Volatility Arbitrage When a Call Is Underpriced
Summary
The document examines a European call priced with volatility below the volatility assumed to govern the underlying asset. Its proposed trade buys the cheaper call and hedges it with a replicating position based on the higher volatility. In the ideal Black–Scholes setting, the call’s positive curvature means that realized volatility above the pricing volatility can produce a positive drift in the discounted hedged position. The answer recasts this as buying an option below its fair value under the assumed true volatility.
The argument depends on strong assumptions: the higher volatility is known, the model and replication are valid, trading is continuous, and hedging and financing are frictionless. Under those conditions it is presented as arbitrage; with real-world estimation error, transaction costs, discrete rebalancing, and changing volatility, the result is better viewed as a statistical volatility trade. The document gives a theoretical derivation, not empirical evidence or an implementation method, and the exact hedge and cash account conventions require care.
Key ideas
- A call priced below its value under the assumed true volatility may be bought and delta hedged.
- Positive option curvature makes the hedged position benefit when realized volatility exceeds pricing volatility.
- The arbitrage claim relies on continuous replication and accurate model assumptions.
- In practical markets, uncertainty and trading frictions make the opportunity statistical rather than risk-free.
- The theoretical derivation does not establish an executable trading strategy.
Tags
Full text
# Method for finding a arbitrage opportunity when market price of call is incorrect
# Method for finding a arbitrage opportunity when market price of call is incorrect
The solution of the Black-scholes equation is the price of a European call. And the option price assumes the underlying stock is a geometric Brownian motion with volatility $\sigma_{1}>0$.
Suppose, however, the underlying asset is really a geometric Brownian motion with volatility $\sigma_{2} > \sigma_{1}$, i.e. \begin{equation} dS(t) = \alpha S(t)dt + \sigma_{2}S(t)dW(t). \end{equation}
Consequently, the market price of the call is incorrect.
Can we set up a portfolio which has an arbitrage opportunity in the market? Furthermore, if there any methods to generate a portfolio arbitrage opportunity (how to consider this problem)?
Inspired by AFK, I try to answer this question by myself in mathematical way.
Firstly, the main idea of generating the portfolio with arbitrage opportunity is to buy a call option and at $\sigma_{1}$, and sell a call priced at $\sigma_{2}$. i.e. \begin{equation} X(t) = c(t,S(t)) - c^{\sigma_{2}}(t,S(t)) \end{equation} where $X(t)$ denote the value of portfolio, $c(t,S(t))$ is the value of the option at time $t$, and $c^{\sigma_{2}}$ is the value of the option priced in $\sigma_{2}$.
Actually, $c^{\sigma_{2}}$ was not exit in the market, but it doesn't matter since you can replicate it by hedging, which means, \begin{equation} X(t) = c(t,S(t)) - c_{x}(t,S(t))S(t) - \Gamma(t)M(t) \end{equation} Now, we want to show that X(t) has arbitrage opportunity.
It is trivial to see that X(0) = 0, then we want to show that dX(t) > 0 (Actually, we finally prove that de^{-rt}X(t) > 0). By Ito formula, we find that $$ dc(t,S(t)) = c_{t}dt + c_{x}dS(t) + 1/2c_{xx}d[S,S](t) $$ and $$ dX(t) = dc(t,S(t)) - c_{x}dS(t) - r(c - X(t) -c_{x}S(t))dt. $$ Then, substitute dc into this equation, we get $$ dX(t) = (c_{t}+1/2c_{xx}\sigma_{2}^{2}S(t)^{2}-rc+rc_{x}S(t))dt + rX(t) $$ Note that c(t,S(t)) follows the Black-scholes equation with $\sigma_{1}$, so we have $$ dX(t) - rX(t) = (c_{t}+1/2c_{xx}\sigma_{1}^{2}S(t)^{2}-rc+rc_{x}S(t))dt + 1/2c_{xx}(\sigma_{2}^{2} - \sigma_{1}^{2})S(t)^{2}dt $$ i.e. $$ de^{-rt}X(t) = 1/2c_{xx}(\sigma_{2}^{2} - \sigma_{1}^{2})S(t)^{2}dt $$ It is always positive ($\sigma_{2}>\sigma{1}$, and $c_{xx}>0$).
In summary, X(t) is a portfolio with X(0) = 0, and de^{-rt}X(t) is always positive, s.t. it has arbitrage opportunity.
If any problem in my idea and my proof, please let me know.
## Answer by airguru (score 0, accepted)
https://quant.stackexchange.com/a/17407
I think the delta-replicating of $\sigma_2$ call is just a fancy way of saying "hedging the call option bought at $\sigma_1$ volatility, with deltas based on $\sigma_2$ volatility". This is full arbitrage in case the hedging/replicating is optimal, and just a statistical arbitrage in real life. You probably do not need such sophisticated proof of why this is an arbitrage, because in case of perfect replication, you replication portfolio will earn just the fair value of the option (based on $\sigma_2$ volatility). And whenever you buy the option at lower price, you have an arbitrage.
This question is probably a duplicate of: Volatility arbitrage - how is the profit extracted? (link is to my answer, see also valuable info in comments).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.