Merge pull request #16 from gregwinther/master

Another update of Black-Scholes project

Author: mhjensen

doc/src/Projects/2020/Project5/BlackScholes/BlackScholes.do.txt

@@ -68,7 +68,7 @@ the stock moves like a geometric Brownian motion,
 Explicit solutions for the Black-Scholes equation,
 called The Black-Scholes formulae, are known only for 
 European call and put options. For other derivatives, such 
-a formula doest not have to exist. However, a numberical solution is 
+a formula doest not have to exist. However, a numerical solution is 
 always possible.
 
 === 5a: Transformation to Heat Equation/Diffusion equation ===
@@ -135,8 +135,8 @@ can adapt the program to solve the Black-Scholes equation.
 Plot $V$ vs $S$ at different values for $t$ ($\tau$).
 
 __Special considerations__
-The variable $x$ is unbounded, i,e. $s\in[-\infty, \infty]$. Numerically,
-we have to pick a bounded interval $[-L, L]$, wher $L$ is a sufficiently 
+The variable $x$ is unbounded, i,e. $x\in[-\infty, \infty]$. Numerically,
+we have to pick a bounded interval $[-L, L]$, where $L$ is a sufficiently 
 large number. This interval remain unchanged when considering an option with a
 different strike price.
 
@@ -156,25 +156,33 @@ You should compare to the analytic solution, i.e. the Black-Scholes Formula:
 
 !bt
 \begin{equation}
-    C(S_t, t) = N(d_1) S_t - N(d_2) PV(K),
+    C(S_t, t) = N(d_1) S_t - N(d_2) PV(E),
 \end{equation}
 !et
-where
+where the present value of the exercise price is
+given by
+!bt
+\begin{equation}
+    PV(E) = Ee^{-rt},
+\end{equation}
+!et
+furthermore we have parameters $d_1$,
 !bt
 \begin{equation}    
     d_1 = \frac{1}{\sigma \sqrt{T - t}}
     \left[
-        \ln \left(\frac{S_t}{K} \right) 
-        + \left(r + \frac{\sigma^2}{2} \right) (T - t)
+        \ln \left(\frac{S_t}{E} \right) 
+        + \left(r + \frac{\sigma^2}{2} \right) (T - t),
     \right] 
 \end{equation}
 !et
-and
+and $d_2$,
 !bt
 \begin{equation}
-    d_2 = d_1 - \sigma \sqrt{t}
+    d_2 = d_1 - \sigma \sqrt{t},
 \end{equation}
 !et
+while $N$ is the cumulative normal distribution function.
 
 
 === 5c: Compute values for first-order Greeks ===
@@ -209,7 +217,7 @@ of the larger listed companies such as Equinor, Norsk Hydro,
 Telenor, Mowi, Orkla etc. See for instance
 URL:"https://www.oslobors.no/markedsaktivitet/#/list/derivatives/quotelist/false"
 
-Since there the marked already prices the options, one can use these options 
+Since there the market already prices the options, one can use these options 
 prices to derive the value of implicit variables. A common practice 
 is to use the market-given prices of options to derive the 
 implied volatility of the option.