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.

  • $\begingroup$ If stock is 105 at maturity, what is payoff of option? What is discounted value of payoff? $\endgroup$ – Alex C Sep 20 '19 at 3:28
  • $\begingroup$ Discounted value of payoff is $\frac{5}{1.05}$? $\endgroup$ – Idonknow Sep 20 '19 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 '19 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 '19 at 4:05

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).

  • $\begingroup$ Am I right to say that any deterministic stock will worth $S(0) e^{rt} $? $\endgroup$ – Idonknow Sep 20 '19 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 $r$. $\endgroup$ – Kevin Sep 20 '19 at 10:59
  • $\begingroup$ I see. Everything is clearer now. Thanks for your lucid explanation. $\endgroup$ – Idonknow Sep 21 '19 at 2:04
  • $\begingroup$ I reread your answer. Correct me if I am wrong, but did you answer second part of my question? $\endgroup$ – Idonknow Dec 4 '19 at 3:36
  • $\begingroup$ @Idonknow You mean the hedging of the call? This is the standard Black-Scholes $\Delta$ hedge as outlined in my answer. Look at the self-financing portfolio $\pi(t,S_t)=C(t,S_t)-\Delta S_t$, apply Ito's Lemma to its change (i.e. $\mathrm{d}\pi(t,S_t)$), notice that this is non-random and use the no-arbitrage principle to arrive at the Black Scholes PDE. In short: In the Black Scholes world, you can hedge (i.e. replicate) a call option by trading into a bond and the underlying stock. The option's sensitivity $\Delta$ tells you how much your hedge portfolio needs to invest into the stock $\endgroup$ – Kevin Dec 4 '19 at 9:36

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