• Highlight the properties of the random walk assumption:
  • Normal distribution and lognormal distribution
  • Expected return, variance and standard deviation
  • Cumulative normal distribution function $N(x)$
• Assumptions underlying the Black-Scholes model
• Calculate the prices of European calls and puts on non-dividend-paying stocks using the Black-Scholes formula
• Volatility:
  • Estimate volatility from historical prices
  • Define implied volatility and discuss its importance
• Extensions:
  • Adjust the Black-Scholes formula to price options on a dividend-paying stock
  • Estimate the prices of American calls on dividend-paying stocks using Black's approximation

• Let $\varphi [m,s]$ denote normal distribution with mean $m$ and standard deviation $s$.
• Consider a stock with price $S_0$, expected return $\mu$ and price volatility $\sigma$.
• Under the random walk assumption, in a short period $\mathrm{\Delta }t$:
  • the change in the stock price, $\mathrm{\Delta }S$, is normally distributed:
    $$\mathrm{\Delta }S\sim \varphi [\mu S_0\mathrm{\Delta }t,\sigma S_0\sqrt{\mathrm{\Delta }t}]$$
  • In other words, the proportional stock return, $\frac{\mathrm{\Delta }S}{S_0}$, is normally distributed:
    $$\frac{\mathrm{\Delta }S}{S_0}\sim \varphi [\mu \mathrm{\Delta }t,\sigma \sqrt{\mathrm{\Delta }t}]$$

• It follows that the stock log-return, $\ln S_T$, is normally distributed:
  $$\ln S_T\sim \varphi [\ln S_0+(\mu -\frac{{\sigma }^2}{2})T,\sigma \sqrt{T}]$$ $$\ln \frac{S_T}{S_0}\sim \varphi [(\mu -\frac{{\sigma }^2}{2})T,\sigma \sqrt{T}]$$
• A random variable whose natural logarithm is normally distributed, by definition, is log-normally distributed.
• Thus if $\ln S_T$ and $\ln \frac{S_T}{S_0}$ are normally distributed, then $S_T$ and $\frac{S_T}{S_0}$ are log-normally distributed:
  
  • The mean stock price is: $E[S_T]=S_0{e}^{\mu T}$
  • Its variance is: $\text{var}[S_T]=S_0^2{e}^{2\mu T}(e^{{\sigma }^{2}T}-1)$

• $\mu$ is the annual expected return on stock:
  • The expected value of the stock price is $E(S_T)=S_0{e}^{\mu T}$
• The actual continuously compounded return on the stock $R$ satisfies the relationship:
  $$S_T=S_0{e}^{RT}⟹R=\frac{1}{T}\ln \frac{S_T}{S_0}$$
• Since $\ln \frac{S_T}{S_0}\sim \varphi [(\mu -\frac{{\sigma }^2}{2})T,\sigma \sqrt{T}]$, then:
  $$R\sim \varphi [\mu -\frac{{\sigma }^2}{2},\frac{\sigma }{\sqrt{T}}]⟹E[R]=\mu -\frac{{\sigma }^2}{2}(\ne \mu )$$
• This is due to the fact that the arithmetic mean, $\mu$, of the stock returns over a period $\mathrm{\Delta }t$ is different to the geometric mean, $\mu -\frac{{\sigma }^2}{2}$, of these returns.

• The volatility $\sigma$ of the stock price is the standard deviation of its annual continuously compounded return.
• The standard deviation of the return over a period $\mathrm{\Delta }t$ is:
  $$\sigma \sqrt{\mathrm{\Delta }t}$$
• Example: If the stock price is $100 and its volatility is 20%, what is the standard deviation of its daily return?
  $$\frac{20\mathrm{\%}}{\sqrt{252}}=\frac{20\mathrm{\%}}{15.87}=1.26\mathrm{\%}\text{ per trading day}$$
  The expected absolute daily price move (1 standard deviation) is:
  $$\mathrm{\$}100\times 0.0126=\mathrm{\$}1.26$$

• The option price and the stock price depend on the same underlying source of uncertainty.
• No-arbitrage arguments: We can form a portfolio consisting of the stock and the option, which eliminates this source of uncertainty:
  • The portfolio is instantaneously riskless and therefore instantaneously earns the risk-free rate $r$.
• The Black-Scholes model is a binomial model where the hedge portfolio is rebalanced infinitely often:
  • The Black-Scholes price is the limit of the binomial tree price as the length of the time step $\mathrm{\Delta }t\to 0$.
• Assumptions of the Black-Scholes Model:
  • The stock price has a lognormal distribution with constant $\mu$ and $\sigma$.
  • The stock pays no dividends during the life of the option.
  • Investors can borrow or lend at the same constant risk-free rate $r$.
  • There are no arbitrage opportunities; no transaction costs or taxes; continuous security trading.

We consider a European call and European put with strike $K$ and maturity $T$:

• The price of the call is:
  $$c=S_{0}N(d_1)-K{e}^{-rT}N(d_2)(1)$$
• The price of the put is:
  $$p=K{e}^{-rT}N(-d_2)-S_{0}N(-d_1)(2)$$
  with:
  $$\begin{array}{rl}{d}_1& =\frac{\ln (S_0/K)+(r+{\sigma }^2/2)T}{\sigma \sqrt{T}}\\ d_2& =\frac{\ln (S_0/K)+(r-{\sigma }^2/2)T}{\sigma \sqrt{T}}=d_1-\sigma \sqrt{T}\end{array}$$
• $N(x)$ is the cumulative distribution function of a standard normal random variable $X\sim \varphi (0,1)$.

• $N(x)$ is the probability that a normally distributed variable with mean 0 and standard deviation 1 is less than $x$: $$N(d)=\text{Prob}\{z<d\},z\sim \varphi (0,1)$$
• Values of $N(x)$ can be obtained by:
  • Using the Excel built-in function NORMSDIST (or NORM.S.DIST)
  • Looking up a standard normal probability table

• Example – Linear Interpolation: What is $N(1.645)$?
  From the table: $N(1.64)=0.9495$ and $N(1.65)=0.9505$, and 1.645 lies exactly halfway between them:
  $$N(1.645)\approx 0.9495+0.5(0.9505-0.9495)=0.9500$$
  So the probability that a standard normal variable is less than 1.645 is about 95% (a quantile commonly used to construct confidence intervals).

Setup: $S_0=\mathrm{\$}100,K=\mathrm{\$}100,r=5\mathrm{\%},\sigma =20\mathrm{\%},T=1$ year. The stock pays no dividends.

This is a simplified example. Original Hull problem with full solution: see Hull Exercises & Solutions, Lecture 8.

• The variable $\mu$ does not feature in the Black-Scholes analysis:
  • The risk preferences of investors do not affect option prices!
• The preference-independent nature of the Black-Scholes formula makes it consistent with the risk-neutral valuation principle:
  • In a “risk-neutral” world, the initial value of an asset is equal to its discounted expected payoff at maturity, discounted at $r$.
• Current asset price: $S_0=e^{-rT}E[S_T]⟹E[S_T]=S_0{e}^{rT}$
• Current value of a (long) forward contract ($f_T$: the contract's value at maturity $T$):
  $$\begin{array}{rl}{f}_0& =e^{-rT}E[f_T]=e^{-rT}E[F_T-F_0]\\ & =e^{-rT}E[S_T]-F_0{e}^{-rT}\\ & =e^{-rT}\cdot \underset{E(S_T)}{\underbrace{S_0{e}^{rT}}}-F_0{e}^{-rT}\\ & =S_0-F_0{e}^{-rT}=(F_T-F_0)e^{-rT}\end{array}$$
• Current value of a call option ($c_T=max(S_T-K,0)$: the option payoff at maturity $T$):
  $$c_0=e^{-rT}E[c_T]=e^{-rT}E[max[S_T-K,0]]=\dots \text{Black-Scholes formula}$$

• Causes of volatility:
  • Under the Efficient Market Hypothesis (EMH), volatility represents new information arriving in the market.
  • Trading itself causes volatility (market microstructure, liquidity shocks).
• Trading Days vs Calendar Days:
  • Volatility is usually much greater when the market is open (i.e. the asset is trading), than when it is closed (weekends, holidays).
  • For this reason, time $T$ is usually measured in trading days (typically 252 days per year), not calendar days (365 days), for the purposes of estimating volatility and pricing options.

• European options on dividend-paying stocks are valued by substituting into the Black-Scholes formula, the stock price $S_0$ with the difference between the stock price and the present value $PV(D)$ of the dividends ($D$: the dividend amount):
  $$S_0\to S_0^{\ast }=S_0-PV(D)$$
  Only dividends with ex-dividend dates during the life of the option should be included.
• Thus the prices of a call and a put on a dividend-paying stock are:
  $$\begin{array}{rl}c& =S_0^{\ast }N(d_1)-K{e}^{-rT}N(d_2)(3)\\ p& =K{e}^{-rT}N(-d_2)-S_0^{\ast }N(-d_1)(4)\end{array}$$
  where:
  $$\begin{array}{rl}{d}_1& =\frac{\ln (S_0^{\ast }/K)+(r+{\sigma }^2/2)T}{\sigma \sqrt{T}}\\ d_2& =\frac{\ln (S_0^{\ast }/K)+(r-{\sigma }^2/2)T}{\sigma \sqrt{T}}=d_1-\sigma \sqrt{T}\end{array}$$

• An American call on a non-dividend-paying stock should never be exercised early ($C=c$).
• An American call on a dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date.
• American puts are typically exercised early under the right conditions, even when the stock pays no dividends.
• The price of the American call is estimated as the maximum of two European option prices (Black's approximation):
  • Option 1: The European call maturing at the same date as the American option (with $S_0^{\ast }=S_0-PV(D)$, where $D$ is the dividend and $PV(D)$ its present value).
  • Option 2: The European call maturing just before the final ex-dividend date (with unadjusted $S_0$).

Setup: $S_0=\mathrm{\$}100,\sigma =20\mathrm{\%},r=5\mathrm{\%}$ (same as the worked example above). A dividend $D=\mathrm{\$}5$ is expected in 6 months ($t_1=0.5$ year). Call strike $K=\mathrm{\$}90$ (deep in the money, so that early exercise can pay), maturity $T=1$ year.