The Black-Scholes formula I

The original derivation of the Black-Scholes [BS] formula is based on a replicating portfolio that ensures that no arbitrage opportunities are allowed. As in the discrete case we consider a portfolio $\mathbf{H}=\left\{ H\left( t\right) \right\} _{t\geq 0}$ which is $\mathscr{F}\left( t\right)$-measurable [we can choose as we go, but at any point in time the choice is deterministic]. $H\left( t\right)$ denotes the number of shares that we are holding at time $t$, the rest of our money will be invested in the money market account, giving us a risk free rate of return, $r$.

The stock, $S\left( t\right)$, follows a geometric Brownian motion

$$\frac{dS\left( t\right) }{S\left( t\right) }=\mu dt+\sigma dB\left( t\right)$$   .

Then the wealth of the investor, $X\left( t\right)$, will follow a diffusion given by [with time suppressed]

$$dX = HdS+r\left( X-HS\right) dt$$

$$= \left[ rX+HS\left( \mu -r\right) \right] dt+HS\sigma dB$$

where one can observe the presence of the risk premium $\mu -r$.

Now we can value a European claim which will have payoffs $g\left[ S\left( T\right) \right]$ at time $T$. Suppose that the value of this claim at time $t$ is given by $G\left( t\right) =f\left( S,t\right)$ when $S=S\left( t\right)$. Applying Itô's formula to this function yields

$$dG=\left[ f_{t}+\mu Sf_{S}+\frac{1}{2}\sigma ^{2}S^{2}f_{SS}\right] dt+\sigma Sf_{S}dB$$.

Now a replicating portfolio $\mathbf{H}$ would track the value of $%% G\left( t\right)$ for all $t\in \left[ 0,T\right]$. We saw above that the value of such a portfolio [the wealth of the investor] will be equal to

$$dX=\left[ rX+HS\left( \mu -r\right) \right] dt+HS\sigma dB$$.

Observe that both diffusions are [as expected of course] driven by the same Brownian motion. In order for the portfolio to track $G\left( t\right)$ at all times [mathematically $X\left( t\right) =G\left( t\right) =f\left( S,t\right)$ for all $t\in \left[ 0,T\right]$] one can just equate the coefficients. The fact that this is feasible implies that the market under these assumptions is complete, since a claim with a generic payoff function $g\left[ S\left( t\right) \right]$ can be replicated. The resulting relations are

$$f_{t}+\mu Sf_{S}+\frac{1}{2}\sigma ^{2}S^{2}f_{SS} = rX+HS\left( \mu -r\right)$$

$$\sigma Sf_{S} = HS\sigma$$

The second equation will give the delta-hedging rule

$$H\left( t\right) =f_{S}\left( S,t\right)$$   ;

while the first equation, using the fact that $X\left( t\right) =f\left( S,t\right)$ yields that

$$f_{t}\left( S,t\right) +rSf_{S}\left( S,t\right) +\frac{1}{2}\sigma ^{2}S^{2}f_{SS}\left( S,t\right) =rf\left( S,t\right)$$   .

This is the BS partial differential equation [PDE]. The terminal condition that has to be satisfied is that

$$f\left( S,T\right) =g\left( S\right)$$   .

Apparently the BS PDE can be used for all European contracts, depending on the payoff function $g\left( S\right)$. For example

$$\begin{tabular}{ll} European Calls & $g\left( S\right) =\left( S-... ...\left( S\right) =V\mathfrak{I}\left[ S\geq K\right] $ \\ etc. & \end{tabular}$$

BS give the solution of the PDE above for the European call $C_{BS}=f\left( S,0\right)$ when $S=S\left( 0\right)$

$$C_{BS}\left( S,K,T,r,\sigma \right) =S\Phi\left( d_{1}\right) -Ke^{-rT}\Phi\left( d_{2}\right)$$

where $\mathcal{N}\left( \cdot \right)$ is the cumulative normal distribution function,

$$d_{1} = \frac{\ln \left( \frac{S}{K}\right) +\left( r+\frac{\sigma ^{2}}{2}%% \right) T}{\sigma \sqrt{T}}\text{, and}$$

$$d_{2} = d_{1}-\sigma \sqrt{T}$$