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).
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.