A binomial‐tree model provides a discrete‐time representation of how an asset price evolves. At each time step, the price can either move “up” or “down” by a specified factor, capturing a simplified version of market uncertainty. Over multiple steps, this forms a recombining tree of possible paths. ## Discrete Steps We assume that at each step $t$, the price $S_t$ relates to the price $S_{t-1}$ with a log-return $r_t$. $ S_t = S_{t-1} *e^{r_t} $ In the binomial tree $r_t$ takes one of two possible values at each step. Either an "up" or "down" return, denoted as follows: $ r_t = \begin{cases} \log u, &\text{with probability} \: p \\ \log d, &\text{with probability} \: (1-p) \end{cases}$ These up and down factors must be calibrated to market data, i.e. the [[Expectation]] and [[Variance]] of the financial asset, which we want to model. ## Bernoulli Formulation A convenient way to encode these two possible states, is to use a [[Bernoulli Distribution|Bernoulli]] [[Random Variable]] $x_t$. $ r_t = a+bx_t, \quad r_t = \begin{cases} \log 1, &\text{with probability} \: p \\ \log 0, &\text{with probability} \: (1-p) \end{cases}$ In this representation: - *Upwards case:* When $x_t=1$, which implies that $r_t = a+b$ - *Downwards case:* When $x_t=0$, which implies that $r_t =a$ ## Calibrating Model Parameters The model parameters $a,b$ govern the size of the up and down factors. We want them to reflect the real mean $\mu$ and variance $\sigma^2$ and $p$ of the financial asset over the given time step $t$. Probability $p$: While there are different ways to estimate the probability of an upwards move, the simplest is to look at historical frequencies. $ \hat p= \frac{\sum_t\mathbb I_{r_t>0}}{N}$ where: - Numerator: The count of positive log-returns, expressed as a sum of indicator variables - Denominator: Number of observed log-returns Mean: $ \begin{align} \mu \equiv \mathbb E[r_t]&=\mathbb E[a+bx_t]\\[2pt] &= a+b\mathbb E[x_t]\\[2pt] &= a+bp\\ \hat \mu &=a+b \hat p \end{align} $ Variance: $ \begin{align} \sigma^2 \equiv \mathrm{Var}(r_t) &= \mathrm{Var}(a+bx_t)\\[2pt] &= b^2 \mathrm{Var}(x_t)\\[2pt] &= b^2p(1-p)\\ \hat \sigma^2 &=b^2 \hat p(1-\hat p) \end{align} $ Use the established relations, we can solve for $b$: $ \begin{align} \sigma^2 &= b^2p(1-p)\\[2pt] b^2 &= \frac{\sigma^2}{p(1-p)}\\[2pt] b&=\frac{\sigma}{\sqrt{p(1-p)}} \end{align} $ Solving for $a$: $ \begin{align} \mu &=a+bp \\[2pt] a &=\mu-bp \\[2pt] a &= \mu - \frac{\sigma}{\sqrt{p(1-p)}}*p\\[4pt] a &= \mu - \sigma\sqrt{\frac{p}{(1-p)}} \end{align} $ ## Relating to Up/Down Factors Once $a$ and $b$ are known, we can interpret them in terms of the up & down factors $\log u$ and $\log d$. Down factor: $ \log d = a = \mu - \sigma\sqrt{\frac{p}{(1-p)}} $ Up-factor: $ \begin{align} \log u &= a+b\\ &= \mu - \sigma\sqrt{\frac{p}{(1-p)}} + \frac{\sigma}{\sqrt{p(1-p)}}\\ &=\mu - \sigma \frac{p}{\sqrt{p(1-p)}}+ \frac{\sigma}{\sqrt{p(1-p)}}\\ &=\mu + \sigma \frac{1-p}{\sqrt{p(1-p)}} \end{align} $