.. slideconf:: :slide_classes: appear ============================================================================== Black-Scholes-Merton Model ============================================================================== Overview ============================================================================== The Black-Scholes-Merton (BSM) model provides a simple formula for computing the price of an option. .. raw:: - It is derived in a fashion similar to the binomial option pricing model. .. raw:: - A riskless portfolio is obtained by buying :math:`\smash{\Delta}` shares of the underlying asset and shorting a single option. .. raw:: - :math:`\smash{\Delta}` represents :math:`\smash{\frac{\partial C}{\partial S}}`, where :math:`\smash{S}` is the price of the underlying and :math:`\smash{C}` is the price of the option. .. raw:: - Unlike the binomial model, the BSM model :math:`\smash{\Delta}` is only valid for an infinitesimal length of time. BSM Assumptions ============================================================================== The BSM model depends on the following assumptions. .. raw:: - The price of the underlying asset follows geometric Brownian motion with parameters :math:`\smash{\mu}` and :math:`\smash{\sigma}`. - There is no restriction on short selling. - There are no transaction costs or taxes. - All assets are perfectly divisible. - The underlying doesn't pay dividends. - There are no riskless arbitrage opportunities. - Asset trading occurs continuously. - The risk-free rate, :math:`\smash{r}` is constant and the same for all maturities. BSM and Ito's Lemma ============================================================================== Suppose the price of an asset follows geometric Brownian motion: .. math:: \begin{align} dS & = \mu S dt + \sigma S dZ. \end{align} .. raw:: According to Ito's lemma, the price of a derivative, :math:`\smash{f}`, follows: .. math:: \begin{align} df & = \left(\frac{\partial f}{\partial S} \mu S + \frac{\partial f}{\partial t} + \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2 \right) dt + \frac{\partial f}{\partial S} \sigma S dZ. \end{align} A Riskless Portfolio ============================================================================== Consider a portfolio that is short one unit of the derivative and long :math:`\smash{\frac{\partial f}{\partial S}}` units of the underlying: .. math:: \begin{align} \Pi & = -f + \frac{\partial f}{\partial S} S \\ \Rightarrow d \Pi & = -df + \frac{\partial f}{\partial S} dS \\ & = \left(-\frac{\partial f}{\partial t} - \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2\right) dt. \end{align} .. raw:: Note that the portfolio is not affected by :math:`\smash{Z}`, so it is riskless. A Riskless Portfolio ============================================================================== - The portfolio must earn the risk-free rate of return over period :math:`\smash{dt}`. .. math:: \begin{gather} d \Pi = r \Pi dt \\ \Rightarrow \left(\frac{\partial f}{\partial t} + \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2\right) dt = r\left(f - \frac{\partial f}{\partial S} S\right) dt \\ \Rightarrow rf = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial S} rS + \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2. \end{gather} .. raw:: - This is the BSM differential equation. Boundary Conditions ============================================================================== The BSM differential equation is true for any derivative that depends on :math:`\smash{S}`. .. raw:: - Boundary conditions determine the price of a particular derivative. .. raw:: - For example, the boundary condition for a call option is its terminal value: :math:`\smash{f = \max(S-X,0)}`. .. raw:: - Likewise, the boundary condition for a put option is its terminal value: :math:`\smash{f = \max(X-S,0)}`. BSM Price of Forward ============================================================================== Suppose that some time ago a forward contract was entered into with delivery price :math:`\smash{K}` and maturity :math:`\smash{T}`. .. raw:: - Recall that at intermediate date :math:`\smash{t}` the value of the forward is :math:`\smash{f = S - K e^{-r(T-t)}}`. .. raw:: - It follows: .. math:: \begin{align} \frac{\partial f}{\partial t} = -r K e^{-r(T-t)} \,\,\,\, \frac{\partial f}{\partial S} = 1 \,\,\,\, \frac{\partial^2 f}{\partial S^2} = 0. \end{align} .. raw:: - Substituting these into the BSM equation, :math:`\smash{rf = rS - rK e^{-r(T-t)}}`, which is true. Risk-Neutral Valuation ============================================================================== A basic principle of asset pricing is that investors demand higher expected return, :math:`\smash{\mu}`, in the presence of higher volatility, :math:`\smash{\sigma}`. .. raw:: - Note that :math:`\smash{\mu}` doesn't appear in the BSM differential equation. .. raw:: - This means we can treat investors *as if* they are risk neutral. .. raw:: - That is, we can value assets under the assumption that investors only demand expected return :math:`\smash{r}`, even though they aren't really risk neutral. .. raw:: - The practical implication is that once we compute future asset payoffs, we can discount them at the risk-free rate (implicitly assuming this is the rate of return that investors demand). BSM Option Pricing Formulas ============================================================================== The BSM option pricing formulas are: .. math:: \begin{align} C & = S_0 \Phi(d_1) - X e^{-rT} \Phi(d_2) \\ P & = X e^{-rT} \Phi(-d_2) - S_0 \Phi(-d_1)\\ d_1 & = \frac{\log(S_0/X) + (r+\sigma^2/2)T}{\sigma \sqrt{T}} \\ d_2 & = \frac{\log(S_0/X) + (r-\sigma^2/2)T}{\sigma \sqrt{T}} = d_1 - \sigma \sqrt{T}. \end{align} .. raw:: - These satisfy the BSM differential equation. .. raw:: - :math:`\smash{\Phi(x)}` represents the standard Normal CDF: :math:`\smash{P(X \leq x)}` when :math:`\smash{X \sim N(0,1)}`. .. raw:: - :math:`\smash{C}` and :math:`\smash{P}` are the prices of European call and put options on the underlying, :math:`\smash{S}`. Normal CDF ============================================================================== .. image:: BlackScholes/normCDF.png :width: 4in :align: center Interpretation ============================================================================== .. math:: \begin{align} P(S_T \geq X) & = P\left(\log(S_T) \geq \log(X)\right) \\ & = P\left(\frac{\log(S_T) - E\left[\log(S_T)\right]}{Sd\left(\log(S_T)\right)} \geq \frac{\log(X) - E\left[\log(S_T)\right]}{Sd\left(\log(S_T)\right)}\right) \\ & = 1 - \Phi\left(\frac{\log(X) - E\left[\log(S_T)\right]}{Sd\left(\log(S_T)\right)}\right) \\ & = \Phi\left(-\frac{\log(X) - E\left[\log(S_T)\right]}{Sd\left(\log(S_T)\right)}\right) \\ & = \Phi(d_2). \end{align} Interpretation ============================================================================== The last equation above follows because, under risk-neutrality :math:`\smash{E\left[\log(S_T)\right] = \log(S_0) + (r - \sigma^2/2)T}` and :math:`\smash{Sd\left(\log(S_T)\right) = \sigma \sqrt{T}}`: .. math:: \begin{align} -\frac{\log(X) - E\left[\log(S_T)\right]}{Sd\left(\log(S_T)\right)} & = \frac{\log(S_0) + (r - \sigma^2/2)T - \log(X)}{\sigma \sqrt{T}} \\ & = \frac{\log(S_0/X) + (r - \sigma^2/2)T}{\sigma \sqrt{T}} \\ & = d_2. \end{align} Interpretation ============================================================================== :math:`\smash{\Phi(d_1)}` is similar to :math:`\smash{\Phi(d_2)}`, but slightly harder to interpret. .. raw:: - :math:`\smash{S_0\Phi(d_1) e^{rT}}` is the expected asset price (under risk neutrality), conditional on the asset expiring with :math:`\smash{S_T \geq X}`. .. raw:: - The expected payoff of the call option is: .. math:: \begin{align} S_0 \Phi(d_1) e^{rT} - X \Phi(d_2). \end{align} .. raw:: - The present value of the call option expected payoff is: .. math:: \begin{align} S_0 \Phi(d_1) - X e^{-rT} \Phi(d_2). \end{align} Extreme Cases ============================================================================== Suppose the current stock price :math:`\smash{S_0}` is very large relative to the strike :math:`\smash{X}`. .. raw:: - In this case :math:`\smash{d_1}` and :math:`\smash{d_2}` are very large. .. raw:: - As a result :math:`\smash{\Phi(d_1)}` and :math:`\smash{\Phi(d_2)}` approach 1. .. raw:: - This causes the call option value to be :math:`\smash{S_0 - X e^{-rT}}`. - This is identical to a forward contract, which makes sense for a deep-in-the-money call. .. raw:: - Likewise :math:`\smash{\Phi(-d_1)}` and :math:`\smash{\Phi(-d_2)}` approach 0. .. raw:: - This causes the put option value to be 0. Extreme Cases ============================================================================== Suppose :math:`\smash{\sigma \to 0}`. .. raw:: - If :math:`\smash{S_0 > X e^{-rT}}`, then :math:`\smash{d_1 \to \infty}` and :math:`\smash{d_2 \to \infty}`, causing :math:`\smash{\Phi(d_1) \to 1}` and :math:`\smash{\Phi(d_2) \to 1}`. .. raw:: - The result is a value of :math:`\smash{S_0 - X e^{-rT} > 0}`, which is identical to the riskless payoff :math:`\smash{\max(S_0 e^{rT} - X,0)}`. .. raw:: - Likewise, if :math:`\smash{S_0 < X e^{-rT}}`, then :math:`\smash{d_1 \to -\infty}` and :math:`\smash{d_2 \to -\infty}`, causing :math:`\smash{\Phi(d_1) \to 0}` and :math:`\smash{\Phi(d_2) \to 0}`. .. raw:: - The result is a value of :math:`\smash{0}`, which is identical to the riskless payoff :math:`\smash{\max(S_0 e^{rT} - X,0)}`. Implied Volatility ============================================================================== Volatility, :math:`\smash{\sigma}`, is the single parameter of the BSM model that is not directly observed. .. raw:: - Typically, the BSM equation is not used to compute an option price, but to compute an *implied volatility*, given an option price. .. raw:: - For example, if :math:`\smash{C = 1.875}`, :math:`\smash{S_0 = 21}`, :math:`\smash{X = 20}`, :math:`\smash{r = 0.1}` and :math:`\smash{T = 0.25}`, what is the value that :math:`\smash{\sigma}` must take for the BSM equation to hold for a European call option? VIX Index ============================================================================== The VIX index is an index of implied volatilities for 30-day S\&P 500 options (calls and puts). .. raw:: - It is interpreted as the (annualized) one standard deviation move (in percentage) of the S&P 500 index over the next 30 days. .. raw:: - Options and futures trade on the VIX itself (as well as the S&P 500). Historical VIX ============================================================================== .. image:: BlackScholes/vixFigure.png :width: 4in :align: center