Why Deep In-the-Money Options Give Unstable Implied Volatility Estimates

# Deep ITM Call Implied Vol via Monte Carlo


# Deep ITM Call Implied Vol via Monte Carlo












Let's say I've computed the price of a call using Monte Carlo with $S_0 = 100$ and $K = 80$, using $T = 0.1$ and $r = 0$ to be $\$20.00095$. This price estimate comes with a $95\%$ confidence interval of $[19.99969 , 20.00221]$. The issue, then, is trying to estimate implied vol, including a confidence interval, since the $95\%$ lower bound on price is below intrinsic and hence the inverse problem has no solution.

Even if we set the confidence interval to be $[20, 20.00221]$, the implied vol estimate would be $0.2172$ with a $95\%$ confidence interval $[0, 0.2324]$, a huge spread.

I'm already using a massive amount of stock prices to get the confidence intervals this tight, so I'd prefer not to just ramp up the number of simulations. All of the paths finish in the money for this case. Is there a variance reduction technique, perhaps, for dealing with these deep ITM issues?

## Answer by Quantuple (score 7, accepted)

https://quant.stackexchange.com/a/27503

[Short answer]

IMHO there is a fundamental problem with wanting to extract a sound implied volatility figure out of a deep ITM option's price. You should use out-of-the-money forward options (OTMF) instead: put options for strikes smaller than the forward price (left wing of the volatility surface) and call options otherwise (right wing of the volatility surface).

[Long answer]

To illustrate my point, let $V$ denote the $t$-value of a European option, which we split into 2 components $$ V = V_i + V_e $$ according the following thought experiment:

- The intrinsic value, $V_i$, is defined as what you would get if you could exercise the option immediately at time $t$, or equivalently, what your final gain would look like should the underlying price be frozen to its current value up to the contract's expiry. $V_i$ is always positive (but can be zero).

- The extrinsic or time value, $V_e$, is the remaining part. It accounts for the fact that the underlying price is expected to evolve and not remain frozen. $V_e$ can be positive or negative.

By construction, we have that the intrinsic value $V_i$ does not depend on the future volatility as it is something we have defined assuming the underlying remained frozen. In contrast, the time value $V_e$ does depend on future volatility, more or less strongly depending on the remaining time to maturity $\tau = T-t$ and where the current spot price $S_t$ is located with respect to the strike $K$.

By definition, the price of a strongly ITM option essentially corresponds to intrinsic value $$ V = V_i + V_e \approx V_i $$ Because $V_i$ does not depend on volatility, it is very difficult to imply a robust volatility figure from an ITM option price (the fraction $V_e$ of the option price which truly depends on volatility is very small relative to the full option price $V$)

On the contrary, strongly OTM options essentially reflect time value $$ V = V_i + V_e \approx V_e $$ which makes it easier to imply volatility (the fraction $V_e$ of the option price which truly depends on volatility is very important relative to the full option price $V$)

Hence you should prefer OTM options to ITM options when it comes to inferring implied volatilities.