I understand the solution to the author's example below, but I can't help but notice that the implied volatility is an imaginary number:

enter image description here

The time-$t$ price of an All-or-nothing Asset Call is $S_t e^{-\delta(T - t)}N(d_1)$

We have $38.66 = S_0e^{-\delta\cdot T}N(d_1) = 60e^{-0.02\cdot0.5} N(d_1)$ and so $N(d_1) = 0.650808991$ and $d_1 = 0.38751$. Therefore

$$0.38751 = \frac{\ln(60/50) + (0.1 - 0.02 + 0.5\sigma^2)\cdot0.5}{\sigma\sqrt{0.5}},$$ which gives

$\sigma \in \{0.548022 - 0.767436i, 0.548022 + 0.767436i\}$.

I don't see how an imaginary implied volatility is possible, so was this option priced incorrectly under the Black-Scholes framework?

The problem is from "Models for Financial Economics" by Abraham S. Weishaus.

  • $\begingroup$ I agree with the answer from LocalVolatility. You can think about why this doesn't work in general. You have an asset that is expected to go up over the next 6 months after accounting for dividends (10% vs 2%), and you are already in the money (60 > 50). It is necessarily more likely than not that you will receive the asset. $N(d_1) > N(d_2)$. So, with a $N(d_1) = 0.38751 > N(d_2) = $ probability you will receive the asset, no real volatility will make a likely scenario unlikely. $\endgroup$
    – RandyF
    Dec 1, 2016 at 22:02

1 Answer 1


I agree with your computations. The problem is that the initial price of the asset-or-nothing call of 38.66 can't arise within the Black-Scholes framework. This seems to be an inconsistency/error in the question.

Below you see a plot of asset-or-nothing call price as a function of the implied volatility. Note that "1" means 100% implied volatility.

enter image description here

Let $A_0$ be the initial option price. It is easy to check that

\begin{equation} \lim_{\sigma \downarrow 0} A_0 = \lim_{\sigma \uparrow \infty} A_0 = S_0 e^{-\delta T} = 59.4030 \end{equation}


\begin{equation} \arg \min_{\sigma \in \mathbb{R}_+} A_0 = \sqrt{\frac{1}{T} \left( \ln \left( \frac{S_0}{K} \right) + (r - \delta) T \right)} = 94.30\% \end{equation}


\begin{equation} \min_{\sigma \in \mathbb{R}_+} A_0 = 44.4070. \end{equation}

However, you can still apply the model-independent put/call parity for asset-or-nothing options as suggested in the answer.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

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