Pricing and hedging a call when volatility is zero

Question

Work in a Black-Scholes world where the stock is at $100$ , the continuously compounded rate is $5\%$ and there are no dividends. If volatility is zero, what is a one-year European call struck at $100$ worth, and how would the writer hedge it?

Short answer

About $4.88$ . With zero volatility the stock grows deterministically to $100e^{0.05} \approx 105.13$ , so the call surely pays $5.13$ , worth $100 - 100e^{-0.05} \approx 4.88$ today. The writer hedges by buying one share (delta $= 1$ ) financed by borrowing $100e^{-0.05} \approx 95.12$ .

Adapted from Crack, Heard on the Street, Q2.1, p.23; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

Start with a free lesson: Options is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free