301-BlackScholes-1.Rmd

# Pricing with Black-Scholes

```{r setup, echo=FALSE, include=FALSE, cache=FALSE}
knitr::opts_chunk$set(
  cache = TRUE,
  dev.args = list(bg = "transparent"),
  tidy=TRUE,
  tidy.opts=list(width.cutoff=45)
)
```

`r tufte::newthought("Recall")` from the previous section that under the risk-neutral
probability, the discounted value is a martingale:

$$S_0 = e^{-rT} E^Q(S_T)$$

where $S_T$ is a log-normal variable that follows the process:

$$S_T = S_0 e^{(r-\frac{\sigma^2}{2})T + \sigma \sqrt{T}Y}$$

with $Y ~ N(0,1)$. In a seminal paper, F. Black and M. Scholes provided an arbitrage-free analytical expression for the value of a European option.

The value of any derivative is the discounted
expected payoff. For a call option, we have:

$$\begin{aligned}
    c(S, 0) &=& e^{-rT} E^Q [ \left( S_0 e^{(r-\frac{\sigma^2}{2})T + \sigma \sqrt{T}Y}-K \right)^+ ] \\
    &=& \frac{e^{-rT}}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \left( S_0 e^{(r-\frac{\sigma^2}{2})T + \sigma \sqrt{T}Y}-K \right)^+ e^{-x^2/2} dx\end{aligned}$$

Now compute the bounds of integration:

$$\begin{aligned}
    & & S_0 e^{(r-\frac{\sigma^2}{2})T + \sigma \sqrt{T}x} \geq K \\
    & <=> & e^{\sigma \sqrt{T} x} \geq \frac{K}{S_0} e^{-(r-\frac{\sigma^2}{2})T} \\
    & <=> & x \geq \frac{1}{\sigma \sqrt{T}} \left( \ln(\frac{K}{S_0}) -(r-\frac{\sigma^2}{2})T \right)\end{aligned}$$

Let

$$T_1 = \frac{1}{\sigma \sqrt{T}} \left( \ln(\frac{K}{S_0}) -(r-\frac{\sigma^2}{2})T \right)$$

$$\begin{aligned}
    c(S, 0) &=&  \frac{e^{-rT}}{\sqrt{2\pi}} \int_{T_1}^{\infty} \left( S_0 e^{(r-\frac{\sigma^2}{2})T + \sigma \sqrt{T}x}-K \right) e^{-x^2/2} dx \\
    &=& \frac{e^{-rT}}{\sqrt{2\pi}} \int_{T_1}^{\infty} S_0 e^{(r-\frac{\sigma^2}{2})T + \sigma \sqrt{T}x} e^{-x^2/2} dx - K  \frac{e^{-rT}}{\sqrt{2\pi}} \int_{T_1}^{\infty}  e^{-x^2/2} dx \\
    &=& A - B\end{aligned}$$

$$B = K e^{-rT} (1-N(T_1))$$

Apply the change of variable $y = x-\sigma \sqrt{T}$ to get

$$\begin{aligned}
    A &=&  \frac{e^{-rT}}{\sqrt{2\pi}} \int_{T_1-\sigma \sqrt{T}}^{\infty} S_0 e^{-y^2/2} dy \\
      &=& S_0 (1-N(T_1 - \sigma \sqrt{T}))\end{aligned}$$

where $N(x)$ is the cumulative normal distribution. Using the identity
$N(x)+N(-x)=1$, one gets the Black-Scholes formula in its standard form.

$$c(S,0) = S_0N(d_1) - Ke^{-rT}N(d_2) \,$$

where

$$\begin{aligned}
    d_1 &=& \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}} \\
    d_2 &=& d_1 - \sigma\sqrt{T}.\end{aligned}$$

The value of the put can be obtained from the call-put parity
relationships.

Relationships Among Greeks
--------------------------

See standard text for derivation of greeks. Recall the Black-Scholes
pricing equation:

$$\frac{\partial f}{\partial t} + rS 
    \frac{\partial f}{\partial S} + 
    \frac{1}{2} \sigma^2 S^2 
    \frac{\partial^2 f}{\partial S^2} = rf$$

This is true of any derivative, and in particular of a delta hedged
portfolio. Set:

$$\begin{aligned}
    \Theta &=& \frac{\partial f}{\partial t} \\
    \Gamma &=& \frac{\partial^2 f}{\partial S^2}\end{aligned}$$

then,

$$\Theta + \frac{1}{2} \sigma^2 S^2 \Gamma = rf$$