Asset-or-Nothing Call Bounds

Article Quant Q&A · Author: user2521987

Summary

The document examines whether an asset-or-nothing call quoted at a stated price can be reconciled with Black–Scholes implied volatility. It applies the pricing formula to infer the normal-distribution input, then solves for volatility and obtains complex roots. The accepted response explains that this result signals an incompatible option price under the model, rather than a valid imaginary volatility.

The response checks the price function’s limits as volatility approaches zero and infinity, and locates its minimum over positive volatility. For the example, the quoted price is below that minimum, so no positive Black–Scholes volatility can produce it. The figures support the diagnosis for the stated inputs; they do not establish whether the source problem contains a typo or whether a different model or market convention applies. The document also notes that model-independent put–call parity for asset-or-nothing options can still be used.

Key ideas

An asset-or-nothing call price need not map to a real implied volatility under Black–Scholes.
The option price approaches the discounted spot value at both low and high volatility limits.
The price function has a positive-volatility minimum, which defines a feasibility bound for quotes.
A quote below that minimum is inconsistent with the model for the stated inputs.
Model-independent parity relations may remain applicable when the Black–Scholes quote is infeasible.

Tags

Options Derivatives pricing Volatility Statistics

Full text

# Is the asset-or-nothing call option in this example valued incorrectly in the Black-Scholes framework?


# Is the asset-or-nothing call option in this example valued incorrectly in the Black-Scholes framework?












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:

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.

## Answer by LocalVolatility (score 5, accepted)

https://quant.stackexchange.com/a/31292

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.

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}

Furthermore

\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}

and

\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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.

Feasibility Bounds for Black–Scholes Asset-or-Nothing Calls

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