Pricing Models Path

The **one-step binomial model** prices an option by restricting the underlying asset to two possible values at expiry, an up state $S_0 u$ and a down state $S_0 d$. A **delta-hedged portfolio** of the underlying and the option is riskless, so it must earn the risk-free rate. This **no-arbitrage** argument uniquely determines the option price. Equivalently, the same price emerges by discounting the expected payoff under the **risk-neutral probability measure**, which assigns probabilities $p$ and $1-p$ so that the stock's expected return equals the risk-free rate. The one-step tree is the building block for multi-step lattices, and in the continuous limit it recovers the **Black–Scholes–Merton formula**.

The **risk-neutral probability** $p$ is the synthetic weight that makes the discounted stock price a martingale. In a one-step binomial model with up and down factors $u$ and $d$, $p = (e^{r\Delta t} - d)/(u - d)$. Under this measure, every traded asset is expected to grow at the **risk-free rate** $r$, which lets us price any derivative as a discounted expectation without estimating real-world drift or investor risk preferences.

**Backward induction** prices a derivative on a tree by starting at the terminal nodes, where the payoff is known, and rolling values back one step at a time. At each interior node the value is a **risk-neutral expectation** of the two child values, discounted at $r$ for the step length $\Delta t$. Repeating the move from leaves to root produces today's fair price.

A **two-step binomial tree** divides the option's life into two equal periods of length $\Delta t$, generating three terminal stock prices $S_0 u^2$, $S_0 u d$, and $S_0 d^2$. Pricing proceeds by **backward induction**. Find the option value at the two intermediate nodes, then discount one more step back to today. Adding steps refines the approximation and, in the Cox–Ross–Rubinstein parameterisation, the tree converges to **Black–Scholes** as the number of steps grows.

At every node of the binomial tree, the holder of an **American option** compares the **continuation value** (the discounted risk-neutral expectation of the next two nodes) against the **immediate exercise value** (intrinsic payoff). The node value is the larger of the two. This $\max$ at every node is the only structural difference between American and European pricing on the same tree.

**Delta** ($\Delta$) is the sensitivity of an option price to a small change in the underlying. In a one-step binomial model, $\Delta = (f_u - f_d)/(S_u - S_d)$, which is also the **hedge ratio**, the number of shares to hold per option to make the portfolio locally riskless. Call deltas lie in $[0, 1]$, put deltas in $[-1, 0]$, and both deltas change as the stock moves, a second-order effect captured by **gamma**.

**Delta hedging** is the practice of holding $\Delta$ shares of the underlying for every option written, so that small movements in the stock cancel out at the portfolio level. Because $\Delta$ itself moves with $S$, $T$, and $\sigma$, the hedge must be **rebalanced** as the market moves. In the Black–Scholes limit of continuous rebalancing with no frictions, delta hedging perfectly **replicates** the option payoff and the hedging cost equals the option price.

When the underlying pays dividends, the binomial tree must be re-parameterised. For a **continuous dividend yield** $q$, replace the growth rate $r$ with $r - q$ in the risk-neutral probability, $p = (e^{(r-q)\Delta t} - d)/(u - d)$. For **discrete cash dividends**, the standard practice is to split the stock into a deterministic dividend stream plus a stochastic component and build the tree on the latter (Hull 2022, §21.3). Dividends reduce expected stock growth, which lowers call prices and raises put prices.

The **Black-Scholes-Merton (BSM) model** gives a closed-form price for European options on a non-dividend paying stock. The call formula is $C = S_0 N(d_1) - K e^{-rT} N(d_2)$. The model rests on five assumptions, continuous trading with no transaction costs, lognormal stock prices, no dividends in the basic form, constant volatility $\sigma$, and a constant risk-free rate $r$ with continuous compounding. Black and Scholes (1973) and Merton (1973) showed that, under these assumptions, a self-financing portfolio of stock and bond can replicate the option, so its price is unique and free of any risk premium.

The **standard normal distribution** is the bell curve with mean zero and variance one. Its cumulative distribution function (CDF) $N(x) = P(Z \leq x)$ gives the probability that a standard normal variable $Z$ falls at or below $x$. In BSM, $N(d_1)$ and $N(d_2)$ are the two pieces of the call formula. $N(d_2)$ is the risk-neutral probability that the call finishes **in the money**, and $N(d_1)$ is the option's **delta**, the hedge ratio in shares of stock per option (Hull, 2022, §15.6).

A **z-table** lists pre-computed values of the standard normal CDF $N(x) = P(Z \leq x)$. In BSM, you compute $d_1$ and $d_2$, then look up $N(d_1)$ and $N(d_2)$ to finish the call or put price. The CDF has no elementary antiderivative, so tabulated values (or numerical routines) are the practical way to evaluate it.

$d_1$ and $d_2$ are the two inputs that feed into $N(\cdot)$ inside the BSM call formula. They are defined by $d_1 = [\ln(S_0/K) + (r + \sigma^2/2)T] / (\sigma\sqrt{T})$ and $d_2 = d_1 - \sigma\sqrt{T}$. $N(d_1)$ is the call's **delta** (shares to hold per option), and $N(d_2)$ is the **risk-neutral probability** the call finishes in the money (Hull, 2022, §15.6).

The **Black-Scholes-Merton (BSM)** European call price is $C = S_0 N(d_1) - K e^{-rT} N(d_2)$, where $N(\cdot)$ is the standard normal CDF, $r$ is the continuously compounded risk-free rate, and $T$ is time to expiry in years. The formula prices a European call under the assumptions of **geometric Brownian motion**, constant volatility, no dividends, and frictionless markets. It was derived by Black and Scholes (1973) and extended by Merton (1973), and remains the benchmark against which all subsequent single-asset option models are measured.

The BSM **European put** price is $P = K e^{-rT} N(-d_2) - S_0 N(-d_1)$, where $d_1$ and $d_2$ are the same as in the call formula. The same result follows from **put-call parity**, $C - P = S_0 - K e^{-rT}$, so $P = C - S_0 + K e^{-rT}$ once the call is known. Either route gives the same number for European options on a non-dividend stock (Hull, 2022, §15.8).

For a stock with **continuous dividend yield** $q$, the BSM call becomes $C = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)$, with $d_1 = [\ln(S_0/K) + (r - q + \sigma^2/2)T] / (\sigma\sqrt{T})$ and $d_2 = d_1 - \sigma\sqrt{T}$. For **discrete dividends**, subtract the present value of expected cash dividends from $S_0$ before applying standard BSM (Hull, 2022, §17.2). Merton (1973) introduced the continuous-yield extension.

**Implied volatility** ($\sigma_{\text{imp}}$) is the value of $\sigma$ that, when plugged into BSM, reproduces the **observed market price** of an option. There is no closed form for $\sigma_{\text{imp}}$, so it is solved numerically with a root-finder such as Newton-Raphson or bisection (Hull, 2022, §15.11). Unlike historical volatility, which looks backward at realised returns, implied volatility is **forward-looking** and reflects what the options market expects future volatility to be.