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In this related question How to derive the implied probability distribution from B-S volatilities?, it is shown how to infer the implied probability density of the future prices of a risky asset from a continuum of call prices written on that asset (Breeden-Litzenberger identity).

The developments, which I invite you to read, basically rely on the fact that the call price writes $$ C=e^{-rT} \int_0^\infty (S-K)^+ p(S) dS $$ or equivalently $$ C = e^{-rT} \Bbb{E} \left[ (S_T-K)^+ \right] \tag{1} $$ under some measure where $S_T$ is a random variable with probability density function: $$ d\Bbb{P}(S_T \leq S)/dS = p(S) $$

Now, the fundamental theorem of asset pricing tells us that equation $(1)$ holds under the so-called risk-neutral measure (a measure equivalent to the real-world measure but under which the $t$-value of any self-financing strategy is a martingale when expressed with respect to the risk-free money market account numéraire).

Consequently, the implied density $p(S)$ you compute by evaluating $$ p(S) = e^{rT} \frac{\partial^2 C}{\partial K^2}(K=S) $$

is indeed a risk-neutral pdf (because it relies on the risk-neutral expression of the call price $(1)$.)

In this related question How to derive the implied probability distribution from B-S volatilities?, it is shown how to infer the implied probability density of the future prices of a risky asset from a continuum of call prices written on that asset (Breeden-Litzenberger identity).

The developments, which I invite you to read, basically rely on the fact that the call price writes $$ C=e^{-rT} \int_0^\infty (S-K)^+ p(S) dS $$ or equivalently $$ C = e^{-rT} \Bbb{E} \left[ (S_T-K)^+ \right] \tag{1} $$ under some measure where $S_T$ is a random variable with probability density function: $$ d\Bbb{P}(S_T \leq S)/dS = p(S) $$

Now, the fundamental theorem of asset pricing tells us that equation $(1)$ holds under the so-called risk-neutral measure (a measure equivalent to the real-world measure but under which the $t$-value of any self-financing strategy is a martingale when expressed with respect to the risk-free money market account numéraire).

Consequently, the implied density $p(S)$ you compute by evaluating $$ p(S) = e^{rT} \frac{\partial^2 C}{\partial K^2}(K=S) $$

is indeed a risk-neutral pdf (because it relies on the risk-neutral expression of the call price $(1)$.)

In this related question How to derive the implied probability distribution from B-S volatilities?, it is shown how to infer the implied probability density of the future prices of a risky asset from a continuum of call prices written on that asset (Breeden-Litzenberger identity).

The developments, which I invite you to read, basically rely on the fact that the call price writes $$ C=e^{-rT} \int_0^\infty (S-K)^+ p(S) dS $$ or equivalently $$ C = e^{-rT} \Bbb{E} \left[ (S_T-K)^+ \right] \tag{1} $$ under some measure where $S_T$ is a random variable with probability density function: $$ d\Bbb{P}(S_T \leq S)/dS = p(S) $$

The fundamental theorem of asset pricing tells us that this measure is precisely the so-called risk-neutral measure (a measure equivalent to the real-world measure but under $t$-value of any self-financing strategy is a martingale when expressed with respect to the risk-free money market account numéraire), such that the implied density $p(S)$ is indeed a risk-neutral pdf.

In this related question How to derive the implied probability distribution from B-S volatilities?, it is shown how to infer the implied probability density of the future prices of a risky asset from a continuum of call prices written on that asset (Breeden-Litzenberger identity).

The developments, which I invite you to read, basically rely on the fact that the call price writes $$ C=e^{-rT} \int_0^\infty (S-K)^+ p(S) dS $$ or equivalently $$ C = e^{-rT} \Bbb{E} \left[ (S_T-K)^+ \right] \tag{1} $$ under some measure where $S_T$ is a random variable with probability density function: $$ d\Bbb{P}(S_T \leq S)/dS = p(S) $$

Now, the fundamental theorem of asset pricing tells us that equation $(1)$ holds under the so-called risk-neutral measure (a measure equivalent to the real-world measure but under which the $t$-value of any self-financing strategy is a martingale when expressed with respect to the risk-free money market account numéraire).

Consequently, the implied density $p(S)$ you compute by evaluating $$ p(S) = e^{rT} \frac{\partial^2 C}{\partial K^2}(K=S) $$

is indeed a risk-neutral pdf (because it relies on the risk-neutral expression of the call price $(1)$.)

In this related question How to derive the implied probability distribution from B-S volatilities?, it is shown how to infer the implied probability density of the future prices of a risky asset from a continuum of call prices written on that asset (Breeden-Litzenberger identity).

The developments, which I invite you to read, basically rely on the fact that the call price writes $$ C=e^{-rT} \int_0^\infty (S-K)^+ p(S) dS $$ or equivalently $$ C = e^{-rT} \Bbb{E} \left[ (S_T-K)^+ \right] \tag{1} $$ under some measure where $S_T$ is a random variable with probability density function: $$ d\Bbb{P}(S_T \leq S)/dS = p(S) $$

The fundamental theorem of asset pricing tells us this measure is precisely the so-called risk-neutral measure (a measure equivalent to the real-world measure under which the discounted $t$-value of any self-financing portfolio is a martingale), such that the implied density $p(S)$ is indeed a risk-neutral pdf.

In this related question How to derive the implied probability distribution from B-S volatilities?, it is shown how to infer the implied probability density of the future prices of a risky asset from a continuum of call prices written on that asset (Breeden-Litzenberger identity).

The developments, which I invite you to read, basically rely on the fact that the call price writes $$ C=e^{-rT} \int_0^\infty (S-K)^+ p(S) dS $$ or equivalently $$ C = e^{-rT} \Bbb{E} \left[ (S_T-K)^+ \right] \tag{1} $$ under some measure where $S_T$ is a random variable with probability density function: $$ d\Bbb{P}(S_T \leq S)/dS = p(S) $$

The fundamental theorem of asset pricing tells us that this measure is precisely the so-called risk-neutral measure (a measure equivalent to the real-world measure but under $t$-value of any self-financing strategy is a martingale when expressed with respect to the risk-free money market account numéraire), such that the implied density $p(S)$ is indeed a risk-neutral pdf.