# Risk neutral probability and forecasting

When our goal is pricing of derivative products we, due to no arbitrage conditions, have to use the risk neutral probability. In other side if we have risk management purpose we have to use the “physical probability” (in this place is better no look in deep what they are). However if our goal, generally speaking, is to do the forecast then what kind of probability we must use ? I think the physical, but I’m not totally sure. Maybe exist different case … I don’t known. What do you think ?

• Forecasting which quantity and for what purpose? Forecasting the spot AAPL over the next 5 mins for high freq trading is different to forecasting the vol of SPX over ten days, different to forecasting the prob that an merger arb deal will fail, different to forecasting the slope of the yield curve in a year's time. – Kiwiakos Jun 2 '16 at 7:54
• In part I'm agree. In fact i wrote "Maybe exist different case … I don’t known". Probably the instrument/variable play a role. In any case I think that time horizon don't play any role. More practically I can stay focused on interest rate and/or stock price level, both in cash and future market. In any case can you give me an example in which we need of risk neutral probabilities and other one where are necessary the real/phisical? – markowitz Jun 2 '16 at 13:09
• Risk-neutral world has the exact same set of events than real world, it's just the probability measures that differ. Deforming the real-world probabilities allows us to write instruments' prices as martingales without needing to care for risk-aversion. This is very convenient and greatly simplifies the developments required to price assets. But an asset price remains an asset price. It is unique whether we choose to express it under risk-neutral probabilities or physical probabilities. (...) – Quantuple Jun 3 '16 at 12:15
• (...) Wrapping up: if you need to find a formula for an asset price, you'd better work in the risk-neutral world where you can get rid of risk-aversion and simplify your developments. However, as explained earlier, this is a pure mathematical construct which amounts to "deforming" the real-world probabilities associated to each event. If you need to generate scenarios or forecasts of asset prices you need to stick with the real-world probability measure. This is why risk-management is performed under $\mathbb{P}$, while derivatives pricing is performed under $\mathbb{Q}$ – Quantuple Jun 3 '16 at 12:17
• When I use "asset price" I mean to price an asset as seen of today, of course if you make forecasts the expected price will change between the real-world and risk-neutral world. – Quantuple Jun 3 '16 at 12:21