Black Scholes Guide#

This is an elegant-design introduction to black scholes model, geared mainly for new users. The end result is a diffusion-type partial differential equation used for derivative pricing. The option value will depend on variables \(S\) and \(t\), parameters \(\sigma\) and \(\mu\) associated with the asset price following GBM, \(E\) and \(T\) associated with the particular contract, and \(r\) associated with currency, namely

\[V(S, t; \sigma, \mu; E, T; r) = V(S, t)\]

Assumptions#

The underlying follows a lognormal random walk with known volatility, i.e.

\[\frac{dS}{S} = \mu dt + \sigma dW, ~ \mu, \sigma, r(t) \in \mathbb{R}\]

No dividends on the underlying or \(D = 0\)
Delta hedging is done continuously
No transaction costs on the underlying
No arbitrage opportunities

A Special Portfolio#

Set up a portfolio \(\Pi\) consisting of one long position in an option and short \(\Delta\) shares in continuous time world

\[\Pi = V(S, t) - \Delta S\]

where \(V\) is value of the option depending on \(S\) and \(t\), \(\Delta\) is the number of shares, and \(S\) is the stock price. Changes in the option value and the underlying both contribute to the change in the portfolio value.

\[d\Pi = dV(S, t) - \Delta dS\]

Note that \(\Delta\) is fixed from time step \(t\) to \(t + dt\), but recalculation is needed at a new time step.

Random terms#

From Ito we have

\[d V = \frac{\partial V}{\partial t} d t+\frac{\partial V}{\partial S} d S+\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} d t\]

The random change in portfolio is therefore

\[d \Pi = \frac{\partial V}{\partial t} d t+\frac{\partial V}{\partial S} d S+\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} d t - \Delta dS\]

and, after plugging \(dS = \mu S dt + \sigma S dW\), the random terms in our portfolio are

\[\left(\sigma S \frac{\partial V}{\partial S} - \Delta \sigma S \right) dW\]

where the risk can be eliminated by carefully choosing \(\Delta\) to be

\[\Delta = \frac{\partial V}{\partial S}\]

in order to reduce the randomness to zero. Delta hedgeing is the perfect elimination of risk, and it is a dynamic hedging strategy where continuous rebalancing is necessary.

Deterministic terms#

After determining the random terms, we are left with deterministic changes which is completely risk-free, namely

\[d \Pi = \left(\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right) dt\]

Intuitively, another risk-free portfolio is just compounding at a risk-free rate, or

\[\left(\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 \S^2 \frac{\partial^2 V}{\partial S^2}\right) dt = d \Pi = r \Pi dt = r \left(V - S \frac{\partial V}{\partial S}\right) dt\]

Dividing above equation by \(dt\) and rearraging we get

\[\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0\]

and this is the Black-Scholes equation, a linear parabolic partial differential equation since it is linear in the first order and ?. It contains all variables and parameters but the drift term \(\mu\), so the volatility \(\sigma\) determines the value of derivatives.

In a complete market, an option can be replicated only with the underlying. If \(\Delta\) is the partial derivative of some option value then the portfolio will yield the same amount as the option payoff.

Note that the Black Scholes equation is defined independent of options. We must specify the \(V(S, T)\) as a final condition.

BSE for option contracts#

Assume the asset receives a continuous and constant dividend yield \(D\), so each time step the asset receives \(D S dt\). Putting into the Black-Scholes equation and choosing \(\Delta = \frac{\partial V}{\partial S}\) we get

\[\begin{split}\begin{align*} d \Pi & = dV - \Delta dS - D \Delta S dt\\ & = \frac{\partial V}{\partial t} d t+\frac{\partial V}{\partial S} d S+\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} d t - \Delta dS - D \Delta S dt\\ \Rightarrow 0 & = \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + (r - D) S \frac{\partial V}{\partial S} - r V \end{align*}\end{split}\]

Options on currencies are handled the same way as dividend-paying stock. We receive interest at a foreign rate of \(r_f\) while holding the foreign currency

\[\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + (r - r_f) S \frac{\partial V}{\partial S} - r V = 0\]

Commodity options require cost of carry \(q\), which is the cost of storage

\[\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + (r + q) S \frac{\partial V}{\partial S} - r V = 0\]

Notice that here we use a positive sign in the Black-Scholes equation since, though we may sell the commodity to the counterparty, it is our responsibility to store and secure.