An option's vega, live¶
A European option's vega, its value's change with the volatility, dV/dσ = S φ(d₁) √(T / P), a call's and a put's alike (lusine.finance.option_vega). σ is the spread over a period P the game states, so vega is per unit of that spread, in euros: with P a year, Hull's S √T N′(d₁), per unit of a volatility a year. Below, an option struck at 50 EUR with a year left, at 5% and a volatility of 20% a year, is swept on your GPU as you read across spots from 10 EUR to 150 EUR, a euro a step, its vega taken per unit of the year's volatility and per unit of a trading day's; thirty-two options are held to Black-Scholes in float64; and sixty-four options are drawn each step across moneyness, rates from −2% to 15%, volatilities from 5% to 90% and lives from a week to five years.
This page runs the program on your GPU through WebGPU, so it needs a browser with WebGPU (a recent Chrome, Edge, Safari or Firefox). Without it the program does not run and nothing is drawn below.
What it shows¶
- The strip above the controls is green while every check holds, and turns red for good once one fails.
- The sweep: the chart draws vega against the spot, a point a step, per unit of the year's volatility, and per unit of a trading day's, σ / √252 over a 252nd of a year, divided by √252: one over the other, a hump about the strike. Below it, the spot and the year's vega.
- The checks: Hull's example, chapter 19's running example (a stock at 49, a strike of 50, 5%, 20%, twenty weeks) a vega of 12.1, to half a unit in the last digit printed (Hull, Options, Futures, and Other Derivatives, 9th ed.); Black-Scholes in float64, the thirty-two options' vegas each within 10⁻⁶ of the spot of Black-Scholes in float64 with scipy's exact normal distribution, at the parameters as binary32 holds them; per unit of the period's spread, for every option drawn and across the sweep, the vega per unit of a trading day's volatility √252 times the year's, within 10⁻⁵, where the year's is above 10⁻³ of the spot. Each is held at every step.
Every number here is computed by the program on the GPU; the page only reads the values back and draws them. The seed, the sweep's option and its spots, the trading day, and Hull's examples are the program's settings, which the host may write; this page offers no control to change them yet.
The program¶
The options, the draws, the sweep and the checks, the panels the page draws, and the one procedure, main, run each step: the spot a euro on, the vegas at it, at the thirty-two options and at the draws, and the checks.
// The option vega's live page: vega swept across the spot with a year's volatility and with a trading day's; thirty-two
// options held to Black-Scholes in float64; sixty-four drawn each step, a day's vega the root of 252 times a year's;
// and Hull's example, with the page's checks computed here on the GPU and shown by the panels the program declares.
namespace option_vega;
lusine.version version = { { number = 1 } };
// The reader's settings: the seed the options are drawn from; the sweep's option, its strike, rate, volatility over a
// year and life; and the spot the sweep starts at and its step.
host struct {
integer seed = 391; float<EUR> strike = 50.0 EUR; float<year^-1> rate = 0.05 year^-1; float volatility = 0.2;
float<year> years = 1.0 year; float<year> period = 1.0 year; float<EUR> first = 10.0 EUR;
float<EUR> stride = 1.0 EUR;
} settings;
// Hull's examples: Example 15.6, a stock at 42 EUR, a strike of 40 EUR, 10% a year, a volatility of 20% a year and six
// months left; and chapter 19's running example, 49 EUR, 50 EUR, 5%, 20% and twenty weeks.
host struct {
float<EUR> spot = 42.0 EUR; float<EUR> strike = 40.0 EUR; float<year^-1> rate = 0.1 year^-1; float volatility = 0.2;
float<year> years = 0.5 year;
} example;
host struct {
float<EUR> spot = 49.0 EUR; float<EUR> strike = 50.0 EUR; float<year^-1> rate = 0.05 year^-1; float volatility = 0.2;
float<year> years = 0.38461538461538464 year;
} running;
// A trading day, a 252nd of a year, the period a day's volatility is over, and the root of 252.
host struct { float<year> day = 0.003968253968253968 year; float root = 15.874507866387544; } trading;
// Thirty-two options, each with its vega, S phi(d1) sqrt(T), by Black-Scholes in float64 with scipy's exact normal
// distribution, at the parameters as binary32 holds them; the kernel's vega, and whether it is within 1e-6 of the spot
// of it.
struct option {
bool call; float<EUR> spot; float<EUR> strike; float<year^-1> rate; float volatility; float<year> years;
float<EUR> reference; float<EUR> vega; bool holds;
};
host table<option> options(capacity = 32) = {
{ call = true, spot = 20.886307268226673 EUR, strike = 58.665533887237984 EUR,
rate = 0.013447202845124347 year^-1, volatility = 0.15902594726357758, years = 4.38507015721194 year,
reference = 0.38936635156198734 EUR },
{ call = false, spot = 188.59482662705486 EUR, strike = 135.69979791601557 EUR,
rate = 0.05057192816027696 year^-1, volatility = 0.3115418699091693, years = 4.222032830755394 year,
reference = 78.17375567653468 EUR },
{ call = true, spot = 118.91243378167061 EUR, strike = 185.2483042029888 EUR,
rate = 0.09011600873524846 year^-1, volatility = 0.31585264386633055, years = 3.721298802897474 year,
reference = 90.77300738444632 EUR },
{ call = true, spot = 163.8803382421281 EUR, strike = 27.370130355000377 EUR,
rate = 0.08699557563034188 year^-1, volatility = 0.3805289170746753, years = 2.9718848042202395 year,
reference = 0.2930320751120414 EUR },
{ call = false, spot = 156.01246541016698 EUR, strike = 98.66825482181102 EUR,
rate = 0.14785047611735416 year^-1, volatility = 0.7774526364639268, years = 2.9756117175286025 year,
reference = 43.73302414275506 EUR },
{ call = true, spot = 120.21559443564263 EUR, strike = 39.4609317943744 EUR,
rate = 0.031629090453178174 year^-1, volatility = 0.41657887517742265, years = 2.621120091497525 year,
reference = 8.349341157546668 EUR },
{ call = true, spot = 125.27283792580695 EUR, strike = 113.42753373611995 EUR,
rate = 0.13960486210071893 year^-1, volatility = 0.32822630365082595, years = 0.663718099301309 year,
reference = 28.330265485522045 EUR },
{ call = true, spot = 111.5146720923226 EUR, strike = 91.70554427769596 EUR,
rate = 0.09035406599957747 year^-1, volatility = 0.6424737027251702, years = 0.9491809736885719 year,
reference = 32.41069477062407 EUR },
{ call = true, spot = 104.7904253041208 EUR, strike = 54.552235260291766 EUR,
rate = 0.035530099141647786 year^-1, volatility = 0.5662054014919639, years = 4.101686172995604 year,
reference = 37.81174754248935 EUR },
{ call = true, spot = 131.9704638511787 EUR, strike = 199.51590533067628 EUR,
rate = 0.12150206984240143 year^-1, volatility = 0.07353334528136175, years = 1.013456937268504 year,
reference = 0.028206378578058387 EUR },
{ call = true, spot = 189.31412648919076 EUR, strike = 101.79493838011325 EUR,
rate = 0.1486055774391051 year^-1, volatility = 0.17001756302296653, years = 3.6964262270876262 year,
reference = 0.1322314999609784 EUR },
{ call = false, spot = 184.9460138697643 EUR, strike = 154.73367705909305 EUR,
rate = 0.06516181953678835 year^-1, volatility = 0.8088997581932528, years = 4.43441302148932 year,
reference = 82.42133621370535 EUR },
{ call = true, spot = 184.149544077138 EUR, strike = 138.86885284372516 EUR,
rate = 0.08453121905364037 year^-1, volatility = 0.8815777025835502, years = 4.812131996853199 year,
reference = 67.15012108037301 EUR },
{ call = false, spot = 179.92226669131634 EUR, strike = 66.09962335996232 EUR,
rate = 0.09058104655070943 year^-1, volatility = 0.4365392531203102, years = 0.6805432126265991 year,
reference = 0.4390792742856121 EUR },
{ call = true, spot = 58.98985403872521 EUR, strike = 69.53690348359211 EUR,
rate = 0.09882627092796568 year^-1, volatility = 0.8984296966945114, years = 4.194961386027854 year,
reference = 27.60052995193209 EUR },
{ call = false, spot = 70.83381725423173 EUR, strike = 196.1692404299365 EUR,
rate = 0.048636007021675504 year^-1, volatility = 0.619002303682292, years = 0.44163010411418885 year,
reference = 1.6034238335428694 EUR },
{ call = true, spot = 182.0562557898536 EUR, strike = 173.70967347304358 EUR,
rate = 0.04764858389327849 year^-1, volatility = 0.6800378619360005, years = 4.203906336381006 year,
reference = 101.59860432332648 EUR },
{ call = false, spot = 66.23051665247758 EUR, strike = 170.51126513679412 EUR,
rate = 0.0634771790745104 year^-1, volatility = 0.7740285718864518, years = 2.4297303435717232 year,
reference = 41.12863252896211 EUR },
{ call = true, spot = 61.040122087469854 EUR, strike = 25.81738810852937 EUR,
rate = -0.01776675074159802 year^-1, volatility = 0.6865046104255538, years = 4.1593741745940465 year,
reference = 22.40167024406485 EUR },
{ call = true, spot = 183.22855401336517 EUR, strike = 180.44889574953785 EUR,
rate = 0.022967575251798763 year^-1, volatility = 0.44718485541057645, years = 4.795045054343254 year,
reference = 132.26613068055755 EUR },
{ call = false, spot = 55.846083764705995 EUR, strike = 53.78166280056927 EUR,
rate = 0.12564516571602508 year^-1, volatility = 0.47603203171217895, years = 0.9746002449187416 year,
reference = 18.635811702247462 EUR },
{ call = false, spot = 49.43203182526905 EUR, strike = 181.98461219516167 EUR,
rate = 0.12752664626939325 year^-1, volatility = 0.10648053312598907, years = 0.6180760422632282 year,
reference = 9.87799622313363e-46 EUR },
{ call = true, spot = 155.62101568917203 EUR, strike = 162.04319920333035 EUR,
rate = 0.10691574785743298 year^-1, volatility = 0.60801310807133, years = 2.030221913104255 year,
reference = 72.21494210349003 EUR },
{ call = false, spot = 143.56758434849291 EUR, strike = 148.75367003537065 EUR,
rate = 0.12858898784657127 year^-1, volatility = 0.7002982559148344, years = 4.144197661866689 year,
reference = 66.360253662026 EUR },
{ call = false, spot = 197.30127196208014 EUR, strike = 87.76729450260797 EUR,
rate = 0.13782776344187664 year^-1, volatility = 0.284284922323631, years = 4.1099414619067645 year,
reference = 4.439319906834982 EUR },
{ call = true, spot = 176.93386638351492 EUR, strike = 24.762009772194894 EUR,
rate = 0.002562164252424219 year^-1, volatility = 0.5123741136193225, years = 0.8223266264332635 year,
reference = 0.0029449947155307872 EUR },
{ call = true, spot = 22.383447260917606 EUR, strike = 87.65377119357558 EUR,
rate = 0.12941660432027086 year^-1, volatility = 0.8920939592482549, years = 4.887267819355272 year,
reference = 16.343305083208 EUR },
{ call = false, spot = 198.92285038640267 EUR, strike = 28.77192047369769 EUR,
rate = 0.07350521936785097 year^-1, volatility = 0.259816172475044, years = 4.990816046864923 year,
reference = 0.020900670121908247 EUR },
{ call = true, spot = 29.333652993672846 EUR, strike = 155.9661717776625 EUR,
rate = 0.13608761873657335 year^-1, volatility = 0.7921456835739259, years = 0.5821716906310608 year,
reference = 0.5897583848027974 EUR },
{ call = true, spot = 96.2014567940601 EUR, strike = 187.00927996286674 EUR,
rate = 0.028803522459524823 year^-1, volatility = 0.3952282877503409, years = 4.334615968395207 year,
reference = 77.5477765034107 EUR },
{ call = false, spot = 84.30148792212937 EUR, strike = 34.445458985634986 EUR,
rate = 0.07128430993582072 year^-1, volatility = 0.24547475493628396, years = 4.191866317697686 year,
reference = 2.1865166848287356 EUR },
{ call = true, spot = 159.96599103543278 EUR, strike = 135.11847347833998 EUR,
rate = 0.09059069229451537 year^-1, volatility = 0.11230768482870263, years = 0.8190242915046948 year,
reference = 2.930737450334188 EUR },
};
// Sixty-four options drawn each step: vega with the volatility over a year, and with a trading day's, sigma / sqrt(252)
// over a 252nd of a year; and whether the day's is sqrt(252) times the year's within 1e-5, where the year's is above
// 1e-3 of the spot.
struct drawn {
integer index; bool call; float<EUR> spot; float<EUR> strike; float<year^-1> rate; float volatility;
float<year> years; float<EUR> year; float<EUR> day; bool holds;
};
table<drawn> draws(capacity = 64) = {
{ index = 0 }, { index = 1 }, { index = 2 }, { index = 3 }, { index = 4 }, { index = 5 }, { index = 6 },
{ index = 7 }, { index = 8 }, { index = 9 }, { index = 10 }, { index = 11 }, { index = 12 }, { index = 13 },
{ index = 14 }, { index = 15 }, { index = 16 }, { index = 17 }, { index = 18 }, { index = 19 }, { index = 20 },
{ index = 21 }, { index = 22 }, { index = 23 }, { index = 24 }, { index = 25 }, { index = 26 }, { index = 27 },
{ index = 28 }, { index = 29 }, { index = 30 }, { index = 31 }, { index = 32 }, { index = 33 }, { index = 34 },
{ index = 35 }, { index = 36 }, { index = 37 }, { index = 38 }, { index = 39 }, { index = 40 }, { index = 41 },
{ index = 42 }, { index = 43 }, { index = 44 }, { index = 45 }, { index = 46 }, { index = 47 }, { index = 48 },
{ index = 49 }, { index = 50 }, { index = 51 }, { index = 52 }, { index = 53 }, { index = 54 }, { index = 55 },
{ index = 56 }, { index = 57 }, { index = 58 }, { index = 59 }, { index = 60 }, { index = 61 }, { index = 62 },
{ index = 63 },
};
// The sweep: the step, from 0 to 140 and round again; the spot, from 10 EUR to 150 EUR; vega per unit of a year's
// volatility, and per unit of a trading day's over the root of 252.
struct { integer step = -1; float<EUR> spot; float<EUR> year; float<EUR> day; } sweep;
// The checks: Hull's chapter 19 example's vega 12.1 to half a unit in the last digit printed; the thirty-two options
// each within 1e-6 of the spot of the float64 vega; and for every option drawn, and across the sweep, a trading day's
// vega sqrt(252) times the year's, within 1e-5, where the year's is above 1e-3 of the spot.
struct { bool hull = true; bool reference = true; bool period = true; } checks;
// A strip of lamps the page draws, lit green while every check holds and red once one fails.
struct lamp { vector<float, 4> colour; };
lattice<lamp, 2> lamps(spacing = 1 m, boundary = wall);
vector<float, 4> lamp_colour(bool holds)
{
return holds ? vector<float, 4>(0.15, 0.6, 0.3, 1.0) : vector<float, 4>(0.75, 0.15, 0.15, 1.0);
}
float eps() { return 1.0 / 8388608.0; }
integer failing(table<bool> holds) { return count(holds) where !holds; }
float uniform_of(integer seed, integer step, bits key, integer index)
{
return lusine.error.uniform(drawn = lusine.error.draw(seed = bits(seed), step = bits(step), key,
index = bits(index)));
}
// An option drawn: a call or a put, a spot and a strike from 20 EUR to 200 EUR, a rate from -2% to 15% a year, a
// volatility from 5% to 90% a year, and a life from a week to five years.
(bool, float<EUR>, float<EUR>, float<year^-1>, float, float<year>) drawing(integer index, integer seed, integer step)
{
return (uniform_of(seed, step, key = fnv1a("call"), index) < 0.5,
20.0 EUR + uniform_of(seed, step, key = fnv1a("spot"), index) * 180.0 EUR,
20.0 EUR + uniform_of(seed, step, key = fnv1a("strike"), index) * 180.0 EUR,
-0.02 year^-1 + uniform_of(seed, step, key = fnv1a("rate"), index) * 0.17 year^-1,
0.05 + 0.85 * uniform_of(seed, step, key = fnv1a("volatility"), index),
0.02 year + uniform_of(seed, step, key = fnv1a("life"), index) * 4.98 year);
}
bool near(float<EUR> vega, float<EUR> reference, float<EUR> spot) { return abs(vega - reference) <= 0.000001 * spot; }
bool scaled(float<EUR> year, float<EUR> day, float<EUR> spot, float root)
{
return year <= 0.001 * spot || abs(day / (year * root) - 1.0) <= 0.00001;
}
output table<lusine.widget> panels = {
lusine.series { { panel = "the sweep", chart = "vega", quantity = sweep.year, steps = 141, against = sweep.spot,
label = "a year's volatility" } },
lusine.series { { panel = "the sweep", chart = "vega", quantity = sweep.day, steps = 141, against = sweep.spot,
label = "a day's, over sqrt(252)" } },
lusine.value { { panel = "the sweep", quantity = sweep.spot, label = "spot" } },
lusine.value { { panel = "the sweep", quantity = sweep.year, label = "a year's" } },
lusine.value { { panel = "the checks", quantity = checks.hull, label = "Hull's example" } },
lusine.value { { panel = "the checks", quantity = checks.reference, label = "Black-Scholes in float64" } },
lusine.value { { panel = "the checks", quantity = checks.period, label = "per unit of the period's spread" } },
};
void main()
{
sweep.step = sweep.step >= 140 ? 0 : sweep.step + 1;
sweep.spot = settings.first + float(sweep.step) * settings.stride;
sweep.year = lusine.finance.option_vega(sweep.spot, settings.strike, settings.rate, settings.volatility,
settings.period, settings.years);
sweep.day = lusine.finance.option_vega(sweep.spot, settings.strike, settings.rate,
volatility = settings.volatility / trading.root, period = trading.day,
settings.years) / trading.root;
options.vega = lusine.finance.option_vega(options.spot, options.strike, options.rate, options.volatility,
settings.period, options.years);
options.holds = near(options.vega, options.reference, options.spot);
(draws.call, draws.spot, draws.strike, draws.rate, draws.volatility, draws.years)
= drawing(draws.index, settings.seed, step = sweep.step);
draws.year = lusine.finance.option_vega(draws.spot, draws.strike, draws.rate, draws.volatility, settings.period,
draws.years);
draws.day = lusine.finance.option_vega(draws.spot, draws.strike, draws.rate,
volatility = draws.volatility / trading.root, period = trading.day,
draws.years);
draws.holds = scaled(draws.year, draws.day, draws.spot, trading.root);
checks.hull = checks.hull && abs(lusine.finance.option_vega(running.spot, running.strike, running.rate,
running.volatility, settings.period, running.years)
- 12.1 EUR) <= 0.05 EUR;
checks.reference = checks.reference && failing(holds = options.holds) == 0;
checks.period = checks.period && failing(holds = draws.holds) == 0
&& scaled(sweep.year, day = sweep.day * trading.root, sweep.spot, trading.root);
lamps.colour = lamp_colour(holds = checks.hull && checks.reference && checks.period);
}
The kernel's other proof (a thousand options against Black-Scholes in float64) is on its proof page.