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```julia
using Gadfly
```
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# Athena/Phoenix Simulation
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## Underlying simulation
In order to price the autocallable bonds, we need to simulate the underlying assets. Let's go ahead and set up the simulation first, as this lays the foundation for what we're trying to do. We're going to use [JNJ ](http://finance.yahoo.com/q?s=jnj ) as the basis for our simulation. This implies the following parameters:
- $S_0$ = \$102.2 (as of time of writing)
- $q$ = 2.84%
- $r$ = [.49, .9, 1.21, 1.45, 1.69] (term structure as of time of writing, linear interpolation)
- $\mu$ = $r - q$ (note that this implies a negative drift because of current low rates)
- $\sigma$ = $\sigma_{imp}$ = 15.62% (from VIX implied volatility)
We additionally define some parameters for simulation:
- `T` : The number of years to simulate
- `m` : The number of paths to simulate
- `n` : The number of steps to simulate in a year
```julia
S0 = 102.2
nominal = 100
q = 2.84 / 100
σ
term = [0, .49, .9, 1.21, 1.45, 1.69] / 100 + 1
###
# Potential: Based on PEP
# S0 = 100.6
# σ
# q = 2.7
###
# Simulation parameters
T = 5 # Using years as the unit of time
n = 250 # simulations per year
m = 100000 # paths
num_simulations = 5; # simulation rounds per price
```
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5
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### Defining the simulation
To make things simpler, we simulate a single year at a time. This allows us to easily add in a dividend policy without too much difficulty, and update the simulation every year to match the term structure. The underlying uses GBM for simulation between years.
```julia
simulate_gbm = function(S0, μ, σ
# Set the initial state
m = length(S0)
t = T / n
motion = zeros(m, n)
motion[:,1] = S0
# Build out all states
for i=1:(n-1)
motion[:,i+1] = motion[:,i] .* exp((μ - σ * σ randn(m))
end
return motion
end
function display_motion(motion, T)
# Given a matrix of paths, display the motion
n = length(motion[1,:])
m = length(motion[:,1])
x = repmat(1:n, m)
# Calculate the ticks we're going to use. We'd like to
# have an xtick every month, so calculate where those
# ticks will actually be at.
if (T > 3)
num_ticks = T
xlabel = "Years"
else
num_ticks = T * 12
xlabel = "Months"
end
tick_width = n / num_ticks
x_ticks = []
for i=1:round(num_ticks)
x_ticks = vcat(x_ticks, i*tick_width)
end
# Use one color for each path. I'm not sure if there's
# a better way to do this without going through DataFrames
colors = []
for i = 1:m
colors = vcat(colors, ones(n)*i)
end
plot(x=x, y=motion', color=colors, Geom.line,
Guide.xticks(ticks=x_ticks, label=false),
Guide.xlabel(xlabel),
Guide.ylabel("Value"))
end;
```
### Example simulation
Let's go ahead and run a sample simulation to see what the functions got us!
```julia
initial = ones(5) * S0
# Using μ=0, T=.25 for now, we'll use the proper values later
motion = simulate_gbm(initial, 0, σ
display_motion(motion, .25)
```
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a.id,e);b?p[l]&&a.animate.apply(a,p[l]):a.animate(d,f,n,r)})}; v.remove=function(){for(;this.length;)this.pop().remove();return this};v.bind=function(a,f,k){var u={};if("function"==typeof f)this.bindings[a]=f;else{var p=k||a;this.bindings[a]=function(a){u[p]=a;f.attr(u)}}return this};v.attr=function(a){var f={},k;for(k in a)if(this.bindings[k])this.bindings[k](a[k]);else f[k]=a[k];a=0;for(k=this.items.length;a< k ; a + + ) this . items [ a ] . attr ( f ) ; return this } ; v . clear = function(){for(;this.length;)this.pop()};
v.splice=function(a,f,k){a=0>a?A(this.length+a,0):a;f=A(0,w(this.length-a,f));var u=[],p=[],b=[],q;for(q=2;q< arguments.length ; q + + ) b . push ( arguments [ q ] ) ; for ( q = 0;q<f;q++)p.push(this[a+q]);for(;q<this.length-a;q++)u.push(this[a+q]);var e = b.length;for(q=0;q<e+u.length;q++)this.items[a+q]=this[a+q]=q<e?b[q]:u[q-e];for(q=this.items.length=this.length-=f-e;this[q];)delete this [ q + + ] ; return new z ( p ) } ; v . exclude = function(a){for(var f = 0,k=this.length;f<k;f++)if(this[f]==a)return this . splice ( f , 1 ) , ! 0 ; return ! 1 } ;
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[];d=a.parseTransformString(d)||[];for(var k=Math.max(b.length,d.length),p=[],v=[],h=0,w,z,y,I;h< k ; h + + ) { y = b[h]||A(d[h]);I=d[h]||A(y);if(y[0]!=I[0]||"r"==y[0].toLowerCase()&&(y[2]!=I[2]||y[3]!=I[3])||"s"==y[0].toLowerCase()&&(y[3]!=I[3]||y[4]!=I[4])){b=a._.transform2matrix(b,f());d=a._.transform2matrix(d,f());p=[["m",b.a,b.b,b.c,b.d,b.e,b.f] ] ; v = [["m",d.a,d.b,d.c,d.d,d.e,d.f] ] ; break } p [ h ] = [ ] ; v [ h ] = [ ] ; w = 0;for(z=Math.max(y.length,I.length);w<z;w++)w in y & & ( p [ h ] [ w ] = y [ w ] ) , w in I & & ( v [ h ] [ w ] = I [ w ] ) } return { from:u ( p ) ,
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]]> < / script >
< script > < ! [ C D A T A [
(function (glob, factory) {
// AMD support
if (typeof define === "function" & & define.amd) {
// Define as an anonymous module
define("Gadfly", ["Snap.svg"], function (Snap) {
return factory(Snap);
});
} else {
// Browser globals (glob is window)
// Snap adds itself to window
glob.Gadfly = factory(glob.Snap);
}
}(this, function (Snap) {
var Gadfly = {};
// Get an x/y coordinate value in pixels
var xPX = function(fig, x) {
var client_box = fig.node.getBoundingClientRect();
return x * fig.node.viewBox.baseVal.width / client_box.width;
};
var yPX = function(fig, y) {
var client_box = fig.node.getBoundingClientRect();
return y * fig.node.viewBox.baseVal.height / client_box.height;
};
Snap.plugin(function (Snap, Element, Paper, global) {
// Traverse upwards from a snap element to find and return the first
// note with the "plotroot" class.
Element.prototype.plotroot = function () {
var element = this;
while (!element.hasClass("plotroot") & & element.parent() != null) {
element = element.parent();
}
return element;
};
Element.prototype.svgroot = function () {
var element = this;
while (element.node.nodeName != "svg" & & element.parent() != null) {
element = element.parent();
}
return element;
};
Element.prototype.plotbounds = function () {
var root = this.plotroot()
var bbox = root.select(".guide.background").node.getBBox();
return {
x0: bbox.x,
x1: bbox.x + bbox.width,
y0: bbox.y,
y1: bbox.y + bbox.height
};
};
Element.prototype.plotcenter = function () {
var root = this.plotroot()
var bbox = root.select(".guide.background").node.getBBox();
return {
x: bbox.x + bbox.width / 2,
y: bbox.y + bbox.height / 2
};
};
// Emulate IE style mouseenter/mouseleave events, since Microsoft always
// does everything right.
// See: http://www.dynamic-tools.net/toolbox/isMouseLeaveOrEnter/
var events = ["mouseenter", "mouseleave"];
for (i in events) {
(function (event_name) {
var event_name = events[i];
Element.prototype[event_name] = function (fn, scope) {
if (Snap.is(fn, "function")) {
var fn2 = function (event) {
if (event.type != "mouseover" & & event.type != "mouseout") {
return;
}
var reltg = event.relatedTarget ? event.relatedTarget :
event.type == "mouseout" ? event.toElement : event.fromElement;
while (reltg & & reltg != this.node) reltg = reltg.parentNode;
if (reltg != this.node) {
return fn.apply(this, event);
}
};
if (event_name == "mouseenter") {
this.mouseover(fn2, scope);
} else {
this.mouseout(fn2, scope);
}
}
return this;
};
})(events[i]);
}
Element.prototype.mousewheel = function (fn, scope) {
if (Snap.is(fn, "function")) {
var el = this;
var fn2 = function (event) {
fn.apply(el, [event]);
};
}
this.node.addEventListener(
/Firefox/i.test(navigator.userAgent) ? "DOMMouseScroll" : "mousewheel",
fn2);
return this;
};
// Snap's attr function can be too slow for things like panning/zooming.
// This is a function to directly update element attributes without going
// through eve.
Element.prototype.attribute = function(key, val) {
if (val === undefined) {
return this.node.getAttribute(key);
} else {
this.node.setAttribute(key, val);
return this;
}
};
Element.prototype.init_gadfly = function() {
this.mouseenter(Gadfly.plot_mouseover)
.mouseleave(Gadfly.plot_mouseout)
.dblclick(Gadfly.plot_dblclick)
.mousewheel(Gadfly.guide_background_scroll)
.drag(Gadfly.guide_background_drag_onmove,
Gadfly.guide_background_drag_onstart,
Gadfly.guide_background_drag_onend);
this.mouseenter(function (event)
init_pan_zoom(this.plotroot());
});
return this;
};
});
// When the plot is moused over, emphasize the grid lines.
Gadfly.plot_mouseover = function(event) {
var root = this.plotroot();
var keyboard_zoom = function(event) {
if (event.which == 187) { // plus
increase_zoom_by_position(root, 0.1, true);
} else if (event.which == 189) { // minus
increase_zoom_by_position(root, -0.1, true);
}
};
root.data("keyboard_zoom", keyboard_zoom);
window.addEventListener("keyup", keyboard_zoom);
var xgridlines = root.select(".xgridlines"),
ygridlines = root.select(".ygridlines");
xgridlines.data("unfocused_strokedash",
xgridlines.attribute("stroke-dasharray").replace(/(\d)(,|$)/g, "$1mm$2"));
ygridlines.data("unfocused_strokedash",
ygridlines.attribute("stroke-dasharray").replace(/(\d)(,|$)/g, "$1mm$2"));
// emphasize grid lines
var destcolor = root.data("focused_xgrid_color");
xgridlines.attribute("stroke-dasharray", "none")
.selectAll("path")
.animate({stroke: destcolor}, 250);
destcolor = root.data("focused_ygrid_color");
ygridlines.attribute("stroke-dasharray", "none")
.selectAll("path")
.animate({stroke: destcolor}, 250);
// reveal zoom slider
root.select(".zoomslider")
.animate({opacity: 1.0}, 250);
};
// Reset pan and zoom on double click
Gadfly.plot_dblclick = function(event) {
set_plot_pan_zoom(this.plotroot(), 0.0, 0.0, 1.0);
};
// Unemphasize grid lines on mouse out.
Gadfly.plot_mouseout = function(event) {
var root = this.plotroot();
window.removeEventListener("keyup", root.data("keyboard_zoom"));
root.data("keyboard_zoom", undefined);
var xgridlines = root.select(".xgridlines"),
ygridlines = root.select(".ygridlines");
var destcolor = root.data("unfocused_xgrid_color");
xgridlines.attribute("stroke-dasharray", xgridlines.data("unfocused_strokedash"))
.selectAll("path")
.animate({stroke: destcolor}, 250);
destcolor = root.data("unfocused_ygrid_color");
ygridlines.attribute("stroke-dasharray", ygridlines.data("unfocused_strokedash"))
.selectAll("path")
.animate({stroke: destcolor}, 250);
// hide zoom slider
root.select(".zoomslider")
.animate({opacity: 0.0}, 250);
};
var set_geometry_transform = function(root, tx, ty, scale) {
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");
var old_scale = root.data("scale");
var xscale = xscalable ? scale : 1.0,
yscale = yscalable ? scale : 1.0;
tx = xscalable ? tx : 0.0;
ty = yscalable ? ty : 0.0;
var t = new Snap.Matrix().translate(tx, ty).scale(xscale, yscale);
root.selectAll(".geometry, image")
.forEach(function (element, i) {
element.transform(t);
});
bounds = root.plotbounds();
if (yscalable) {
var xfixed_t = new Snap.Matrix().translate(0, ty).scale(1.0, yscale);
root.selectAll(".xfixed")
.forEach(function (element, i) {
element.transform(xfixed_t);
});
root.select(".ylabels")
.transform(xfixed_t)
.selectAll("text")
.forEach(function (element, i) {
if (element.attribute("gadfly:inscale") == "true") {
var cx = element.asPX("x"),
cy = element.asPX("y");
var st = element.data("static_transform");
unscale_t = new Snap.Matrix();
unscale_t.scale(1, 1/scale, cx, cy).add(st);
element.transform(unscale_t);
var y = cy * scale + ty;
element.attr("visibility",
bounds.y0 < = y & & y < = bounds.y1 ? "visible" : "hidden");
}
});
}
if (xscalable) {
var yfixed_t = new Snap.Matrix().translate(tx, 0).scale(xscale, 1.0);
var xtrans = new Snap.Matrix().translate(tx, 0);
root.selectAll(".yfixed")
.forEach(function (element, i) {
element.transform(yfixed_t);
});
root.select(".xlabels")
.transform(yfixed_t)
.selectAll("text")
.forEach(function (element, i) {
if (element.attribute("gadfly:inscale") == "true") {
var cx = element.asPX("x"),
cy = element.asPX("y");
var st = element.data("static_transform");
unscale_t = new Snap.Matrix();
unscale_t.scale(1/scale, 1, cx, cy).add(st);
element.transform(unscale_t);
var x = cx * scale + tx;
element.attr("visibility",
bounds.x0 < = x & & x < = bounds.x1 ? "visible" : "hidden");
}
});
}
// we must unscale anything that is scale invariance: widths, raiduses, etc.
var size_attribs = ["font-size"];
var unscaled_selection = ".geometry, .geometry *";
if (xscalable) {
size_attribs.push("rx");
unscaled_selection += ", .xgridlines";
}
if (yscalable) {
size_attribs.push("ry");
unscaled_selection += ", .ygridlines";
}
root.selectAll(unscaled_selection)
.forEach(function (element, i) {
// circle need special help
if (element.node.nodeName == "circle") {
var cx = element.attribute("cx"),
cy = element.attribute("cy");
unscale_t = new Snap.Matrix().scale(1/xscale, 1/yscale,
cx, cy);
element.transform(unscale_t);
return;
}
for (i in size_attribs) {
var key = size_attribs[i];
var val = parseFloat(element.attribute(key));
if (val !== undefined & & val != 0 & & !isNaN(val)) {
element.attribute(key, val * old_scale / scale);
}
}
});
};
// Find the most appropriate tick scale and update label visibility.
var update_tickscale = function(root, scale, axis) {
if (!root.hasClass(axis + "scalable")) return;
var tickscales = root.data(axis + "tickscales");
var best_tickscale = 1.0;
var best_tickscale_dist = Infinity;
for (tickscale in tickscales) {
var dist = Math.abs(Math.log(tickscale) - Math.log(scale));
if (dist < best_tickscale_dist ) {
best_tickscale_dist = dist;
best_tickscale = tickscale;
}
}
if (best_tickscale != root.data(axis + "tickscale")) {
root.data(axis + "tickscale", best_tickscale);
var mark_inscale_gridlines = function (element, i) {
var inscale = element.attr("gadfly:scale") == best_tickscale;
element.attribute("gadfly:inscale", inscale);
element.attr("visibility", inscale ? "visible" : "hidden");
};
var mark_inscale_labels = function (element, i) {
var inscale = element.attr("gadfly:scale") == best_tickscale;
element.attribute("gadfly:inscale", inscale);
element.attr("visibility", inscale ? "visible" : "hidden");
};
root.select("." + axis + "gridlines").selectAll("path").forEach(mark_inscale_gridlines);
root.select("." + axis + "labels").selectAll("text").forEach(mark_inscale_labels);
}
};
var set_plot_pan_zoom = function(root, tx, ty, scale) {
var old_scale = root.data("scale");
var bounds = root.plotbounds();
var width = bounds.x1 - bounds.x0,
height = bounds.y1 - bounds.y0;
// compute the viewport derived from tx, ty, and scale
var x_min = -width * scale - (scale * width - width),
x_max = width * scale,
y_min = -height * scale - (scale * height - height),
y_max = height * scale;
var x0 = bounds.x0 - scale * bounds.x0,
y0 = bounds.y0 - scale * bounds.y0;
var tx = Math.max(Math.min(tx - x0, x_max), x_min),
ty = Math.max(Math.min(ty - y0, y_max), y_min);
tx += x0;
ty += y0;
// when the scale change, we may need to alter which set of
// ticks is being displayed
if (scale != old_scale) {
update_tickscale(root, scale, "x");
update_tickscale(root, scale, "y");
}
set_geometry_transform(root, tx, ty, scale);
root.data("scale", scale);
root.data("tx", tx);
root.data("ty", ty);
};
var scale_centered_translation = function(root, scale) {
var bounds = root.plotbounds();
var width = bounds.x1 - bounds.x0,
height = bounds.y1 - bounds.y0;
var tx0 = root.data("tx"),
ty0 = root.data("ty");
var scale0 = root.data("scale");
// how off from center the current view is
var xoff = tx0 - (bounds.x0 * (1 - scale0) + (width * (1 - scale0)) / 2),
yoff = ty0 - (bounds.y0 * (1 - scale0) + (height * (1 - scale0)) / 2);
// rescale offsets
xoff = xoff * scale / scale0;
yoff = yoff * scale / scale0;
// adjust for the panel position being scaled
var x_edge_adjust = bounds.x0 * (1 - scale),
y_edge_adjust = bounds.y0 * (1 - scale);
return {
x: xoff + x_edge_adjust + (width - width * scale) / 2,
y: yoff + y_edge_adjust + (height - height * scale) / 2
};
};
// Initialize data for panning zooming if it isn't already.
var init_pan_zoom = function(root) {
if (root.data("zoompan-ready")) {
return;
}
// The non-scaling-stroke trick. Rather than try to correct for the
// stroke-width when zooming, we force it to a fixed value.
var px_per_mm = root.node.getCTM().a;
// Drag events report deltas in pixels, which we'd like to convert to
// millimeters.
root.data("px_per_mm", px_per_mm);
root.selectAll("path")
.forEach(function (element, i) {
sw = element.asPX("stroke-width") * px_per_mm;
if (sw > 0) {
element.attribute("stroke-width", sw);
element.attribute("vector-effect", "non-scaling-stroke");
}
});
// Store ticks labels original tranformation
root.selectAll(".xlabels > text, .ylabels > text")
.forEach(function (element, i) {
var lm = element.transform().localMatrix;
element.data("static_transform",
new Snap.Matrix(lm.a, lm.b, lm.c, lm.d, lm.e, lm.f));
});
var xgridlines = root.select(".xgridlines");
var ygridlines = root.select(".ygridlines");
var xlabels = root.select(".xlabels");
var ylabels = root.select(".ylabels");
if (root.data("tx") === undefined) root.data("tx", 0);
if (root.data("ty") === undefined) root.data("ty", 0);
if (root.data("scale") === undefined) root.data("scale", 1.0);
if (root.data("xtickscales") === undefined) {
// index all the tick scales that are listed
var xtickscales = {};
var ytickscales = {};
var add_x_tick_scales = function (element, i) {
xtickscales[element.attribute("gadfly:scale")] = true;
};
var add_y_tick_scales = function (element, i) {
ytickscales[element.attribute("gadfly:scale")] = true;
};
if (xgridlines) xgridlines.selectAll("path").forEach(add_x_tick_scales);
if (ygridlines) ygridlines.selectAll("path").forEach(add_y_tick_scales);
if (xlabels) xlabels.selectAll("text").forEach(add_x_tick_scales);
if (ylabels) ylabels.selectAll("text").forEach(add_y_tick_scales);
root.data("xtickscales", xtickscales);
root.data("ytickscales", ytickscales);
root.data("xtickscale", 1.0);
}
var min_scale = 1.0, max_scale = 1.0;
for (scale in xtickscales) {
min_scale = Math.min(min_scale, scale);
max_scale = Math.max(max_scale, scale);
}
for (scale in ytickscales) {
min_scale = Math.min(min_scale, scale);
max_scale = Math.max(max_scale, scale);
}
root.data("min_scale", min_scale);
root.data("max_scale", max_scale);
// store the original positions of labels
if (xlabels) {
xlabels.selectAll("text")
.forEach(function (element, i) {
element.data("x", element.asPX("x"));
});
}
if (ylabels) {
ylabels.selectAll("text")
.forEach(function (element, i) {
element.data("y", element.asPX("y"));
});
}
// mark grid lines and ticks as in or out of scale.
var mark_inscale = function (element, i) {
element.attribute("gadfly:inscale", element.attribute("gadfly:scale") == 1.0);
};
if (xgridlines) xgridlines.selectAll("path").forEach(mark_inscale);
if (ygridlines) ygridlines.selectAll("path").forEach(mark_inscale);
if (xlabels) xlabels.selectAll("text").forEach(mark_inscale);
if (ylabels) ylabels.selectAll("text").forEach(mark_inscale);
// figure out the upper ond lower bounds on panning using the maximum
// and minum grid lines
var bounds = root.plotbounds();
var pan_bounds = {
x0: 0.0,
y0: 0.0,
x1: 0.0,
y1: 0.0
};
if (xgridlines) {
xgridlines
.selectAll("path")
.forEach(function (element, i) {
if (element.attribute("gadfly:inscale") == "true") {
var bbox = element.node.getBBox();
if (bounds.x1 - bbox.x < pan_bounds.x0 ) {
pan_bounds.x0 = bounds.x1 - bbox.x;
}
if (bounds.x0 - bbox.x > pan_bounds.x1) {
pan_bounds.x1 = bounds.x0 - bbox.x;
}
element.attr("visibility", "visible");
}
});
}
if (ygridlines) {
ygridlines
.selectAll("path")
.forEach(function (element, i) {
if (element.attribute("gadfly:inscale") == "true") {
var bbox = element.node.getBBox();
if (bounds.y1 - bbox.y < pan_bounds.y0 ) {
pan_bounds.y0 = bounds.y1 - bbox.y;
}
if (bounds.y0 - bbox.y > pan_bounds.y1) {
pan_bounds.y1 = bounds.y0 - bbox.y;
}
element.attr("visibility", "visible");
}
});
}
// nudge these values a little
pan_bounds.x0 -= 5;
pan_bounds.x1 += 5;
pan_bounds.y0 -= 5;
pan_bounds.y1 += 5;
root.data("pan_bounds", pan_bounds);
root.data("zoompan-ready", true)
};
// drag actions, i.e. zooming and panning
var pan_action = {
start: function(root, x, y, event) {
root.data("dx", 0);
root.data("dy", 0);
root.data("tx0", root.data("tx"));
root.data("ty0", root.data("ty"));
},
update: function(root, dx, dy, x, y, event) {
var px_per_mm = root.data("px_per_mm");
dx /= px_per_mm;
dy /= px_per_mm;
var tx0 = root.data("tx"),
ty0 = root.data("ty");
var dx0 = root.data("dx"),
dy0 = root.data("dy");
root.data("dx", dx);
root.data("dy", dy);
dx = dx - dx0;
dy = dy - dy0;
var tx = tx0 + dx,
ty = ty0 + dy;
set_plot_pan_zoom(root, tx, ty, root.data("scale"));
},
end: function(root, event) {
},
cancel: function(root) {
set_plot_pan_zoom(root, root.data("tx0"), root.data("ty0"), root.data("scale"));
}
};
var zoom_box;
var zoom_action = {
start: function(root, x, y, event) {
var bounds = root.plotbounds();
var width = bounds.x1 - bounds.x0,
height = bounds.y1 - bounds.y0;
var ratio = width / height;
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");
var px_per_mm = root.data("px_per_mm");
x = xscalable ? x / px_per_mm : bounds.x0;
y = yscalable ? y / px_per_mm : bounds.y0;
var w = xscalable ? 0 : width;
var h = yscalable ? 0 : height;
zoom_box = root.rect(x, y, w, h).attr({
"fill": "#000",
"opacity": 0.25
});
zoom_box.data("ratio", ratio);
},
update: function(root, dx, dy, x, y, event) {
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");
var px_per_mm = root.data("px_per_mm");
var bounds = root.plotbounds();
if (yscalable) {
y /= px_per_mm;
y = Math.max(bounds.y0, y);
y = Math.min(bounds.y1, y);
} else {
y = bounds.y1;
}
if (xscalable) {
x /= px_per_mm;
x = Math.max(bounds.x0, x);
x = Math.min(bounds.x1, x);
} else {
x = bounds.x1;
}
dx = x - zoom_box.attr("x");
dy = y - zoom_box.attr("y");
if (xscalable & & yscalable) {
var ratio = zoom_box.data("ratio");
var width = Math.min(Math.abs(dx), ratio * Math.abs(dy));
var height = Math.min(Math.abs(dy), Math.abs(dx) / ratio);
dx = width * dx / Math.abs(dx);
dy = height * dy / Math.abs(dy);
}
var xoffset = 0,
yoffset = 0;
if (dx < 0 ) {
xoffset = dx;
dx = -1 * dx;
}
if (dy < 0 ) {
yoffset = dy;
dy = -1 * dy;
}
if (isNaN(dy)) {
dy = 0.0;
}
if (isNaN(dx)) {
dx = 0.0;
}
zoom_box.transform("T" + xoffset + "," + yoffset);
zoom_box.attr("width", dx);
zoom_box.attr("height", dy);
},
end: function(root, event) {
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");
var zoom_bounds = zoom_box.getBBox();
if (zoom_bounds.width * zoom_bounds.height < = 0) {
return;
}
var plot_bounds = root.plotbounds();
var zoom_factor = 1.0;
if (yscalable) {
zoom_factor = (plot_bounds.y1 - plot_bounds.y0) / zoom_bounds.height;
} else {
zoom_factor = (plot_bounds.x1 - plot_bounds.x0) / zoom_bounds.width;
}
var tx = (root.data("tx") - zoom_bounds.x) * zoom_factor + plot_bounds.x0,
ty = (root.data("ty") - zoom_bounds.y) * zoom_factor + plot_bounds.y0;
set_plot_pan_zoom(root, tx, ty, root.data("scale") * zoom_factor);
zoom_box.remove();
},
cancel: function(root) {
zoom_box.remove();
}
};
Gadfly.guide_background_drag_onstart = function(x, y, event) {
var root = this.plotroot();
var scalable = root.hasClass("xscalable") || root.hasClass("yscalable");
var zoomable = !event.altKey & & !event.ctrlKey & & event.shiftKey & & scalable;
var panable = !event.altKey & & !event.ctrlKey & & !event.shiftKey & & scalable;
var drag_action = zoomable ? zoom_action :
panable ? pan_action :
undefined;
root.data("drag_action", drag_action);
if (drag_action) {
var cancel_drag_action = function(event) {
if (event.which == 27) { // esc key
drag_action.cancel(root);
root.data("drag_action", undefined);
}
};
window.addEventListener("keyup", cancel_drag_action);
root.data("cancel_drag_action", cancel_drag_action);
drag_action.start(root, x, y, event);
}
};
Gadfly.guide_background_drag_onmove = function(dx, dy, x, y, event) {
var root = this.plotroot();
var drag_action = root.data("drag_action");
if (drag_action) {
drag_action.update(root, dx, dy, x, y, event);
}
};
Gadfly.guide_background_drag_onend = function(event) {
var root = this.plotroot();
window.removeEventListener("keyup", root.data("cancel_drag_action"));
root.data("cancel_drag_action", undefined);
var drag_action = root.data("drag_action");
if (drag_action) {
drag_action.end(root, event);
}
root.data("drag_action", undefined);
};
Gadfly.guide_background_scroll = function(event) {
if (event.shiftKey) {
increase_zoom_by_position(this.plotroot(), 0.001 * event.wheelDelta);
event.preventDefault();
}
};
Gadfly.zoomslider_button_mouseover = function(event) {
this.select(".button_logo")
.animate({fill: this.data("mouseover_color")}, 100);
};
Gadfly.zoomslider_button_mouseout = function(event) {
this.select(".button_logo")
.animate({fill: this.data("mouseout_color")}, 100);
};
Gadfly.zoomslider_zoomout_click = function(event) {
increase_zoom_by_position(this.plotroot(), -0.1, true);
};
Gadfly.zoomslider_zoomin_click = function(event) {
increase_zoom_by_position(this.plotroot(), 0.1, true);
};
Gadfly.zoomslider_track_click = function(event) {
// TODO
};
// Map slider position x to scale y using the function y = a*exp(b*x)+c.
// The constants a, b, and c are solved using the constraint that the function
// should go through the points (0; min_scale), (0.5; 1), and (1; max_scale).
var scale_from_slider_position = function(position, min_scale, max_scale) {
var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2),
b = 2 * Math.log((max_scale - 1) / (1 - min_scale)),
c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2);
return a * Math.exp(b * position) + c;
}
// inverse of scale_from_slider_position
var slider_position_from_scale = function(scale, min_scale, max_scale) {
var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2),
b = 2 * Math.log((max_scale - 1) / (1 - min_scale)),
c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2);
return 1 / b * Math.log((scale - c) / a);
}
var increase_zoom_by_position = function(root, delta_position, animate) {
var scale = root.data("scale"),
min_scale = root.data("min_scale"),
max_scale = root.data("max_scale");
var position = slider_position_from_scale(scale, min_scale, max_scale);
position += delta_position;
scale = scale_from_slider_position(position, min_scale, max_scale);
set_zoom(root, scale, animate);
}
var set_zoom = function(root, scale, animate) {
var min_scale = root.data("min_scale"),
max_scale = root.data("max_scale"),
old_scale = root.data("scale");
var new_scale = Math.max(min_scale, Math.min(scale, max_scale));
if (animate) {
Snap.animate(
old_scale,
new_scale,
function (new_scale) {
update_plot_scale(root, new_scale);
},
200);
} else {
update_plot_scale(root, new_scale);
}
}
var update_plot_scale = function(root, new_scale) {
var trans = scale_centered_translation(root, new_scale);
set_plot_pan_zoom(root, trans.x, trans.y, new_scale);
root.selectAll(".zoomslider_thumb")
.forEach(function (element, i) {
var min_pos = element.data("min_pos"),
max_pos = element.data("max_pos"),
min_scale = root.data("min_scale"),
max_scale = root.data("max_scale");
var xmid = (min_pos + max_pos) / 2;
var xpos = slider_position_from_scale(new_scale, min_scale, max_scale);
element.transform(new Snap.Matrix().translate(
Math.max(min_pos, Math.min(
max_pos, min_pos + (max_pos - min_pos) * xpos)) - xmid, 0));
});
};
Gadfly.zoomslider_thumb_dragmove = function(dx, dy, x, y, event) {
var root = this.plotroot();
var min_pos = this.data("min_pos"),
max_pos = this.data("max_pos"),
min_scale = root.data("min_scale"),
max_scale = root.data("max_scale"),
old_scale = root.data("old_scale");
var px_per_mm = root.data("px_per_mm");
dx /= px_per_mm;
dy /= px_per_mm;
var xmid = (min_pos + max_pos) / 2;
var xpos = slider_position_from_scale(old_scale, min_scale, max_scale) +
dx / (max_pos - min_pos);
// compute the new scale
var new_scale = scale_from_slider_position(xpos, min_scale, max_scale);
new_scale = Math.min(max_scale, Math.max(min_scale, new_scale));
update_plot_scale(root, new_scale);
event.stopPropagation();
};
Gadfly.zoomslider_thumb_dragstart = function(x, y, event) {
this.animate({fill: this.data("mouseover_color")}, 100);
var root = this.plotroot();
// keep track of what the scale was when we started dragging
root.data("old_scale", root.data("scale"));
event.stopPropagation();
};
Gadfly.zoomslider_thumb_dragend = function(event) {
this.animate({fill: this.data("mouseout_color")}, 100);
event.stopPropagation();
};
var toggle_color_class = function(root, color_class, ison) {
var guides = root.selectAll(".guide." + color_class + ",.guide ." + color_class);
var geoms = root.selectAll(".geometry." + color_class + ",.geometry ." + color_class);
if (ison) {
guides.animate({opacity: 0.5}, 250);
geoms.animate({opacity: 0.0}, 250);
} else {
guides.animate({opacity: 1.0}, 250);
geoms.animate({opacity: 1.0}, 250);
}
};
Gadfly.colorkey_swatch_click = function(event) {
var root = this.plotroot();
var color_class = this.data("color_class");
if (event.shiftKey) {
root.selectAll(".colorkey text")
.forEach(function (element) {
var other_color_class = element.data("color_class");
if (other_color_class != color_class) {
toggle_color_class(root, other_color_class,
element.attr("opacity") == 1.0);
}
});
} else {
toggle_color_class(root, color_class, this.attr("opacity") == 1.0);
}
};
return Gadfly;
}));
//@ sourceURL=gadfly.js
(function (glob, factory) {
// AMD support
if (typeof require === "function" & & typeof define === "function" & & define.amd) {
require(["Snap.svg", "Gadfly"], function (Snap, Gadfly) {
factory(Snap, Gadfly);
});
} else {
factory(glob.Snap, glob.Gadfly);
}
})(window, function (Snap, Gadfly) {
var fig = Snap("#fig-3a6dd25ad25c4037a166889ee51bb151");
fig.select("#fig-3a6dd25ad25c4037a166889ee51bb151-element-8")
.init_gadfly();
fig.select("#fig-3a6dd25ad25c4037a166889ee51bb151-element-11")
.plotroot().data("unfocused_ygrid_color", "#D0D0E0")
;
fig.select("#fig-3a6dd25ad25c4037a166889ee51bb151-element-11")
.plotroot().data("focused_ygrid_color", "#A0A0A0")
;
fig.select("#fig-3a6dd25ad25c4037a166889ee51bb151-element-12")
.plotroot().data("unfocused_xgrid_color", "#D0D0E0")
;
fig.select("#fig-3a6dd25ad25c4037a166889ee51bb151-element-12")
.plotroot().data("focused_xgrid_color", "#A0A0A0")
;
fig.select("#fig-3a6dd25ad25c4037a166889ee51bb151-element-16")
.data("mouseover_color", "#CD5C5C")
;
fig.select("#fig-3a6dd25ad25c4037a166889ee51bb151-element-16")
.data("mouseout_color", "#6A6A6A")
;
fig.select("#fig-3a6dd25ad25c4037a166889ee51bb151-element-16")
.click(Gadfly.zoomslider_zoomin_click)
.mouseenter(Gadfly.zoomslider_button_mouseover)
.mouseleave(Gadfly.zoomslider_button_mouseout)
;
fig.select("#fig-3a6dd25ad25c4037a166889ee51bb151-element-18")
.data("max_pos", 111.58)
;
fig.select("#fig-3a6dd25ad25c4037a166889ee51bb151-element-18")
.data("min_pos", 94.58)
;
fig.select("#fig-3a6dd25ad25c4037a166889ee51bb151-element-18")
.click(Gadfly.zoomslider_track_click);
fig.select("#fig-3a6dd25ad25c4037a166889ee51bb151-element-19")
.data("max_pos", 111.58)
;
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2024-10-13 21:17:09 -04:00
### Computing the term structure
Now that we've got the basic motion set up, let's start making things a bit more sophisticated for the model. We're going to assume that the drift of the stock is the difference between the implied forward rate and the quarterly dividend rate.
We're given the yearly term structure, and need to calculate the quarterly forward rate to match this structure. The term structure is assumed to follow:
$d(0, t) = d(0,t-1)\cdot f_{i-1, i}$
Where $f_{i-1, i}$ is the quarterly forward rate.
```julia
forward_term = function(yearly_term)
# It is assumed that we have a yearly term structure passed in, and starts at year 0
# This implies a nominal rate above 0 for the first year!
years = length(term)-1 # because we start at 0
structure = [(term[i+1] / term[i]) for i=1:years]
end;
```
### Illustrating the term structure
Now that we've got our term structure, let's validate that we're getting the correct results! If we've done this correctly, then:
```
term[2] == term[1] * structure[1]
```
```julia
# Example term structure taken from:
# http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield
# Linear interpolation used years in-between periods, assuming real-dollar
# interest rates
forward_yield = forward_term(term)
calculated_term2 = term[1] * forward_yield[1]
println("Actual term[2]: $(term[2]); Calculated term[2]: $(calculated_term2)")
```
2024-10-21 21:55:59 -04:00
Actual term[2]: 1.0049; Calculated term[2]: 1.0049
2024-10-13 21:17:09 -04:00
### The full underlying simulation
Now that we have the term structure set up, we can actually start doing some real simulation! Let's construct some paths through the full 5-year time frame. In order to do this, we will simulate 1 year at a time, and use the forward rates at those times to compute the drift. Thus, there will be 5 total simulations batched together.
```julia
full_motion = ones(5) * S0
full_term = vcat(term[1], forward_yield)
for i=1:T
μ = (full_term[i] - 1 - q)
year_motion = simulate_gbm(full_motion[:,end], μ, σ
full_motion = hcat(full_motion, year_motion)
end
display_motion(full_motion, T)
```
2024-10-21 21:55:59 -04:00
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return a};v.toString=function(){return"Snap\u2018s set"};v.type="set";a.set=function(){var a=new z;arguments.length&&a.push.apply(a,Array.prototype.slice.call(arguments,0)); return a}});C.plugin(function(a,v,y,C){function A(a){var b=a[0];switch(b.toLowerCase()){case "t":return[b,0,0];case "m":return[b,1,0,0,1,0,0];case "r":return 4==a.length?[b,0,a[2],a[3] ]:[b,0];case "s":return 5==a.length?[b,1,1,a[3],a[4] ]:3==a.length?[b,1,1]:[b,1]}}function w(b,d,f){d=q(d).replace(/\.{3}|\u2026/g,b);b=a.parseTransformString(b)||
[];d=a.parseTransformString(d)||[];for(var k=Math.max(b.length,d.length),p=[],v=[],h=0,w,z,y,I;h< k ; h + + ) { y = b[h]||A(d[h]);I=d[h]||A(y);if(y[0]!=I[0]||"r"==y[0].toLowerCase()&&(y[2]!=I[2]||y[3]!=I[3])||"s"==y[0].toLowerCase()&&(y[3]!=I[3]||y[4]!=I[4])){b=a._.transform2matrix(b,f());d=a._.transform2matrix(d,f());p=[["m",b.a,b.b,b.c,b.d,b.e,b.f] ] ; v = [["m",d.a,d.b,d.c,d.d,d.e,d.f] ] ; break } p [ h ] = [ ] ; v [ h ] = [ ] ; w = 0;for(z=Math.max(y.length,I.length);w<z;w++)w in y & & ( p [ h ] [ w ] = y [ w ] ) , w in I & & ( v [ h ] [ w ] = I [ w ] ) } return { from:u ( p ) ,
to:u(v),f:n(p)}}function z(a){return a}function d(a){return function(b){return+b.toFixed(3)+a}}function f(b){return a.rgb(b[0],b[1],b[2])}function n(a){var b=0,d,f,k,n,h,p,q=[];d=0;for(f=a.length;d< f ; d + + ) { h = "[" ; p = ['"'+a[d][0]+'"'];k=1;for(n=a[d].length;k<n;k++)p[k]="val["+b++ + " ] " ; h + = p + " ] " ; q [ d ] = h } return Function ( " val " , " return Snap . path . toString . call ( [ " + q + " ] ) " ) } function u ( a ) { for ( var b = [],d=0,f=a.length;d<f;d++)for(var k = 1,n=a[d].length;k<n;k++)b.push(a[d][k]);return b } var p = {},b=/[a-z]+$/i,q=String;
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]]> < / script >
< script > < ! [ C D A T A [
(function (glob, factory) {
// AMD support
if (typeof define === "function" & & define.amd) {
// Define as an anonymous module
define("Gadfly", ["Snap.svg"], function (Snap) {
return factory(Snap);
});
} else {
// Browser globals (glob is window)
// Snap adds itself to window
glob.Gadfly = factory(glob.Snap);
}
}(this, function (Snap) {
var Gadfly = {};
// Get an x/y coordinate value in pixels
var xPX = function(fig, x) {
var client_box = fig.node.getBoundingClientRect();
return x * fig.node.viewBox.baseVal.width / client_box.width;
};
var yPX = function(fig, y) {
var client_box = fig.node.getBoundingClientRect();
return y * fig.node.viewBox.baseVal.height / client_box.height;
};
Snap.plugin(function (Snap, Element, Paper, global) {
// Traverse upwards from a snap element to find and return the first
// note with the "plotroot" class.
Element.prototype.plotroot = function () {
var element = this;
while (!element.hasClass("plotroot") & & element.parent() != null) {
element = element.parent();
}
return element;
};
Element.prototype.svgroot = function () {
var element = this;
while (element.node.nodeName != "svg" & & element.parent() != null) {
element = element.parent();
}
return element;
};
Element.prototype.plotbounds = function () {
var root = this.plotroot()
var bbox = root.select(".guide.background").node.getBBox();
return {
x0: bbox.x,
x1: bbox.x + bbox.width,
y0: bbox.y,
y1: bbox.y + bbox.height
};
};
Element.prototype.plotcenter = function () {
var root = this.plotroot()
var bbox = root.select(".guide.background").node.getBBox();
return {
x: bbox.x + bbox.width / 2,
y: bbox.y + bbox.height / 2
};
};
// Emulate IE style mouseenter/mouseleave events, since Microsoft always
// does everything right.
// See: http://www.dynamic-tools.net/toolbox/isMouseLeaveOrEnter/
var events = ["mouseenter", "mouseleave"];
for (i in events) {
(function (event_name) {
var event_name = events[i];
Element.prototype[event_name] = function (fn, scope) {
if (Snap.is(fn, "function")) {
var fn2 = function (event) {
if (event.type != "mouseover" & & event.type != "mouseout") {
return;
}
var reltg = event.relatedTarget ? event.relatedTarget :
event.type == "mouseout" ? event.toElement : event.fromElement;
while (reltg & & reltg != this.node) reltg = reltg.parentNode;
if (reltg != this.node) {
return fn.apply(this, event);
}
};
if (event_name == "mouseenter") {
this.mouseover(fn2, scope);
} else {
this.mouseout(fn2, scope);
}
}
return this;
};
})(events[i]);
}
Element.prototype.mousewheel = function (fn, scope) {
if (Snap.is(fn, "function")) {
var el = this;
var fn2 = function (event) {
fn.apply(el, [event]);
};
}
this.node.addEventListener(
/Firefox/i.test(navigator.userAgent) ? "DOMMouseScroll" : "mousewheel",
fn2);
return this;
};
// Snap's attr function can be too slow for things like panning/zooming.
// This is a function to directly update element attributes without going
// through eve.
Element.prototype.attribute = function(key, val) {
if (val === undefined) {
return this.node.getAttribute(key);
} else {
this.node.setAttribute(key, val);
return this;
}
};
Element.prototype.init_gadfly = function() {
this.mouseenter(Gadfly.plot_mouseover)
.mouseleave(Gadfly.plot_mouseout)
.dblclick(Gadfly.plot_dblclick)
.mousewheel(Gadfly.guide_background_scroll)
.drag(Gadfly.guide_background_drag_onmove,
Gadfly.guide_background_drag_onstart,
Gadfly.guide_background_drag_onend);
this.mouseenter(function (event)
init_pan_zoom(this.plotroot());
});
return this;
};
});
// When the plot is moused over, emphasize the grid lines.
Gadfly.plot_mouseover = function(event) {
var root = this.plotroot();
var keyboard_zoom = function(event) {
if (event.which == 187) { // plus
increase_zoom_by_position(root, 0.1, true);
} else if (event.which == 189) { // minus
increase_zoom_by_position(root, -0.1, true);
}
};
root.data("keyboard_zoom", keyboard_zoom);
window.addEventListener("keyup", keyboard_zoom);
var xgridlines = root.select(".xgridlines"),
ygridlines = root.select(".ygridlines");
xgridlines.data("unfocused_strokedash",
xgridlines.attribute("stroke-dasharray").replace(/(\d)(,|$)/g, "$1mm$2"));
ygridlines.data("unfocused_strokedash",
ygridlines.attribute("stroke-dasharray").replace(/(\d)(,|$)/g, "$1mm$2"));
// emphasize grid lines
var destcolor = root.data("focused_xgrid_color");
xgridlines.attribute("stroke-dasharray", "none")
.selectAll("path")
.animate({stroke: destcolor}, 250);
destcolor = root.data("focused_ygrid_color");
ygridlines.attribute("stroke-dasharray", "none")
.selectAll("path")
.animate({stroke: destcolor}, 250);
// reveal zoom slider
root.select(".zoomslider")
.animate({opacity: 1.0}, 250);
};
// Reset pan and zoom on double click
Gadfly.plot_dblclick = function(event) {
set_plot_pan_zoom(this.plotroot(), 0.0, 0.0, 1.0);
};
// Unemphasize grid lines on mouse out.
Gadfly.plot_mouseout = function(event) {
var root = this.plotroot();
window.removeEventListener("keyup", root.data("keyboard_zoom"));
root.data("keyboard_zoom", undefined);
var xgridlines = root.select(".xgridlines"),
ygridlines = root.select(".ygridlines");
var destcolor = root.data("unfocused_xgrid_color");
xgridlines.attribute("stroke-dasharray", xgridlines.data("unfocused_strokedash"))
.selectAll("path")
.animate({stroke: destcolor}, 250);
destcolor = root.data("unfocused_ygrid_color");
ygridlines.attribute("stroke-dasharray", ygridlines.data("unfocused_strokedash"))
.selectAll("path")
.animate({stroke: destcolor}, 250);
// hide zoom slider
root.select(".zoomslider")
.animate({opacity: 0.0}, 250);
};
var set_geometry_transform = function(root, tx, ty, scale) {
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");
var old_scale = root.data("scale");
var xscale = xscalable ? scale : 1.0,
yscale = yscalable ? scale : 1.0;
tx = xscalable ? tx : 0.0;
ty = yscalable ? ty : 0.0;
var t = new Snap.Matrix().translate(tx, ty).scale(xscale, yscale);
root.selectAll(".geometry, image")
.forEach(function (element, i) {
element.transform(t);
});
bounds = root.plotbounds();
if (yscalable) {
var xfixed_t = new Snap.Matrix().translate(0, ty).scale(1.0, yscale);
root.selectAll(".xfixed")
.forEach(function (element, i) {
element.transform(xfixed_t);
});
root.select(".ylabels")
.transform(xfixed_t)
.selectAll("text")
.forEach(function (element, i) {
if (element.attribute("gadfly:inscale") == "true") {
var cx = element.asPX("x"),
cy = element.asPX("y");
var st = element.data("static_transform");
unscale_t = new Snap.Matrix();
unscale_t.scale(1, 1/scale, cx, cy).add(st);
element.transform(unscale_t);
var y = cy * scale + ty;
element.attr("visibility",
bounds.y0 < = y & & y < = bounds.y1 ? "visible" : "hidden");
}
});
}
if (xscalable) {
var yfixed_t = new Snap.Matrix().translate(tx, 0).scale(xscale, 1.0);
var xtrans = new Snap.Matrix().translate(tx, 0);
root.selectAll(".yfixed")
.forEach(function (element, i) {
element.transform(yfixed_t);
});
root.select(".xlabels")
.transform(yfixed_t)
.selectAll("text")
.forEach(function (element, i) {
if (element.attribute("gadfly:inscale") == "true") {
var cx = element.asPX("x"),
cy = element.asPX("y");
var st = element.data("static_transform");
unscale_t = new Snap.Matrix();
unscale_t.scale(1/scale, 1, cx, cy).add(st);
element.transform(unscale_t);
var x = cx * scale + tx;
element.attr("visibility",
bounds.x0 < = x & & x < = bounds.x1 ? "visible" : "hidden");
}
});
}
// we must unscale anything that is scale invariance: widths, raiduses, etc.
var size_attribs = ["font-size"];
var unscaled_selection = ".geometry, .geometry *";
if (xscalable) {
size_attribs.push("rx");
unscaled_selection += ", .xgridlines";
}
if (yscalable) {
size_attribs.push("ry");
unscaled_selection += ", .ygridlines";
}
root.selectAll(unscaled_selection)
.forEach(function (element, i) {
// circle need special help
if (element.node.nodeName == "circle") {
var cx = element.attribute("cx"),
cy = element.attribute("cy");
unscale_t = new Snap.Matrix().scale(1/xscale, 1/yscale,
cx, cy);
element.transform(unscale_t);
return;
}
for (i in size_attribs) {
var key = size_attribs[i];
var val = parseFloat(element.attribute(key));
if (val !== undefined & & val != 0 & & !isNaN(val)) {
element.attribute(key, val * old_scale / scale);
}
}
});
};
// Find the most appropriate tick scale and update label visibility.
var update_tickscale = function(root, scale, axis) {
if (!root.hasClass(axis + "scalable")) return;
var tickscales = root.data(axis + "tickscales");
var best_tickscale = 1.0;
var best_tickscale_dist = Infinity;
for (tickscale in tickscales) {
var dist = Math.abs(Math.log(tickscale) - Math.log(scale));
if (dist < best_tickscale_dist ) {
best_tickscale_dist = dist;
best_tickscale = tickscale;
}
}
if (best_tickscale != root.data(axis + "tickscale")) {
root.data(axis + "tickscale", best_tickscale);
var mark_inscale_gridlines = function (element, i) {
var inscale = element.attr("gadfly:scale") == best_tickscale;
element.attribute("gadfly:inscale", inscale);
element.attr("visibility", inscale ? "visible" : "hidden");
};
var mark_inscale_labels = function (element, i) {
var inscale = element.attr("gadfly:scale") == best_tickscale;
element.attribute("gadfly:inscale", inscale);
element.attr("visibility", inscale ? "visible" : "hidden");
};
root.select("." + axis + "gridlines").selectAll("path").forEach(mark_inscale_gridlines);
root.select("." + axis + "labels").selectAll("text").forEach(mark_inscale_labels);
}
};
var set_plot_pan_zoom = function(root, tx, ty, scale) {
var old_scale = root.data("scale");
var bounds = root.plotbounds();
var width = bounds.x1 - bounds.x0,
height = bounds.y1 - bounds.y0;
// compute the viewport derived from tx, ty, and scale
var x_min = -width * scale - (scale * width - width),
x_max = width * scale,
y_min = -height * scale - (scale * height - height),
y_max = height * scale;
var x0 = bounds.x0 - scale * bounds.x0,
y0 = bounds.y0 - scale * bounds.y0;
var tx = Math.max(Math.min(tx - x0, x_max), x_min),
ty = Math.max(Math.min(ty - y0, y_max), y_min);
tx += x0;
ty += y0;
// when the scale change, we may need to alter which set of
// ticks is being displayed
if (scale != old_scale) {
update_tickscale(root, scale, "x");
update_tickscale(root, scale, "y");
}
set_geometry_transform(root, tx, ty, scale);
root.data("scale", scale);
root.data("tx", tx);
root.data("ty", ty);
};
var scale_centered_translation = function(root, scale) {
var bounds = root.plotbounds();
var width = bounds.x1 - bounds.x0,
height = bounds.y1 - bounds.y0;
var tx0 = root.data("tx"),
ty0 = root.data("ty");
var scale0 = root.data("scale");
// how off from center the current view is
var xoff = tx0 - (bounds.x0 * (1 - scale0) + (width * (1 - scale0)) / 2),
yoff = ty0 - (bounds.y0 * (1 - scale0) + (height * (1 - scale0)) / 2);
// rescale offsets
xoff = xoff * scale / scale0;
yoff = yoff * scale / scale0;
// adjust for the panel position being scaled
var x_edge_adjust = bounds.x0 * (1 - scale),
y_edge_adjust = bounds.y0 * (1 - scale);
return {
x: xoff + x_edge_adjust + (width - width * scale) / 2,
y: yoff + y_edge_adjust + (height - height * scale) / 2
};
};
// Initialize data for panning zooming if it isn't already.
var init_pan_zoom = function(root) {
if (root.data("zoompan-ready")) {
return;
}
// The non-scaling-stroke trick. Rather than try to correct for the
// stroke-width when zooming, we force it to a fixed value.
var px_per_mm = root.node.getCTM().a;
// Drag events report deltas in pixels, which we'd like to convert to
// millimeters.
root.data("px_per_mm", px_per_mm);
root.selectAll("path")
.forEach(function (element, i) {
sw = element.asPX("stroke-width") * px_per_mm;
if (sw > 0) {
element.attribute("stroke-width", sw);
element.attribute("vector-effect", "non-scaling-stroke");
}
});
// Store ticks labels original tranformation
root.selectAll(".xlabels > text, .ylabels > text")
.forEach(function (element, i) {
var lm = element.transform().localMatrix;
element.data("static_transform",
new Snap.Matrix(lm.a, lm.b, lm.c, lm.d, lm.e, lm.f));
});
var xgridlines = root.select(".xgridlines");
var ygridlines = root.select(".ygridlines");
var xlabels = root.select(".xlabels");
var ylabels = root.select(".ylabels");
if (root.data("tx") === undefined) root.data("tx", 0);
if (root.data("ty") === undefined) root.data("ty", 0);
if (root.data("scale") === undefined) root.data("scale", 1.0);
if (root.data("xtickscales") === undefined) {
// index all the tick scales that are listed
var xtickscales = {};
var ytickscales = {};
var add_x_tick_scales = function (element, i) {
xtickscales[element.attribute("gadfly:scale")] = true;
};
var add_y_tick_scales = function (element, i) {
ytickscales[element.attribute("gadfly:scale")] = true;
};
if (xgridlines) xgridlines.selectAll("path").forEach(add_x_tick_scales);
if (ygridlines) ygridlines.selectAll("path").forEach(add_y_tick_scales);
if (xlabels) xlabels.selectAll("text").forEach(add_x_tick_scales);
if (ylabels) ylabels.selectAll("text").forEach(add_y_tick_scales);
root.data("xtickscales", xtickscales);
root.data("ytickscales", ytickscales);
root.data("xtickscale", 1.0);
}
var min_scale = 1.0, max_scale = 1.0;
for (scale in xtickscales) {
min_scale = Math.min(min_scale, scale);
max_scale = Math.max(max_scale, scale);
}
for (scale in ytickscales) {
min_scale = Math.min(min_scale, scale);
max_scale = Math.max(max_scale, scale);
}
root.data("min_scale", min_scale);
root.data("max_scale", max_scale);
// store the original positions of labels
if (xlabels) {
xlabels.selectAll("text")
.forEach(function (element, i) {
element.data("x", element.asPX("x"));
});
}
if (ylabels) {
ylabels.selectAll("text")
.forEach(function (element, i) {
element.data("y", element.asPX("y"));
});
}
// mark grid lines and ticks as in or out of scale.
var mark_inscale = function (element, i) {
element.attribute("gadfly:inscale", element.attribute("gadfly:scale") == 1.0);
};
if (xgridlines) xgridlines.selectAll("path").forEach(mark_inscale);
if (ygridlines) ygridlines.selectAll("path").forEach(mark_inscale);
if (xlabels) xlabels.selectAll("text").forEach(mark_inscale);
if (ylabels) ylabels.selectAll("text").forEach(mark_inscale);
// figure out the upper ond lower bounds on panning using the maximum
// and minum grid lines
var bounds = root.plotbounds();
var pan_bounds = {
x0: 0.0,
y0: 0.0,
x1: 0.0,
y1: 0.0
};
if (xgridlines) {
xgridlines
.selectAll("path")
.forEach(function (element, i) {
if (element.attribute("gadfly:inscale") == "true") {
var bbox = element.node.getBBox();
if (bounds.x1 - bbox.x < pan_bounds.x0 ) {
pan_bounds.x0 = bounds.x1 - bbox.x;
}
if (bounds.x0 - bbox.x > pan_bounds.x1) {
pan_bounds.x1 = bounds.x0 - bbox.x;
}
element.attr("visibility", "visible");
}
});
}
if (ygridlines) {
ygridlines
.selectAll("path")
.forEach(function (element, i) {
if (element.attribute("gadfly:inscale") == "true") {
var bbox = element.node.getBBox();
if (bounds.y1 - bbox.y < pan_bounds.y0 ) {
pan_bounds.y0 = bounds.y1 - bbox.y;
}
if (bounds.y0 - bbox.y > pan_bounds.y1) {
pan_bounds.y1 = bounds.y0 - bbox.y;
}
element.attr("visibility", "visible");
}
});
}
// nudge these values a little
pan_bounds.x0 -= 5;
pan_bounds.x1 += 5;
pan_bounds.y0 -= 5;
pan_bounds.y1 += 5;
root.data("pan_bounds", pan_bounds);
root.data("zoompan-ready", true)
};
// drag actions, i.e. zooming and panning
var pan_action = {
start: function(root, x, y, event) {
root.data("dx", 0);
root.data("dy", 0);
root.data("tx0", root.data("tx"));
root.data("ty0", root.data("ty"));
},
update: function(root, dx, dy, x, y, event) {
var px_per_mm = root.data("px_per_mm");
dx /= px_per_mm;
dy /= px_per_mm;
var tx0 = root.data("tx"),
ty0 = root.data("ty");
var dx0 = root.data("dx"),
dy0 = root.data("dy");
root.data("dx", dx);
root.data("dy", dy);
dx = dx - dx0;
dy = dy - dy0;
var tx = tx0 + dx,
ty = ty0 + dy;
set_plot_pan_zoom(root, tx, ty, root.data("scale"));
},
end: function(root, event) {
},
cancel: function(root) {
set_plot_pan_zoom(root, root.data("tx0"), root.data("ty0"), root.data("scale"));
}
};
var zoom_box;
var zoom_action = {
start: function(root, x, y, event) {
var bounds = root.plotbounds();
var width = bounds.x1 - bounds.x0,
height = bounds.y1 - bounds.y0;
var ratio = width / height;
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");
var px_per_mm = root.data("px_per_mm");
x = xscalable ? x / px_per_mm : bounds.x0;
y = yscalable ? y / px_per_mm : bounds.y0;
var w = xscalable ? 0 : width;
var h = yscalable ? 0 : height;
zoom_box = root.rect(x, y, w, h).attr({
"fill": "#000",
"opacity": 0.25
});
zoom_box.data("ratio", ratio);
},
update: function(root, dx, dy, x, y, event) {
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");
var px_per_mm = root.data("px_per_mm");
var bounds = root.plotbounds();
if (yscalable) {
y /= px_per_mm;
y = Math.max(bounds.y0, y);
y = Math.min(bounds.y1, y);
} else {
y = bounds.y1;
}
if (xscalable) {
x /= px_per_mm;
x = Math.max(bounds.x0, x);
x = Math.min(bounds.x1, x);
} else {
x = bounds.x1;
}
dx = x - zoom_box.attr("x");
dy = y - zoom_box.attr("y");
if (xscalable & & yscalable) {
var ratio = zoom_box.data("ratio");
var width = Math.min(Math.abs(dx), ratio * Math.abs(dy));
var height = Math.min(Math.abs(dy), Math.abs(dx) / ratio);
dx = width * dx / Math.abs(dx);
dy = height * dy / Math.abs(dy);
}
var xoffset = 0,
yoffset = 0;
if (dx < 0 ) {
xoffset = dx;
dx = -1 * dx;
}
if (dy < 0 ) {
yoffset = dy;
dy = -1 * dy;
}
if (isNaN(dy)) {
dy = 0.0;
}
if (isNaN(dx)) {
dx = 0.0;
}
zoom_box.transform("T" + xoffset + "," + yoffset);
zoom_box.attr("width", dx);
zoom_box.attr("height", dy);
},
end: function(root, event) {
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");
var zoom_bounds = zoom_box.getBBox();
if (zoom_bounds.width * zoom_bounds.height < = 0) {
return;
}
var plot_bounds = root.plotbounds();
var zoom_factor = 1.0;
if (yscalable) {
zoom_factor = (plot_bounds.y1 - plot_bounds.y0) / zoom_bounds.height;
} else {
zoom_factor = (plot_bounds.x1 - plot_bounds.x0) / zoom_bounds.width;
}
var tx = (root.data("tx") - zoom_bounds.x) * zoom_factor + plot_bounds.x0,
ty = (root.data("ty") - zoom_bounds.y) * zoom_factor + plot_bounds.y0;
set_plot_pan_zoom(root, tx, ty, root.data("scale") * zoom_factor);
zoom_box.remove();
},
cancel: function(root) {
zoom_box.remove();
}
};
Gadfly.guide_background_drag_onstart = function(x, y, event) {
var root = this.plotroot();
var scalable = root.hasClass("xscalable") || root.hasClass("yscalable");
var zoomable = !event.altKey & & !event.ctrlKey & & event.shiftKey & & scalable;
var panable = !event.altKey & & !event.ctrlKey & & !event.shiftKey & & scalable;
var drag_action = zoomable ? zoom_action :
panable ? pan_action :
undefined;
root.data("drag_action", drag_action);
if (drag_action) {
var cancel_drag_action = function(event) {
if (event.which == 27) { // esc key
drag_action.cancel(root);
root.data("drag_action", undefined);
}
};
window.addEventListener("keyup", cancel_drag_action);
root.data("cancel_drag_action", cancel_drag_action);
drag_action.start(root, x, y, event);
}
};
Gadfly.guide_background_drag_onmove = function(dx, dy, x, y, event) {
var root = this.plotroot();
var drag_action = root.data("drag_action");
if (drag_action) {
drag_action.update(root, dx, dy, x, y, event);
}
};
Gadfly.guide_background_drag_onend = function(event) {
var root = this.plotroot();
window.removeEventListener("keyup", root.data("cancel_drag_action"));
root.data("cancel_drag_action", undefined);
var drag_action = root.data("drag_action");
if (drag_action) {
drag_action.end(root, event);
}
root.data("drag_action", undefined);
};
Gadfly.guide_background_scroll = function(event) {
if (event.shiftKey) {
increase_zoom_by_position(this.plotroot(), 0.001 * event.wheelDelta);
event.preventDefault();
}
};
Gadfly.zoomslider_button_mouseover = function(event) {
this.select(".button_logo")
.animate({fill: this.data("mouseover_color")}, 100);
};
Gadfly.zoomslider_button_mouseout = function(event) {
this.select(".button_logo")
.animate({fill: this.data("mouseout_color")}, 100);
};
Gadfly.zoomslider_zoomout_click = function(event) {
increase_zoom_by_position(this.plotroot(), -0.1, true);
};
Gadfly.zoomslider_zoomin_click = function(event) {
increase_zoom_by_position(this.plotroot(), 0.1, true);
};
Gadfly.zoomslider_track_click = function(event) {
// TODO
};
// Map slider position x to scale y using the function y = a*exp(b*x)+c.
// The constants a, b, and c are solved using the constraint that the function
// should go through the points (0; min_scale), (0.5; 1), and (1; max_scale).
var scale_from_slider_position = function(position, min_scale, max_scale) {
var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2),
b = 2 * Math.log((max_scale - 1) / (1 - min_scale)),
c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2);
return a * Math.exp(b * position) + c;
}
// inverse of scale_from_slider_position
var slider_position_from_scale = function(scale, min_scale, max_scale) {
var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2),
b = 2 * Math.log((max_scale - 1) / (1 - min_scale)),
c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2);
return 1 / b * Math.log((scale - c) / a);
}
var increase_zoom_by_position = function(root, delta_position, animate) {
var scale = root.data("scale"),
min_scale = root.data("min_scale"),
max_scale = root.data("max_scale");
var position = slider_position_from_scale(scale, min_scale, max_scale);
position += delta_position;
scale = scale_from_slider_position(position, min_scale, max_scale);
set_zoom(root, scale, animate);
}
var set_zoom = function(root, scale, animate) {
var min_scale = root.data("min_scale"),
max_scale = root.data("max_scale"),
old_scale = root.data("scale");
var new_scale = Math.max(min_scale, Math.min(scale, max_scale));
if (animate) {
Snap.animate(
old_scale,
new_scale,
function (new_scale) {
update_plot_scale(root, new_scale);
},
200);
} else {
update_plot_scale(root, new_scale);
}
}
var update_plot_scale = function(root, new_scale) {
var trans = scale_centered_translation(root, new_scale);
set_plot_pan_zoom(root, trans.x, trans.y, new_scale);
root.selectAll(".zoomslider_thumb")
.forEach(function (element, i) {
var min_pos = element.data("min_pos"),
max_pos = element.data("max_pos"),
min_scale = root.data("min_scale"),
max_scale = root.data("max_scale");
var xmid = (min_pos + max_pos) / 2;
var xpos = slider_position_from_scale(new_scale, min_scale, max_scale);
element.transform(new Snap.Matrix().translate(
Math.max(min_pos, Math.min(
max_pos, min_pos + (max_pos - min_pos) * xpos)) - xmid, 0));
});
};
Gadfly.zoomslider_thumb_dragmove = function(dx, dy, x, y, event) {
var root = this.plotroot();
var min_pos = this.data("min_pos"),
max_pos = this.data("max_pos"),
min_scale = root.data("min_scale"),
max_scale = root.data("max_scale"),
old_scale = root.data("old_scale");
var px_per_mm = root.data("px_per_mm");
dx /= px_per_mm;
dy /= px_per_mm;
var xmid = (min_pos + max_pos) / 2;
var xpos = slider_position_from_scale(old_scale, min_scale, max_scale) +
dx / (max_pos - min_pos);
// compute the new scale
var new_scale = scale_from_slider_position(xpos, min_scale, max_scale);
new_scale = Math.min(max_scale, Math.max(min_scale, new_scale));
update_plot_scale(root, new_scale);
event.stopPropagation();
};
Gadfly.zoomslider_thumb_dragstart = function(x, y, event) {
this.animate({fill: this.data("mouseover_color")}, 100);
var root = this.plotroot();
// keep track of what the scale was when we started dragging
root.data("old_scale", root.data("scale"));
event.stopPropagation();
};
Gadfly.zoomslider_thumb_dragend = function(event) {
this.animate({fill: this.data("mouseout_color")}, 100);
event.stopPropagation();
};
var toggle_color_class = function(root, color_class, ison) {
var guides = root.selectAll(".guide." + color_class + ",.guide ." + color_class);
var geoms = root.selectAll(".geometry." + color_class + ",.geometry ." + color_class);
if (ison) {
guides.animate({opacity: 0.5}, 250);
geoms.animate({opacity: 0.0}, 250);
} else {
guides.animate({opacity: 1.0}, 250);
geoms.animate({opacity: 1.0}, 250);
}
};
Gadfly.colorkey_swatch_click = function(event) {
var root = this.plotroot();
var color_class = this.data("color_class");
if (event.shiftKey) {
root.selectAll(".colorkey text")
.forEach(function (element) {
var other_color_class = element.data("color_class");
if (other_color_class != color_class) {
toggle_color_class(root, other_color_class,
element.attr("opacity") == 1.0);
}
});
} else {
toggle_color_class(root, color_class, this.attr("opacity") == 1.0);
}
};
return Gadfly;
}));
//@ sourceURL=gadfly.js
(function (glob, factory) {
// AMD support
if (typeof require === "function" & & typeof define === "function" & & define.amd) {
require(["Snap.svg", "Gadfly"], function (Snap, Gadfly) {
factory(Snap, Gadfly);
});
} else {
factory(glob.Snap, glob.Gadfly);
}
})(window, function (Snap, Gadfly) {
var fig = Snap("#fig-0378e04b897742b597befd2e8e1c169e");
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-8")
.init_gadfly();
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-11")
.plotroot().data("unfocused_ygrid_color", "#D0D0E0")
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-11")
.plotroot().data("focused_ygrid_color", "#A0A0A0")
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-12")
.plotroot().data("unfocused_xgrid_color", "#D0D0E0")
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-12")
.plotroot().data("focused_xgrid_color", "#A0A0A0")
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-16")
.data("mouseover_color", "#CD5C5C")
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-16")
.data("mouseout_color", "#6A6A6A")
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-16")
.click(Gadfly.zoomslider_zoomin_click)
.mouseenter(Gadfly.zoomslider_button_mouseover)
.mouseleave(Gadfly.zoomslider_button_mouseout)
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-18")
.data("max_pos", 111.58)
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-18")
.data("min_pos", 94.58)
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-18")
.click(Gadfly.zoomslider_track_click);
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-19")
.data("max_pos", 111.58)
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-19")
.data("min_pos", 94.58)
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-19")
.data("mouseover_color", "#CD5C5C")
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-19")
.data("mouseout_color", "#6A6A6A")
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-19")
.drag(Gadfly.zoomslider_thumb_dragmove,
Gadfly.zoomslider_thumb_dragstart,
Gadfly.zoomslider_thumb_dragend)
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-20")
.data("mouseover_color", "#CD5C5C")
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-20")
.data("mouseout_color", "#6A6A6A")
;
fig.select("#fig-0378e04b897742b597befd2e8e1c169e-element-20")
.click(Gadfly.zoomslider_zoomout_click)
.mouseenter(Gadfly.zoomslider_button_mouseover)
.mouseleave(Gadfly.zoomslider_button_mouseout)
;
});
]]> < / script >
< / svg >
2024-10-13 21:17:09 -04:00
### Final simulation
We're now going to actually build out the full motion that we'll use for computing the pricing of our autocallable products. It will be largely the same, but we will use far more sample paths for the simulation.
```julia
full_simulation = function(S0, T, n, m, term)
forward = vcat(term[1], forward_term(term))
# And an S0 to kick things off.
final_motion = ones(m) * S0
for i=1:T
μ = (forward[i] - 1 - q)
year_motion = simulate_gbm(final_motion[:,end], μ, σ
final_motion = hcat(final_motion, year_motion)
end
return final_motion
end
tic()
full_simulation(S0, T, n, m, term)
time = toq()
@printf ("Time to run simulation: %.2fs", time)
```
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Time to run simulation: 5.34s
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## Athena Simulation
Now that we've defined our underlying simulation, let's actually try and price an Athena note. Athena has the following characteristics:
- Automatically called if the underlying is above the **call barrier** at observation
- Accelerated coupon paid if the underlying is above the **call barrier** at observation
- The coupon paid is $c \cdot i$ with $i$ as the current year, and $c$ the coupon rate
- Principle protection up until a **protection barrier** at observation; All principle at risk if this barrier not met
- Observed yearly
```julia
call_barrier = S0
strike = S0
protection_barrier = S0 * .6
coupon = nominal * .07
price_athena = function(initial_price, year_prices, call_barrier,
protection_barrier, coupon, forward_structure)
total_coupons = 0
t = length(year_prices)
for i=1:t
price = year_prices[i]
if price ≥ call_barrier
return (nominal + coupon*i) * exp((prod(forward_structure[i:end])-1)* (t-i))
end
end
# We've reached maturity, time to check capital protection
if year_prices[end] > protection_barrier
return nominal
else
put = (strike - year_prices[end]) / strike
return nominal*(1-put)
end
end
forward_structure = forward_term(term)
price_function = (year_prices) -> price_athena(S0, year_prices,
call_barrier, protection_barrier, coupon, forward_structure)
athena = function()
year_indexes = [n*i for i=1:T]
motion = full_simulation(S0, T, n, m, term)
payoffs = [price_function(motion[i, year_indexes]) for i=1:m]
return mean(payoffs)
end
mean_payoffs = zeros(num_simulations)
for i=1:num_simulations
tic()
mean_payoffs[i] = athena()
time = toq()
@printf ("Mean of simulation %i: \$%.4f; Simulation time: %.2fs\n", i, mean_payoffs[i], time)
end
final_mean = mean(mean_payoffs)
println("Mean over $num_simulations simulations: $(mean(mean_payoffs))")
pv = final_mean * (exp(-(prod(forward_structure)-1)*T))
@printf ("Present value of Athena note: \$%.2f, notional: \$%.2f", pv, nominal)
```
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Mean of simulation 1: $103.2805; Simulation time: 5.59s
Mean of simulation 2: $103.3796; Simulation time: 5.05s
Mean of simulation 3: $103.4752; Simulation time: 5.18s
Mean of simulation 4: $103.4099; Simulation time: 5.37s
Mean of simulation 5: $103.3260; Simulation time: 5.32s
Mean over 5 simulations: 103.37421610015554
Present value of Athena note: $95.00, notional: $100.00
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## Phoenix without Memory Simulation
Let's move into pricing a Phoenix without memory. It's very similar to the Athena production, with the exception that we introduce a coupon barrier so coupons are paid even when the underlying is below the initial price.
The Phoenix product has the following characteristics (example [here ](https://www.rbccm.com/usstructurednotes/file-780079.pdf )):
- Automatically called if the underlying is above the **call barrier** at observation
- Coupon paid if the underlying is above a **coupon barrier** at observation
- Principle protection up until a **protection barrier** at observation; All principle at risk if this barrier not met
- Observed yearly
Some example paths (all assume that a call barrier of the current price, and coupon barrier some level below that):
- At the end of year 1, the stock is above the call barrier; the note is called and you receive the value of the stock plus the coupon being paid.
- At the end of year 1, the stock is above the coupon barrier, but not the call barrier; you receive the coupon. At the end of year 2, the stock is below the coupon barrier; you receive nothing. At the end of year 3, the stock is above the call barrier; the note is called and you receive the value of the stock plus a coupon for year 3.
We're going to re-use the same simulation, with the following parameters:
- Call barrier: 100%
- Coupon barrier: 70%
- Coupon: 6%
- Capital protection until 70% (at maturity)
```julia
call_barrier = S0
coupon_barrier = S0 * .8
protection_barrier = S0 * .6
coupon = nominal * .06
price_phoenix_no_memory = function(initial_price, year_prices, call_barrier, coupon_barrier,
protection_barrier, coupon, forward_structure)
total_coupons = 0
t = length(year_prices)
for i=1:t
price = year_prices[i]
if price ≥ call_barrier
return (nominal + coupon + total_coupons)*exp((prod(forward_structure[i:end])-1)*(t-i))
elseif price ≥ coupon_barrier
total_coupons = total_coupons * exp(forward_structure[i]-1) + coupon
else
total_coupons *= exp(forward_structure[i]-1)
end
end
# We've reached maturity, time to check capital protection
if year_prices[end] > protection_barrier
return nominal + total_coupons
else
put = (strike - year_prices[end]) / strike
return nominal*(1-put)
end
end
forward_structure = forward_term(term)
price_function = (year_prices) -> price_phoenix_no_memory(S0, year_prices,
call_barrier, coupon_barrier, protection_barrier, coupon, forward_structure)
phoenix_no_memory = function()
year_indexes = [n*i for i=1:T]
motion = full_simulation(S0, T, n, m, term)
payoffs = [price_function(motion[i, year_indexes]) for i=1:m]
return mean(payoffs)
end
mean_payoffs = zeros(num_simulations)
for i=1:num_simulations
tic()
mean_payoffs[i] = phoenix_no_memory()
time = toq()
@printf ("Mean of simulation %i: \$%.4f; Simulation time: %.2fs\n", i, mean_payoffs[i], time)
end
final_mean = mean(mean_payoffs)
println("Mean over $num_simulations simulations: $(mean(mean_payoffs))")
pv = final_mean * exp(-(prod(forward_structure)-1)* (T))
@printf ("Present value of Phoenix without memory note: \$%.2f", pv)
```
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Mean of simulation 1: $106.0562; Simulation time: 5.72s
Mean of simulation 2: $106.0071; Simulation time: 5.85s
Mean of simulation 3: $105.9959; Simulation time: 5.87s
Mean of simulation 4: $106.0665; Simulation time: 5.93s
Mean of simulation 5: $106.0168; Simulation time: 5.81s
Mean over 5 simulations: 106.02850857209883
Present value of Phoenix without memory note: $97.44
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## Phoenix with Memory Simulation
The Phoenix with Memory structure is very similar to the Phoenix, but as the name implies, has a special "memory" property: **It remembers any coupons that haven't been paid at prior observation times, and pays them all if the underlying crosses the coupon barrier** . For example:
- Note issued with 100% call barrier, 70% coupon barrier. At year 1, the underlying is at 50%, so no coupons are paid. At year 2, the underlying is at 80%, so coupons for both year 1 and 2 are paid, resulting in a double coupon.
You can also find an example [here ](https://www.rbccm.com/usstructurednotes/file-781232.pdf ).
Let's go ahead and set up the simulation! The parameters will be the same, but we can expect that the value will go up because of the memory attribute
```julia
call_barrier = S0
coupon_barrier = S0 * .8
protection_barrier = S0 * .6
coupon = nominal * .07
price_phoenix_with_memory = function(initial_price, year_prices, call_barrier,
coupon_barrier, protection_barrier, coupon, forward_structure)
last_coupon = 0
total_coupons = 0
t = length(year_prices)
for i=1:t
price = year_prices[i]
if price > call_barrier
return (nominal + coupon + total_coupons)*exp((prod(forward_structure[i:end])-1)*(t-i))
elseif price > coupon_barrier
####################################################################
# The only difference between with/without memory is the below lines
memory_coupons = (i - last_coupon) * coupon
last_coupon = i
total_coupons = total_coupons * exp(forward_structure[i]-1) + memory_coupons
####################################################################
else
total_coupons *= exp(forward_structure[i]-1)
end
end
# We've reached maturity, time to check capital protection
if year_prices[end] > protection_barrier
return nominal + total_coupons
else
put = (strike - year_prices[end]) / strike
return nominal*(1-put)
end
end
forward_structure = forward_term(term)
price_function = (year_prices) -> price_phoenix_with_memory(S0, year_prices,
call_barrier, coupon_barrier, protection_barrier, coupon, forward_structure)
phoenix_with_memory = function()
year_indexes = [n*i for i=1:T]
motion = full_simulation(S0, T, n, m, term)
payoffs = [price_function(motion[i, year_indexes]) for i=1:m]
return mean(payoffs)
end
mean_payoffs = zeros(num_simulations)
for i=1:num_simulations
tic()
mean_payoffs[i] = phoenix_with_memory()
time = toq()
@printf ("Mean of simulation %i: \$%.4f; Simulation time: %.2fs\n",
i, mean_payoffs[i], time)
end
final_mean = mean(mean_payoffs)
println("Mean over $num_simulations simulations: $(mean(mean_payoffs))")
pv = final_mean * exp(-(prod(forward_structure)-1)* (T))
@printf ("Present value of Phoenix with memory note: \$%.2f", pv)
```
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Mean of simulation 1: $108.8612; Simulation time: 5.89s
Mean of simulation 2: $109.0226; Simulation time: 5.90s
Mean of simulation 3: $108.9175; Simulation time: 5.92s
Mean of simulation 4: $108.9426; Simulation time: 5.94s
Mean of simulation 5: $108.8087; Simulation time: 6.06s
Mean over 5 simulations: 108.91052564051816
Present value of Phoenix with memory note: $100.09
