Option Price Derivatives Reveal Forward-Measure Probabilities, Not Real-World Odds

# Risk Neutrality Necessary for Dual Delta Calculation?


# Risk Neutrality Necessary for Dual Delta Calculation?












I have an option chain for a specific expiry date.

Then calculate dP/dK numerically for each pair of strikes.

My hunch is that this calculation is not risk neutral in the strictest sense of the word (i.e. in relation to the risk free rate of return) because I am comparing options with forward price strikes.

It does assume no arbitrage between options obviously.

This upshot is, can I say that the probability of ending up in the money is the 'actual' probability rather than 'just' a risk neutral probability? Or is there some implicit assumption which I am missing?

Thanks!

## Answer by AFK (score 2, accepted)

https://quant.stackexchange.com/a/17333

After a lot of guess work, I think can try and answer what I think might be your question.

First, note that at maturity the forward equals the spot: $F_T^T = S_T$ so I am not sure what you mean by "forward price strike". I think you mean that your have forward prices of calls and puts.

If you chose a model for your index $S$ and the rates, then the present value of a call is $$ C(T,K) = E^{\mathbb{Q}}[e^{-\int_0^Tr_s ds}(S_T-K)_+] = P(0,T)E^{\mathbb{Q}_T}[(F^T_T-K)_+] $$ where $\mathbb{Q}_T$ is the forward measure. This means that $$ \mathbb{Q}_T[S_T>K] = -P(0,T)^{-1}\partial_KC(T,K) $$ The forward price of the call is simply the undiscounted value $$ \widehat{C}(T,K) = E^{\mathbb{Q}_T}[(F^T_T-K)_+] $$ so $$ \mathbb{Q}_T[S_T>K] = -\partial_K\widehat{C}(T,K) $$

Since you only computed $-\partial_KC(T,K)$ and yet your probabilities sum to $1$, I will guess that your quotes are for forward prices of your options. This is confirmed by using put call parity, $\widehat{C}(T,K)-\widehat{P}(T,K) = F_T-K$, the forward $F_T = K + \widehat{C}(T,K)-\widehat{P}(T,K)$ is independent of the strike (in your case equal to $\approx 2060.7$).

Now to answer your question. The probabilities you implied from market prices are $T$-forward probabilities. If you chose to model rates as deterministic, the $T$-forward measure is the same as the standard risk-neutral measure, else they are different. But whatever you chose, you are not computing distribution of the S&P under the actual = real = historical probability $\mathbb{P}$. Derivatives can be priced easily because it requires the risk free rate and the vol without knowing the drift i.e. the actual noise free trend of the underlying. Conversly you cannot imply this drift from option prices so you cannot imply the real world probabilities of the S&P going up or down.