> [!tldr] > In a no-arb market with no trading costs and a deterministic interest rate, there exists a **risk neutral measure** $\mathbb{Q}$ such that we can fairly price any product by taking its expected value. > > Mathematically, if a product is worth $P_{T}$ at time $T$, then $P_{t}=e^{-r(T-t)} \mathbb{E}_{\mathbb{Q}}[P_{T} ~|~ \mathcal{F}_{t}].$Here $\mathcal{F}_{t}$ is the information available at time $t$. Note that $\mathbb{Q}$ is in general different from the usual probability measure $\mathbb{P}$. It is unique if and only if all products traded can be perfectly replicated by other products in the market. > [!idea] > One interpretation of $\mathbb{Q}$ is the argument that *by hedging the product with $\Delta_{t}:= \frac{ \partial P }{ \partial S }$ shares of the stock, the actual drift $\mu$ of the stock is irrelevant*: picking which every value yields the same pricing, and we might as well choose a value that makes computations convenient (i.e. $r$). > Therefore, $\mu$-ignorant arguments (like pricing hedged portfolios) made in the risk-neutral world still hold in the original Black-Scholes world. However, those that depend on the drift does not (for example, we can't compute the probability of an option being in the money, which is dependent on the distribution $S_{T}$, and by extension $\mu$). A hedge that works almost surely under $\mathbb{Q}$ works almost surely under $\mathbb{P}$ (and vice versa) as well, since the two probability measures are equivalent, i.e. they agree on what events are $\mathrm{a.s.}$ (the hedge being one of them). ### Probabilistic Pricing in Black-Scholes Model Under the [[Black-Scholes Equation|Black-Scholes Model]], the stock price $S_{t}$ evolves with $\frac{dS_{t}}{S_{t}}=\mu ~ dt + \sigma~ dW_{t},$with $W_{t}$ being a Brownian motion under the (usual) measure $\mathbb{P}$. Under the risk-neutral measure $\mathbb{Q}$, the dynamics is instead $\frac{dS_{t}}{S_{t}}= r~dt + \sigma~d \tilde{W}_{t},$where $\tilde{W}_{t}$ is a Brownian motion under $\mathbb{Q}$. Therefore, $S_{t}$ follows a [[Geometric Brownian Motion|geometric Brownian motion]] with the [[Log-Normal Distribution|log-normal distribution]] $\begin{align*} \frac{S_{T}}{S_{t}} ~|~ S_{t}&\sim \mathrm{LogNorm}\big(( r - \sigma^{2} / 2 )\tau, ~ \sigma^{2}\tau\big),\\ \log S_{T} ~|~ S_{t} &\sim \log S_{t} + N\big((r-\sigma^{2} / 2)\tau , ~ \sigma^{2}\tau\big). \end{align*}$ Now the **fundamental theorem of asset pricing** then gives the price of a product $P_{t}$ as: $P_{t}=e^{-r\tau}\cdot \mathbb{E}_{\mathbb{Q}}[P_{T} ~|~ \mathcal{F}_{t}].$and the expectation can be evaluated by integrating over the density of $S_{T}$. ### Examples of Probabilistic Pricing > [!idea] Computing the probability under the risk-neutral measure > There are two ways to find such probability: > - To transform the problem into the scale of $\log (S_{T} / S_{t}) ~|~ S_{t}$, and use its Gaussian density, > - Or to directly integrate $P_{T}(S_{T})$ using the log-normal density of $S_{T} ~|~ S_{t}$. > [!examples] Example using the Gaussian density > If $P_{T}=\mathbf{1}(S_{T} \ge K)$ is a binary European call, then $\begin{align*} \mathbb{E}_{\mathbb{Q}}[P_{T} ~|~ \mathcal{F}_{t}]&= \mathrm{Prob}_{\mathbb{Q}}[S_{T} \ge K ~|~ \mathcal{F_{t}}]\\ &= \mathrm{Prob}_{\mathbb{Q}}\Bigg[ ~\underbrace{\log\left( \frac{S_{T}}{S_{t}} \right)}_{\text{Gaussian}} \ge \log\left( \frac{K}{S_{t}} \right) ~|~ \mathcal{F_{t}}\Bigg]\\ &= 1-\Phi\left(\frac{\log(K / S_{t}) - (r - \sigma^{2} / 2)\tau}{\sqrt{ \tau } \sigma}\right)\\ &= \Phi\left(\frac{\log(S_{t} / K) + (r-\sigma ^{2} / 2)\tau}{\sqrt{ \tau }\sigma }\right). \end{align*}$Now discounting by the interest rate gives the fair price at time $t$. > [!examples] Example using log-normal density > Here, using the density found in [[Log-Normal Distribution#Deriving the Distribution and Density|the log-normal note]], $S_{T} ~|~ S_{t}$ has the density $\begin{align*} f_{S_{T} ~|~ S_{t}}(s)&= \frac{1}{S_{t}}f_{S_{T} / S_{t} ~|~ S_{t}}\left( \frac{s}{S_{t}} \right)\\ &= \frac{1}{s}\phi\left( \log \frac{s}{S_{t}}~;~\left( r-\frac{\sigma^{2}}{2} \right)\tau, \sigma^{2}\tau \right) \end{align*}$