1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ If stock is 105 at maturity, what is payoff of option? What is discounted value of payoff? $\endgroup$ – Alex C Sep 20 at 3:28
  • $\begingroup$ Discounted value of payoff is $\frac{5}{1.05}$? $\endgroup$ – Idonknow Sep 20 at 3:33
  • $\begingroup$ Yes, so you could buy this much worth of stock today and you would be hedged. $\endgroup$ – Alex C Sep 20 at 3:54
  • $\begingroup$ But I do not know how to hedge. Perhaps my background in hedging is not firm yet. Do you have any recommendation? $\endgroup$ – Idonknow Sep 20 at 4:05
3
$\begingroup$

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).

$\endgroup$
  • $\begingroup$ Am I right to say that any deterministic stock will worth $S(0) e^{rt} $? $\endgroup$ – Idonknow Sep 20 at 10:32
  • 1
    $\begingroup$ 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 $r$. $\endgroup$ – KeSchn Sep 20 at 10:59
  • $\begingroup$ I see. Everything is clearer now. Thanks for your lucid explanation. $\endgroup$ – Idonknow Sep 21 at 2:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.