The original question is quoted below.

The underlying stock price is now \$100, and tomorrow it will be either \$101 (with probability $p$) or \$99 (with probability $1-p$). A call option with value $c$ which expire tomorrow has exercise price \$100. Find the the value of $c$ under the Black-Scholes model. Ignore interest rate.

When I tackle this question, I first derive $p=\frac{1}{2}$. To use call option price formula, we need $S, E, r, T-t, \sigma$. From the question, it is clear that $$S=100, E=100, r=0, T-t=\frac{1}{365}$$ So we only need $\sigma$. Since $\sigma$ is measured by the standard deviation of the return $\frac{dS}{S}$, I proceed as follow: $$E(return)=(1/100)(0.5)+(-1/100)(0.5)=0$$ $$Var(return)=(1/100-0)^2(0.5)+(-1/100-0)^2(0.5)=0.0001$$ $$sd(return)=\sqrt{0.0001}=0.01$$ Apply the explicit price formula for call option under the Black-Scholes model, I found $$d_1=0.00026171, d_2=-0.00026171, N(d_1) = 0.500104407965456, N(d_2)=0.499895592034544$$ Thus, the desired price is $$SN(d_1)-Ee^{-r(T-t)}N(d_2)=0.0209$$ The procedure seems logical to me. However,since the profit from the call option is $$(1)(0.5)+(0)(0.5)=0.5$$ I expected the price to be close or equal to \$0.5. How come they differ so much?

  • $\begingroup$ Check the time units, I think you are using in one place 1 year and in another 1 day. Be consistent. $\endgroup$ – Alex C Oct 27 '16 at 7:42
  • $\begingroup$ @AlexC Thank you for your comment. I tried to divide my standard deviation by 365 and $\sqrt{365}$, but the resulting price is even smaller. $\endgroup$ – IDontKnowMath Oct 27 '16 at 8:23
  • 1
    $\begingroup$ It is not clear what model you want to use: binomial model (as in the title of your question) or Black-Scholes (as in the question within your posting). One is discrete time/spot space the other is continuous time/spot space. You're basically mixing both approaches here. In a 1 period model with two states, it is possible to perfectly replicate an option with the risk-free asset and the underlying, this is what you should do to find the price IMHO. $\endgroup$ – Quantuple Oct 27 '16 at 9:05
  • $\begingroup$ @Quantuple I think that the question that I received is weird too. But I believe that I should estimate the volatility from the binomial model and use it in the BS option price formula. $\endgroup$ – IDontKnowMath Oct 27 '16 at 11:03

From your answer to my comment, here is what I would do.

Over the horizon $[0,\Delta t]$, the BS model tells you that the expected log-return is $$ \Bbb{E}\left[ \ln\left(\frac{S_{t+\Delta t}}{S_t}\right) \right] = \left(\mu-\frac{1}{2}\sigma^2\right)\Delta t$$ with a variance $$ \Bbb{V}\left[ \ln\left(\frac{S_{t+\Delta t}}{S_t}\right) \right] = \sigma^2 \Delta t$$

Over the same horizon, your binomial model tells you that: $$ \Bbb{E}\left[ \ln\left(\frac{S_{t+\Delta t}}{S_t}\right) \right] = 0.5 \ln(101/100)+0.5\ln(99/100) \approx -5e^{-5}$$ \begin{align} \Bbb{V}\left[ \ln\left(\frac{S_{t+\Delta t}}{S_t}\right) \right] &=(0.5\ln^2(101/100) + 0.5\ln^2(99/100)) - (-5e^{-5})^2 \approx 1e^{-4} \end{align}

If you want your models to be mutually consistent, they should at least agree on the two first moments of the log-return's distribution. This constrains you to choose: $$ (\mu-\frac{1}{2}\sigma^2)\Delta t = -5e^{-5},\quad \sigma^2 \Delta t = 1e^{-4} $$ which gives, with $\Delta t=1/365$ $$\sigma = \sqrt{1e^{-4}\dot\,365} = 1e^{-2}\sqrt{365} = 0.1911$$ $$\mu = -5e^{-5}\dot\,365+\frac{1}{2}(0.1911)^2 \approx 1e^{-5}$$ Now you can use BS formula as you propose to find a price around $0.4$.

| improve this answer | |

Black Scholes can be seen as the continuous limit of a binomial model when the number of steps go to infinity.

(It can be seen as a result of Donsker's theorem)

Thus it is normal that your call price in the one-period model is different than the one in the BS model.

If you have $n$ steps in your binomial to describe the period $[0,T]$ and if your increment on one step in $\pm h$, then the equivalent volatility is $h\sqrt{\frac{n}{T}}$.

So here $n=1$, $T=\frac{1}{365}$ and $h=1$ so $\sigma=\sqrt{365}$.

| improve this answer | |
  • $\begingroup$ Taking $\sigma = \sqrt{365}$, I obtained the option price \$38.2925, which doesn't make sense, since the maximum payoff is only $\$101-\$100=\$1$. $\endgroup$ – IDontKnowMath Oct 27 '16 at 10:59
  • $\begingroup$ The move is 1% so the std dev is $\sqrt{365}=19.10497\%$ so 0.1910497 goes in the BS formula I think, giving a call price of approx 0.4 $\endgroup$ – noob2 Oct 27 '16 at 12:16

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