OM-W6

Authors: Developed by Black and Scholes (1973) and Merton (1973).
Purpose: To price stock options based on stock price dynamics.
Key Equation: The stock price dynamics is modeled by the stochastic differential equation:
- $dSt = \mu St dt + \sigma St dWt$
- where dSt is the change in stock price, μ is the drift, σ is the volatility, and W_t is a Wiener process or Brownian motion.

Binomial Model: Works with discrete states and discrete time (finite possible stock prices and time steps).
BMS Model: Based on continuous states (the stock price can be any value between 0 and ∞) and continuous time.

The model has been expanded to price:
- Currency options (Garman and Kohlhagen)
- Options on futures (Black).
All variations are treated under the umbrella of the BMS model.

Equation: $dSt = \mu St dt + \sigma St dWt$
Brownian Motion: The source of randomness, which generates a normal distribution with mean 0 and variance t, denoted as $\mathcal{N}(0, t)$ .
- dWt are independent increments, each $dWt \sim \mathcal{N}(0, dt)$ .
Time Increment Definitions:
- dt: An infinitesimally small time increment, smaller than 1 millisecond.
- dWt: The instantaneous increment of a Brownian motion, defined as: $dWt = W{t+dt} - Wt$ .

If $X \sim \mathcal{N}(0, 1)$ , then any linear transformation $a + bx \sim \mathcal{N}(a, b^2)$ .
For the BMS model, the instantaneous return can be expressed as:
- $dSt = \mu St dt + \sigma St dWt \sim \mathcal{N}(\mu St dt, \sigma^2 St^2 dt)$ .
Definitions of key parameters:
- μ: The annualized mean return.
- σ²: Annualized variance.
- σ: Annualized standard deviation (volatility).

The stock price follows a geometric Brownian motion represented as:
- $rac{dSt}{St} = \mu dt + \sigma dW_t$ .
Drift ($μ$) and Diffusion ($σ$) characterize the stock price dynamics:
- Instantaneous changes are normally distributed and longer horizons yield log-normally distributed changes in price.
- As derived through Itô's lemma:
- $d \ln S_t \sim \mathcal{N}(\mu dt - 0.5 \sigma^2 dt, \sigma^2 dt)$ .

Using the equation:
- $dSt = \mu St dt + \sigma St dWt$ with parameters μ = 10%, σ = 20%, S = 100, t = 1.
Discretizing to daily intervals, approximate as:
- dt \
  eq A_t = rac{1}{252}.
Generate standard normal random variables:
- $\varepsilon(100 \times 252) \sim \mathcal{N}(0, 1)$ .
Log returns converted to stock price paths are:
- $St = S0 e^{\sum R_d}$ .

Hedging Argument: Combining the option with stock results in a risk-free portfolio.
- Define parameters to ensure the portfolio's value is riskless and earns the risk-free rate.
- The only parameter that matters in this context is σ, allowing for hedging away risk.

From the hedging argument, the following PDE is obtained:
- $\frac{\partial f}{\partial t} + (r - q)S \frac{\partial f}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 f}{\partial S^2} = r f$
Parameters: Only volatility (σ) is present in the derivation; risk premium (μ) is excluded from valuation.
Analytical formulas for call and put options can be derived using terminal payout conditions.

Call Option Value ($C_t$):
- $Ct = St e^{-(r)(T-t)} N(d1) - K e^{-r(T-t)} N(d2)$
- where $d1 = \frac{\ln(St/K) + (r-q)(T-t) + 0.5\sigma^2(T-t)}{\sigma \sqrt{T-t}}$ and
- $d2 = d1 - \sigma \sqrt{T-t}$ .
Put Option Value ($P_t$):
- $Pt = e^{-r(T-t)} [K N(-d2) - St N(-d1)]$ .
Cumulative Normal Distribution:
- $N(x)$ represents the cumulative probability up to $x$ for a standard normal variable, used in option pricing calculations.

The BMS model adjusts for the underlying security price, accounting for dividends or carrying costs through forward pricing models.
Using the indirect method, implied volatility can be calculated to match observed option prices.

The only unknown variable in the BMS context is σ, which is inferred by matching observed option prices.
- Implied Volatility (IV) is derived from market prices and has a monotonic correspondence to observed option prices.
- Traders often quote implied volatilities, as it provides better information than dollar prices, excluding arbitrage opportunities.

Analyzing how the value of both call and put options responds to volatility levels:
- Value decomposition into intrinsic value and time value:
- Intrinsic Value: Value if the underlying price remains constant.
- Time Value: Value arising from potential price movements, with larger movements reflecting higher volatility leading to greater time values.

The expected outcomes based on implied volatility rather than constant volatility indicate unequal pricing across strikes/premiums for options.
Short-term options exhibit volatility smiles, whereas longer-term options generally present a skew, providing insights into market expectations of price distributions.
Negative skewness often indicates a higher probability of downward price movements, deviating from normally distributed reality.

Market realities such as jumps or stochastic volatility contradict BMS assumptions, indicating that returns deviate from normally distributed outcomes.
- BMS assumes small diffusions over time, whereas actual observed price movements may involve sudden jumps, leading to fat-tailed distributions.
Differences in implied volatilities across different market conditions hint towards implicit jumps and varying market dynamics over time.

Second-generation models leverage Lévy processes to explain both continuous and jump dynamics, helping to address BMS deviations and stochastic volatility over time.
Recognition of non-normality in returns promotes the development of new models to enhance option valuation accuracy.

Understand fundamental properties of normally distributed variables and distinguish between stochastic processes and random variables.
Connect BMS with the binomial model and commit key formulas to memory, relating option pricing to forward pricing and the put-call parity.
Acknowledge the implications of observed implied volatility plots, recognizing their means of reflecting market sentiment and efficiency, potentially introducing future pricing models.