factor e^{-rT} = e^{-0.0225}. Using a numerical approximation, e^{-0.0225} ≈ 0.9775 (rounded to four decimal places).

3) Intermediate calculations
- Present value of the strike: K e^{-rT} ≈ 38 × 0.9775 = 37.145.
- Now apply put–call parity: c = p + S − K e^{-rT} = 1.42 + 45.00 − 37.145 = 9.275.

4) Why this is correct
- This value directly follows from the arbitrage-free put–call parity relation and the continuous discounting of the strike. The only inputs required (S, p, K, r, T) are given, so the computed c is the unique no-arbitrage price for the corresponding European call.

5) Why other approaches are incorrect or unnecessary
- Using Black–Scholes without the volatility is impossible; parity uses only observable prices and the risk-free rate. Ignoring discounting (i.e., using K instead of K e^{-rT}) would overstate the present value of the strike and give an incorrect call price.
- Any small differences in the third decimal come from rounding the exponential; keeping more precision in e^{-0.0225} gives the same result to two decimal places.

6) Conclusion
- Rounding c = 9.275 to two decimal places gives 9.27. This matches the provided answer: 9.27.

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Question at position 1 Other things equal, a put option's value increases as the strike price increases TrueFalse

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