If the volatility is zero (i.e. σ=0), what is the call worth? After valuing the call, how to hedge the call (assuming you sold it)

Question: All Black-Scholes assumptions hold. Assume no dividends. The stock price is $100. The riskless interest rate is 5% per annum. Consider a one-year European call option struck at-the-money (i.e. strike equals current spot). $$(1)$$ If the volatility is zero (i.e. σ=0), what is the call worth? $$(2)$$ After valuing the call, how to hedge the call (assuming you sold it). My attempt to $$(1)$$: Since volatility is zero, it means that return does not deviate from riskless return, that is, $$100 \times 1.05 = 105.$$ So the call worth $$\105.$$ But I have no idea on how to hedge the call. Any idea would be appreciated. • If stock is 105 at maturity, what is payoff of option? What is discounted value of payoff? – Alex C Sep 20 at 3:28 • Discounted value of payoff is$\frac{5}{1.05}$? – Idonknow Sep 20 at 3:33 • Yes, so you could buy this much worth of stock today and you would be hedged. – Alex C Sep 20 at 3:54 • But I do not know how to hedge. Perhaps my background in hedging is not firm yet. Do you have any recommendation? – Idonknow Sep 20 at 4:05 1 Answer If $$\sigma=0$$, the stock price is deterministic and grows at rate $$r$$. In one year, it is thus worth $$100\cdot e^{0.05}\approx 105.13$$. The strike is $$K=100$$. Your payoff is thus $$5.13$$. Discounting at rate $$r$$, you get as today’s fair option price $$5.13\cdot e^{-0.05}\approx4.88$$. Note that there is no randomness and the stock price is perfectly predictable. Hedging such a known payoff can be done by simply investing money into a bond. More interestingly, if $$\sigma\neq0$$, then there is no static hedge and you need to dynamically hedge the option. Black and Scholes (1973) show that the portfolio $$C-\Delta S$$ is locally risk-free and hence equals the risk-free bond. Here, $$\Delta=\frac{\partial C}{\partial S}$$. This gives you a way of hedging the call option by investing in the stock and a (default free zero-coupon) bond (which matures when the option expires). From that relationship, Black and Scholes (1973) also derive their famous PDE which gives a way of finding a closed-form solution for the option price. In a nutshell: when hedging, you replicate the payoff of the derivative. For options, you need to continuously adjust your hedging portfolio (because $$\Delta$$ keeps changing). • Am I right to say that any deterministic stock will worth$S(0) e^{rt} $? – Idonknow Sep 20 at 10:32 • In the Black-Scholes model$\mathrm{d}S_t=rS_t\mathrm{d}t+\sigma S_t\mathrm{d}W_t$and if$\sigma=0$, you are left with the ODE$\mathrm{d}S_t=rS_t\mathrm{d}t$which gives rise to$S_t=S_0e^{rt}$. You could include dividends with yield$q$which gives$S_t=S_0e^{(r-q)t}$. Furthermore, one could introduce time-dependent interest rates (and dividends) and you would get$S_t=S_0e^{\int_0^t r_s\mathrm{d}s}$which is still deterministic. Think of it that way: with$\sigma=0$, the stock is a risk-free asset and hence needs to return the risk-free rate$r\$. – KeSchn Sep 20 at 10:59
• I see. Everything is clearer now. Thanks for your lucid explanation. – Idonknow Sep 21 at 2:04