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‎doc/Projects/2020/Project5/BlackScholes/html/._BlackScholes-bs000.html‎

Lines changed: 23 additions & 12 deletions
Original file line numberDiff line numberDiff line change
@@ -224,7 +224,7 @@ <h3 id="___sec2" class="anchor">The Black-Scholes equation </h3>
224224
Explicit solutions for the Black-Scholes equation,
225225
called The Black-Scholes formulae, are known only for
226226
European call and put options. For other derivatives, such
227-
a formula doest not have to exist. However, a numberical solution is
227+
a formula doest not have to exist. However, a numerical solution is
228228
always possible.
229229

230230
<h3 id="___sec3" class="anchor">5a: Transformation to Heat Equation/Diffusion equation </h3>
@@ -305,8 +305,8 @@ <h3 id="___sec4" class="anchor">5b: Create solver(s) for the 1D diffusion equati
305305

306306
<p>
307307
<b>Special considerations</b>
308-
The variable \( x \) is unbounded, i,e. \( s\in[-\infty, \infty] \). Numerically,
309-
we have to pick a bounded interval \( [-L, L] \), wher \( L \) is a sufficiently
308+
The variable \( x \) is unbounded, i,e. \( x\in[-\infty, \infty] \). Numerically,
309+
we have to pick a bounded interval \( [-L, L] \), where \( L \) is a sufficiently
310310
large number. This interval remain unchanged when considering an option with a
311311
different strike price.
312312

@@ -330,31 +330,42 @@ <h3 id="___sec4" class="anchor">5b: Create solver(s) for the 1D diffusion equati
330330

331331
$$
332332
\begin{equation}
333-
C(S_t, t) = N(d_1) S_t - N(d_2) PV(K),
333+
C(S_t, t) = N(d_1) S_t - N(d_2) PV(E),
334334
\tag{5}
335335
\end{equation}
336336
$$
337337

338-
where
338+
where the present value of the exercise price is
339+
given by
340+
$$
341+
\begin{equation}
342+
PV(E) = Ee^{-rt},
343+
\tag{6}
344+
\end{equation}
345+
$$
346+
347+
furthermore we have parameters \( d_1 \),
339348
$$
340349
\begin{equation}
341350
d_1 = \frac{1}{\sigma \sqrt{T - t}}
342351
\left[
343-
\ln \left(\frac{S_t}{K} \right)
344-
+ \left(r + \frac{\sigma^2}{2} \right) (T - t)
352+
\ln \left(\frac{S_t}{E} \right)
353+
+ \left(r + \frac{\sigma^2}{2} \right) (T - t),
345354
\right]
346-
\tag{6}
355+
\tag{7}
347356
\end{equation}
348357
$$
349358

350-
and
359+
and \( d_2 \),
351360
$$
352361
\begin{equation}
353-
d_2 = d_1 - \sigma \sqrt{t}
354-
\tag{7}
362+
d_2 = d_1 - \sigma \sqrt{t},
363+
\tag{8}
355364
\end{equation}
356365
$$
357366

367+
while \( N \) is the cumulative normal distribution function.
368+
358369
<h3 id="___sec5" class="anchor">5c: Compute values for first-order Greeks </h3>
359370

360371
<p>
@@ -395,7 +406,7 @@ <h3 id="___sec6" class="anchor">5d: Find data and compute implied volatility </h
395406
<a href="https://www.oslobors.no/markedsaktivitet/#/list/derivatives/quotelist/false" target="_self"><tt>https://www.oslobors.no/markedsaktivitet/#/list/derivatives/quotelist/false</tt></a>
396407

397408
<p>
398-
Since there the marked already prices the options, one can use these options
409+
Since there the market already prices the options, one can use these options
399410
prices to derive the value of implicit variables. A common practice
400411
is to use the market-given prices of options to derive the
401412
implied volatility of the option.

‎doc/Projects/2020/Project5/BlackScholes/html/BlackScholes-bs.html‎

Lines changed: 23 additions & 12 deletions
Original file line numberDiff line numberDiff line change
@@ -224,7 +224,7 @@ <h3 id="___sec2" class="anchor">The Black-Scholes equation </h3>
224224
Explicit solutions for the Black-Scholes equation,
225225
called The Black-Scholes formulae, are known only for
226226
European call and put options. For other derivatives, such
227-
a formula doest not have to exist. However, a numberical solution is
227+
a formula doest not have to exist. However, a numerical solution is
228228
always possible.
229229

230230
<h3 id="___sec3" class="anchor">5a: Transformation to Heat Equation/Diffusion equation </h3>
@@ -305,8 +305,8 @@ <h3 id="___sec4" class="anchor">5b: Create solver(s) for the 1D diffusion equati
305305

306306
<p>
307307
<b>Special considerations</b>
308-
The variable \( x \) is unbounded, i,e. \( s\in[-\infty, \infty] \). Numerically,
309-
we have to pick a bounded interval \( [-L, L] \), wher \( L \) is a sufficiently
308+
The variable \( x \) is unbounded, i,e. \( x\in[-\infty, \infty] \). Numerically,
309+
we have to pick a bounded interval \( [-L, L] \), where \( L \) is a sufficiently
310310
large number. This interval remain unchanged when considering an option with a
311311
different strike price.
312312

@@ -330,31 +330,42 @@ <h3 id="___sec4" class="anchor">5b: Create solver(s) for the 1D diffusion equati
330330

331331
$$
332332
\begin{equation}
333-
C(S_t, t) = N(d_1) S_t - N(d_2) PV(K),
333+
C(S_t, t) = N(d_1) S_t - N(d_2) PV(E),
334334
\tag{5}
335335
\end{equation}
336336
$$
337337

338-
where
338+
where the present value of the exercise price is
339+
given by
340+
$$
341+
\begin{equation}
342+
PV(E) = Ee^{-rt},
343+
\tag{6}
344+
\end{equation}
345+
$$
346+
347+
furthermore we have parameters \( d_1 \),
339348
$$
340349
\begin{equation}
341350
d_1 = \frac{1}{\sigma \sqrt{T - t}}
342351
\left[
343-
\ln \left(\frac{S_t}{K} \right)
344-
+ \left(r + \frac{\sigma^2}{2} \right) (T - t)
352+
\ln \left(\frac{S_t}{E} \right)
353+
+ \left(r + \frac{\sigma^2}{2} \right) (T - t),
345354
\right]
346-
\tag{6}
355+
\tag{7}
347356
\end{equation}
348357
$$
349358

350-
and
359+
and \( d_2 \),
351360
$$
352361
\begin{equation}
353-
d_2 = d_1 - \sigma \sqrt{t}
354-
\tag{7}
362+
d_2 = d_1 - \sigma \sqrt{t},
363+
\tag{8}
355364
\end{equation}
356365
$$
357366

367+
while \( N \) is the cumulative normal distribution function.
368+
358369
<h3 id="___sec5" class="anchor">5c: Compute values for first-order Greeks </h3>
359370

360371
<p>
@@ -395,7 +406,7 @@ <h3 id="___sec6" class="anchor">5d: Find data and compute implied volatility </h
395406
<a href="https://www.oslobors.no/markedsaktivitet/#/list/derivatives/quotelist/false" target="_self"><tt>https://www.oslobors.no/markedsaktivitet/#/list/derivatives/quotelist/false</tt></a>
396407

397408
<p>
398-
Since there the marked already prices the options, one can use these options
409+
Since there the market already prices the options, one can use these options
399410
prices to derive the value of implicit variables. A common practice
400411
is to use the market-given prices of options to derive the
401412
implied volatility of the option.

‎doc/Projects/2020/Project5/BlackScholes/html/BlackScholes.html‎

Lines changed: 23 additions & 12 deletions
Original file line numberDiff line numberDiff line change
@@ -182,7 +182,7 @@ <h3 id="___sec2">The Black-Scholes equation </h3>
182182
Explicit solutions for the Black-Scholes equation,
183183
called The Black-Scholes formulae, are known only for
184184
European call and put options. For other derivatives, such
185-
a formula doest not have to exist. However, a numberical solution is
185+
a formula doest not have to exist. However, a numerical solution is
186186
always possible.
187187

188188
<h3 id="___sec3">5a: Transformation to Heat Equation/Diffusion equation </h3>
@@ -263,8 +263,8 @@ <h3 id="___sec4">5b: Create solver(s) for the 1D diffusion equation. </h3>
263263

264264
<p>
265265
<b>Special considerations</b>
266-
The variable \( x \) is unbounded, i,e. \( s\in[-\infty, \infty] \). Numerically,
267-
we have to pick a bounded interval \( [-L, L] \), wher \( L \) is a sufficiently
266+
The variable \( x \) is unbounded, i,e. \( x\in[-\infty, \infty] \). Numerically,
267+
we have to pick a bounded interval \( [-L, L] \), where \( L \) is a sufficiently
268268
large number. This interval remain unchanged when considering an option with a
269269
different strike price.
270270

@@ -288,31 +288,42 @@ <h3 id="___sec4">5b: Create solver(s) for the 1D diffusion equation. </h3>
288288

289289
$$
290290
\begin{equation}
291-
C(S_t, t) = N(d_1) S_t - N(d_2) PV(K),
291+
C(S_t, t) = N(d_1) S_t - N(d_2) PV(E),
292292
\label{_auto5}
293293
\end{equation}
294294
$$
295295

296-
where
296+
where the present value of the exercise price is
297+
given by
298+
$$
299+
\begin{equation}
300+
PV(E) = Ee^{-rt},
301+
\label{_auto6}
302+
\end{equation}
303+
$$
304+
305+
furthermore we have parameters \( d_1 \),
297306
$$
298307
\begin{equation}
299308
d_1 = \frac{1}{\sigma \sqrt{T - t}}
300309
\left[
301-
\ln \left(\frac{S_t}{K} \right)
302-
+ \left(r + \frac{\sigma^2}{2} \right) (T - t)
310+
\ln \left(\frac{S_t}{E} \right)
311+
+ \left(r + \frac{\sigma^2}{2} \right) (T - t),
303312
\right]
304-
\label{_auto6}
313+
\label{_auto7}
305314
\end{equation}
306315
$$
307316

308-
and
317+
and \( d_2 \),
309318
$$
310319
\begin{equation}
311-
d_2 = d_1 - \sigma \sqrt{t}
312-
\label{_auto7}
320+
d_2 = d_1 - \sigma \sqrt{t},
321+
\label{_auto8}
313322
\end{equation}
314323
$$
315324

325+
while \( N \) is the cumulative normal distribution function.
326+
316327
<h3 id="___sec5">5c: Compute values for first-order Greeks </h3>
317328

318329
<p>
@@ -353,7 +364,7 @@ <h3 id="___sec6">5d: Find data and compute implied volatility </h3>
353364
<a href="https://www.oslobors.no/markedsaktivitet/#/list/derivatives/quotelist/false" target="_blank"><tt>https://www.oslobors.no/markedsaktivitet/#/list/derivatives/quotelist/false</tt></a>
354365

355366
<p>
356-
Since there the marked already prices the options, one can use these options
367+
Since there the market already prices the options, one can use these options
357368
prices to derive the value of implicit variables. A common practice
358369
is to use the market-given prices of options to derive the
359370
implied volatility of the option.

‎doc/Projects/2020/Project5/BlackScholes/ipynb/ipynb-BlackScholes-src.tar.gz‎

0 Bytes
Binary file not shown.

‎doc/Projects/2020/Project5/BlackScholes/pdf/BlackScholes.p.tex‎

Lines changed: 16 additions & 10 deletions
Original file line numberDiff line numberDiff line change
@@ -191,7 +191,7 @@ \subsection{Solving the Black-Scholes Equation Numerically}
191191
Explicit solutions for the Black-Scholes equation,
192192
called The Black-Scholes formulae, are known only for
193193
European call and put options. For other derivatives, such
194-
a formula doest not have to exist. However, a numberical solution is
194+
a formula doest not have to exist. However, a numerical solution is
195195
always possible.
196196

197197
\paragraph{5a: Transformation to Heat Equation/Diffusion equation.}
@@ -252,8 +252,8 @@ \subsection{Solving the Black-Scholes Equation Numerically}
252252
Plot $V$ vs $S$ at different values for $t$ ($\tau$).
253253

254254
\paragraph{Special considerations}
255-
The variable $x$ is unbounded, i,e. $s\in[-\infty, \infty]$. Numerically,
256-
we have to pick a bounded interval $[-L, L]$, wher $L$ is a sufficiently
255+
The variable $x$ is unbounded, i,e. $x\in[-\infty, \infty]$. Numerically,
256+
we have to pick a bounded interval $[-L, L]$, where $L$ is a sufficiently
257257
large number. This interval remain unchanged when considering an option with a
258258
different strike price.
259259

@@ -272,20 +272,26 @@ \subsection{Solving the Black-Scholes Equation Numerically}
272272
You should compare to the analytic solution, i.e.~the Black-Scholes Formula:
273273

274274
\begin{equation}
275-
C(S_t, t) = N(d_1) S_t - N(d_2) PV(K),
275+
C(S_t, t) = N(d_1) S_t - N(d_2) PV(E),
276276
\end{equation}
277-
where
277+
where the present value of the exercise price is
278+
given by
279+
\begin{equation}
280+
PV(E) = Ee^{-rt},
281+
\end{equation}
282+
furthermore we have parameters $d_1$,
278283
\begin{equation}
279284
d_1 = \frac{1}{\sigma \sqrt{T - t}}
280285
\left[
281-
\ln \left(\frac{S_t}{K} \right)
282-
+ \left(r + \frac{\sigma^2}{2} \right) (T - t)
286+
\ln \left(\frac{S_t}{E} \right)
287+
+ \left(r + \frac{\sigma^2}{2} \right) (T - t),
283288
\right]
284289
\end{equation}
285-
and
290+
and $d_2$,
286291
\begin{equation}
287-
d_2 = d_1 - \sigma \sqrt{t}
292+
d_2 = d_1 - \sigma \sqrt{t},
288293
\end{equation}
294+
while $N$ is the cumulative normal distribution function.
289295

290296

291297
\paragraph{5c: Compute values for first-order Greeks.}
@@ -318,7 +324,7 @@ \subsection{Solving the Black-Scholes Equation Numerically}
318324
Telenor, Mowi, Orkla etc. See for instance
319325
\href{{https://www.oslobors.no/markedsaktivitet/#/list/derivatives/quotelist/false}}{\nolinkurl{https://www.oslobors.no/markedsaktivitet/\#/list/derivatives/quotelist/false}}
320326

321-
Since there the marked already prices the options, one can use these options
327+
Since there the market already prices the options, one can use these options
322328
prices to derive the value of implicit variables. A common practice
323329
is to use the market-given prices of options to derive the
324330
implied volatility of the option.

‎doc/Projects/2020/Project5/BlackScholes/pdf/BlackScholes.pdf‎

2.49 KB
Binary file not shown.

‎doc/Projects/2020/Project5/BlackScholes/pdf/BlackScholes.tex‎

Lines changed: 16 additions & 10 deletions
Original file line numberDiff line numberDiff line change
@@ -165,7 +165,7 @@ \subsection*{Solving the Black-Scholes Equation Numerically}
165165
Explicit solutions for the Black-Scholes equation,
166166
called The Black-Scholes formulae, are known only for
167167
European call and put options. For other derivatives, such
168-
a formula doest not have to exist. However, a numberical solution is
168+
a formula doest not have to exist. However, a numerical solution is
169169
always possible.
170170

171171
\paragraph{5a: Transformation to Heat Equation/Diffusion equation.}
@@ -226,8 +226,8 @@ \subsection*{Solving the Black-Scholes Equation Numerically}
226226
Plot $V$ vs $S$ at different values for $t$ ($\tau$).
227227

228228
\paragraph{Special considerations}
229-
The variable $x$ is unbounded, i,e. $s\in[-\infty, \infty]$. Numerically,
230-
we have to pick a bounded interval $[-L, L]$, wher $L$ is a sufficiently
229+
The variable $x$ is unbounded, i,e. $x\in[-\infty, \infty]$. Numerically,
230+
we have to pick a bounded interval $[-L, L]$, where $L$ is a sufficiently
231231
large number. This interval remain unchanged when considering an option with a
232232
different strike price.
233233

@@ -246,20 +246,26 @@ \subsection*{Solving the Black-Scholes Equation Numerically}
246246
You should compare to the analytic solution, i.e.~the Black-Scholes Formula:
247247

248248
\begin{equation}
249-
C(S_t, t) = N(d_1) S_t - N(d_2) PV(K),
249+
C(S_t, t) = N(d_1) S_t - N(d_2) PV(E),
250250
\end{equation}
251-
where
251+
where the present value of the exercise price is
252+
given by
253+
\begin{equation}
254+
PV(E) = Ee^{-rt},
255+
\end{equation}
256+
furthermore we have parameters $d_1$,
252257
\begin{equation}
253258
d_1 = \frac{1}{\sigma \sqrt{T - t}}
254259
\left[
255-
\ln \left(\frac{S_t}{K} \right)
256-
+ \left(r + \frac{\sigma^2}{2} \right) (T - t)
260+
\ln \left(\frac{S_t}{E} \right)
261+
+ \left(r + \frac{\sigma^2}{2} \right) (T - t),
257262
\right]
258263
\end{equation}
259-
and
264+
and $d_2$,
260265
\begin{equation}
261-
d_2 = d_1 - \sigma \sqrt{t}
266+
d_2 = d_1 - \sigma \sqrt{t},
262267
\end{equation}
268+
while $N$ is the cumulative normal distribution function.
263269

264270

265271
\paragraph{5c: Compute values for first-order Greeks.}
@@ -292,7 +298,7 @@ \subsection*{Solving the Black-Scholes Equation Numerically}
292298
Telenor, Mowi, Orkla etc. See for instance
293299
\href{{https://www.oslobors.no/markedsaktivitet/#/list/derivatives/quotelist/false}}{\nolinkurl{https://www.oslobors.no/markedsaktivitet/\#/list/derivatives/quotelist/false}}
294300

295-
Since there the marked already prices the options, one can use these options
301+
Since there the market already prices the options, one can use these options
296302
prices to derive the value of implicit variables. A common practice
297303
is to use the market-given prices of options to derive the
298304
implied volatility of the option.

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