BlackScholes

MFE Topic List

Black-Scholes formula

Given
{$ S $} = current price
{$ K $} = strike price
{$ \sigma $} = volitility
{$ r $} = risk-free interest rate
{$ T $} = expiration time
{$ t $} = current time (so that {$ T-t $} is time until expiration)
{$ \delta $} = continuous dividend rate
the Black-Scholes formula for the price of a European call option is {$$ C = C(S, K, \sigma, r, T-t, \delta)= Se^{-\delta (T-t)} N(d_1) - Ke^{-r(T-t)} N(d_2), $$} where {$N$} is the standard normal distribution function, {$$ d_1= \frac{\ln(S/K) + (r-\delta +\frac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}}=\frac{\ln(Se^{-\delta (T-t)}/Ke^{-r(T-t)})+\frac{1}{2}\sigma^2 (T-t)}{\sigma\sqrt{T-t}},$$} {$$ d_2 = d_1 -\sigma \sqrt{T-t}.$$}

The price of a European put option is {$$ P = P(S, K, \sigma, r, T-t, \delta)= -Se^{-\delta (T-t)} N(-d_1) + Ke^{-r(T-t)} N(-d_2). $$} Note that put-call parity holds.

Black-Scholes equation

Given
{$ \displaystyle{ \Delta = \frac{\partial C}{\partial S}} $}
{$ \displaystyle{ \Gamma = \frac{\partial^2 C}{\partial S^2}} $}
{$ \displaystyle{ \theta = \frac{\partial C}{\partial t}} $}
the Black-Scholes equation is {$$ \frac{1}{2}\sigma^2S^2\Gamma +(r-\delta)S\Delta + \theta = rC.$$}

We can verify that the functions for call and put prices are solutions to the Black-Scholes differential equation. The key is the following equality, which seems to come up frequently in calculations: {$$ Se^{-\delta(T-t)} N'(d_1) = Ke^{-r(T-t)}N'(d_2). $$}

To understand the equation, we need two tools: the movement of stock prices based on Brownian motion, and delta-hedging.

Delta-hedging

This is the idea behind the Black-Scholes equation. We look at things from the market-maker's perspective.

A market maker sells an option, and hedge his position (against stock prices rising) by buying some stocks. By definition, {$\Delta$} is the change of the option price per dollar change in the stock price, so if we want the price sensitivity to be the same for the option and the stock, we would invest in {$\Delta$} shares of stock.

Selling one option and buying {$\Delta$} shares of stock means the investment is {$$ \Delta S - C. $$} Let {$dS$} be a small change in stock price, {$dt$} a small change in time, and {$dC$} the corresponding change in option price. The change in the value of the portfolio is {$$ \text{change in the value of stock - change in the value of option - interest } $$} {$$ = \Delta \,dS +\delta \,dt\, \Delta S - dC - r\,dt(\Delta S - C). $$} By Taylor series approximation, we know that {$$ dC = dS \,\Delta + \frac{1}{2} (dS)^2 \,\Gamma + dt \, \theta.$$} Substituting gives {$$ \Delta \,dS +\delta \,dt\, \Delta S - (dS \,\Delta + \frac{1}{2} (dS)^2 \,\Gamma + dt \, \theta) - r\,dt(\Delta S - C) $$} {$$= \delta \, dt\, \Delta S - \frac{1}{2} (dS)^2 \,\Gamma - dt \, \theta - r\,dt(\Delta S - C). $$} Assume now that if the stock moves one standard deviation, then this delta-hedged solution breaks even, that is, the change of the value of portfolio = 0 when {$$ (dS)^2 = \sigma^2 S^2 \,dt. $$} Thus {$$ \delta \, dt\, \Delta S - \frac{1}{2} \sigma^2 S^2 \,dt \,\Gamma - dt \, \theta - r\,dt(\Delta S - C) = 0.$$} Divide by {$-dt$} gives the Black-Scholes equation.

Other Assets

Replace {$ Se^{-\delta T} $} with {$ F^P_{0,T}(S), $} the prepaid forward price of underlying asset. Replace {$Ke^{-rT}$} with {$ F^P_{0,T}(K), $} the prepaid forward price of the strike asset.

• {$ F^P_{0,T}(S) = S_0 - PV(Div)$} for stock with discrete dividend.
• {$ F^P_{0,T}(x) = x_0 e^{-r_f T}$} for foreign currency with spot exchange {$x$} and

interest rate {$r_f$}.

• {$ F^P_{0,T}(S) = Fe^{-rT}$} for futures with price {$F$}; the {$e^{-rT}$} term cancels to give the Black formula.

Quantities depending on Greeks

The Greek measure of a portfolio is the sum of the Greeks of the individual portfolio component.

Suppose that the stock price changes by {$ \varepsilon $}.

• The change in option price is {$ \varepsilon \Delta $}.
• The option elasticity is

{$$ \Omega = \frac{\text{% change in option price}}{\text{% change in stock price}}=\frac{\frac{\varepsilon \Delta}{C}}{\frac{\varepsilon}{S}} = \frac{S\Delta}{C}.$$}

• {$ \Omega $} is a measure of leverage. {$ \Omega \geq 1 $} for a call, because a call is a levered investment in the stock, which is riskier. {$ \Omega \leq 0$} for a put because we're shorting a stock.
• The volatility of an option is

{$$ \sigma_{option} = \sigma_{stock} \cdot |\Omega|. $$}

Let {$ \alpha $} be the expected rate of return on the stock (so the stock is worth {$S(1+\alpha)$} at a future date).

• The risk premium of the stock is {$ \alpha - r $}.
• The risk premium of an option is {$ (\alpha -r)\Omega $}.

The Sharpe ratio for any asset is the ratio of risk premium to volatility. The Sharpe ratios for a call and its underlying stock are the same (the {$ \Omega $} is divided out).

Perpetual Options