# Risk-Neutrality: Discount factors of the $P$ world according to risk preferences?

I am coming to terms with the connections between the so-called $$P$$ world and the $$Q$$ world. In my understanding, the risk-neutral measure $$Q$$ induces a probability space under which investors are indifferent to risk. For example, if we have two instruments $$S^{1},S^{2}$$ in a one-period model with $$Q_{S^{1}_{1}}=0.5\delta_{50}+0.5\delta_{100}$$ and $$Q_{S^{2}_{1}}=75$$, i.e. the expected payoffs under $$Q$$ of $$S^{1},S^{2}$$ are identitical, then the instruments will be of equal value.

Pricing $$S^{1},S^{2}$$ in the $$P$$ world is more difficult since it is not risk-neutral such that every state of the world needs to be investigated according to the risk preference of the investor. If the investor is risk-averse, we need to discount the price by some particular factor, otherwise in the case of risk seeking agents the price will increase.

An example of the discount factors going into the calculations of prices within the $$P$$ world in the case of a risk averse agent would be what? I mean the risk-free rate is the same for all market participants (in theory), right?

Is this the basic idea of the difference between the $$P$$ and $$Q$$ worlds?

You're right. Euler's equation states $$p_t=\mathbb E^\mathbb P_t[M_{t+1}X_{t+1}],$$ that is pricing under $$\mathbb P$$ requires you to know the stochastic discount factor (SDF, aka pricing kernel) $$M$$. $$M$$ is (typically) found in a general equilibrium setting, depending on the marginal utility of investors. (Note: a strictly positive $$M$$ exists if the market is free of arbitrage and does not require a general equilibrium.) You can easily see that the covariance between $$M$$ and $$X$$ determines the systematic risk of the payoff $$X$$.
Using a change a measure (Radon Nikodym derivative), we can write $$p_t=e^{-r\Delta t}\mathbb E^\mathbb Q_t[X_{t+1}].$$ This gives an alternative (yet fully equivalent) way of computing the price of an asset. [Note: There's a one-to-one relationship between $$M$$ and $$\mathbb Q$$.] Under $$\mathbb Q$$, we can simply discount the expected payoff at the risk-free rate. Thus, the preferences of investors don't matter. Risk premia are zero (risk-neutral world''). This makes pricing much easier because we don't need to figure out what $$M$$ is -- and $$r$$ is observable as you said.
In an informal sense, you simply merge $$M$$ and $$\mathbb{P}$$ together to obtain a new artificial probability measure, $$\mathbb{Q}$$ (risk-neutral measure or equivalent martingale measure). You then only need to figure out what the expectation of the payoff under $$\mathbb{Q}$$ is and then you get the price of an option. Alternatively, you need real-world probabilities ($$\mathbb P$$) and investor's attitude towards risk ($$M$$), see also this answer.
The difference between pricing under $$\mathbb Q$$ and $$\mathbb P$$ is normally the difference between absolute pricing and relative pricing. You typically use $$p_t=\mathbb E^\mathbb P_t[M_{t+1}X_{t+1}]$$ to price basic assets (e.g. stocks) depending on the risk attitude of investors (risk-aversion, EIS, etc.). Pricing under $$\mathbb Q$$ normally relates to relative pricing (pricing by no-arbitrage or replication). This is weaker than an equilibrium approach. You basically take some prices as given (underlying) and value new assets (derivatives) relative to these prices.