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An option's theta, live

A European option's theta, its value's change as calendar time passes, everything else held: −S φ(d₁) σ / (2√T) − r K e^(−rT) N(d₂) for a call, and + r K e^(−rT) N(−d₂) in place of the second term for a put, in euros a year of calendar time (lusine.finance.option_theta). Below, a call and a put struck at 50 EUR with a year left, at 5% and a volatility of 20%, are swept on your GPU as you read across spots from 10 EUR to 150 EUR, a euro a step; and thirty-two options are held to Black-Scholes in float64, and to the Black-Scholes equation.

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 the call's theta and the put's against the spot, a point a step: both most negative near the strike, where time costs an option most, the put's above the call's by the carry, r K e^(−rT). Below it, the spot and the two.
  • The checks: Hull's example, chapter 19's running example (a stock at 49, a strike of 50, 5%, 20%, twenty weeks) a call's theta of −4.31 a year, 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' thetas each within 10⁻⁶ of the spot a year of Black-Scholes in float64 with scipy's exact normal distribution, at the parameters as binary32 holds them; the Black-Scholes equation, θ + r S Δ + σ² S² Γ / 2 = r V for each of the thirty-two, its theta the kernel's and its delta, gamma and value in float64, within the same. 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 sweep's option and its spots, 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 sweep and the checks, the panels the page draws, and the one procedure, main, run each step: the spot a euro on, the thetas at it and at the thirty-two options, and the checks.

// The option theta's live page: a call's theta and a put's swept across the spot; thirty-two options held to
// Black-Scholes in float64 and to the Black-Scholes equation; and Hull's example, with the page's checks computed here
// on the GPU and shown by the panels the program declares.

namespace option_theta;

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;

// Thirty-two options, calls and puts, each with its theta by Black-Scholes in float64 with scipy's exact normal
// distribution, at the parameters as binary32 holds them, and the equation's other terms in float64, r S delta +
// sigma^2 S^2 gamma / 2 - r V; the kernel's theta; whether it is within 1e-6 of the spot a year of the float64 theta,
// and whether it and the other terms add to nothing within the same.
struct option {
  bool call; float<EUR> spot; float<EUR> strike; float<year^-1> rate; float volatility; float<year> years;
  float<EUR/year> reference; float<EUR/year> rest; float<EUR/year> theta; bool holds; bool equation;
};
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 = -0.0078027330416833955 EUR/year, rest = 0.0078027330416833955 EUR/year },
  { call = false, spot = 188.59482662705486 EUR, strike = 135.69979791601557 EUR,
    rate = 0.05057192816027696 year^-1, volatility = 0.3115418699091693, years = 4.222032830755394 year,
    reference = -1.2275299773284845 EUR/year, rest = 1.2275299773284845 EUR/year },
  { call = true, spot = 118.91243378167061 EUR, strike = 185.2483042029888 EUR,
    rate = 0.09011600873524846 year^-1, volatility = 0.31585264386633055, years = 3.721298802897474 year,
    reference = -7.612130912814374 EUR/year, rest = 7.612130912814374 EUR/year },
  { call = true, spot = 163.8803382421281 EUR, strike = 27.370130355000377 EUR,
    rate = 0.08699557563034188 year^-1, volatility = 0.3805289170746753, years = 2.9718848042202395 year,
    reference = -1.8525954065741073 EUR/year, rest = 1.8525954065741068 EUR/year },
  { call = false, spot = 156.01246541016698 EUR, strike = 98.66825482181102 EUR,
    rate = 0.14785047611735416 year^-1, volatility = 0.7774526364639268, years = 2.9756117175286025 year,
    reference = -1.01202323701375 EUR/year, rest = 1.01202323701375 EUR/year },
  { call = true, spot = 120.21559443564263 EUR, strike = 39.4609317943744 EUR,
    rate = 0.031629090453178174 year^-1, volatility = 0.41657887517742265, years = 2.621120091497525 year,
    reference = -1.7257955488800734 EUR/year, rest = 1.7257955488800727 EUR/year },
  { call = true, spot = 125.27283792580695 EUR, strike = 113.42753373611995 EUR,
    rate = 0.13960486210071893 year^-1, volatility = 0.32822630365082595, years = 0.663718099301309 year,
    reference = -17.40426433265798 EUR/year, rest = 17.40426433265798 EUR/year },
  { call = true, spot = 111.5146720923226 EUR, strike = 91.70554427769596 EUR,
    rate = 0.09035406599957747 year^-1, volatility = 0.6424737027251702, years = 0.9491809736885719 year,
    reference = -15.184260961762027 EUR/year, rest = 15.184260961762028 EUR/year },
  { call = true, spot = 104.7904253041208 EUR, strike = 54.552235260291766 EUR,
    rate = 0.035530099141647786 year^-1, volatility = 0.5662054014919639, years = 4.101686172995604 year,
    reference = -3.5295183889584503 EUR/year, rest = 3.5295183889584507 EUR/year },
  { call = true, spot = 131.9704638511787 EUR, strike = 199.51590533067628 EUR,
    rate = 0.12150206984240143 year^-1, volatility = 0.07353334528136175, years = 1.013456937268504 year,
    reference = -0.0018368201502858574 EUR/year, rest = 0.0018368201502858572 EUR/year },
  { call = true, spot = 189.31412648919076 EUR, strike = 101.79493838011325 EUR,
    rate = 0.1486055774391051 year^-1, volatility = 0.17001756302296653, years = 3.6964262270876262 year,
    reference = -8.733975257391437 EUR/year, rest = 8.733975257391439 EUR/year },
  { call = false, spot = 184.9460138697643 EUR, strike = 154.73367705909305 EUR,
    rate = 0.06516181953678835 year^-1, volatility = 0.8088997581932528, years = 4.43441302148932 year,
    reference = -2.0936464997614577 EUR/year, rest = 2.0936464997614577 EUR/year },
  { call = true, spot = 184.149544077138 EUR, strike = 138.86885284372516 EUR,
    rate = 0.08453121905364037 year^-1, volatility = 0.8815777025835502, years = 4.812131996853199 year,
    reference = -8.26668194965753 EUR/year, rest = 8.26668194965753 EUR/year },
  { call = false, spot = 179.92226669131634 EUR, strike = 66.09962335996232 EUR,
    rate = 0.09058104655070943 year^-1, volatility = 0.4365392531203102, years = 0.6805432126265991 year,
    reference = -0.12513026053246779 EUR/year, rest = 0.12513026053246706 EUR/year },
  { call = true, spot = 58.98985403872521 EUR, strike = 69.53690348359211 EUR,
    rate = 0.09882627092796568 year^-1, volatility = 0.8984296966945114, years = 4.194961386027854 year,
    reference = -3.938340133000204 EUR/year, rest = 3.9383401330002035 EUR/year },
  { call = false, spot = 70.83381725423173 EUR, strike = 196.1692404299365 EUR,
    rate = 0.048636007021675504 year^-1, volatility = 0.619002303682292, years = 0.44163010411418885 year,
    reference = 8.174542813301859 EUR/year, rest = -8.17454281330186 EUR/year },
  { call = true, spot = 182.0562557898536 EUR, strike = 173.70967347304358 EUR,
    rate = 0.04764858389327849 year^-1, volatility = 0.6800378619360005, years = 4.203906336381006 year,
    reference = -10.260596912818725 EUR/year, rest = 10.260596912818725 EUR/year },
  { call = false, spot = 66.23051665247758 EUR, strike = 170.51126513679412 EUR,
    rate = 0.0634771790745104 year^-1, volatility = 0.7740285718864518, years = 2.4297303435717232 year,
    reference = 1.7609659630971564 EUR/year, rest = -1.7609659630971572 EUR/year },
  { call = true, spot = 61.040122087469854 EUR, strike = 25.81738810852937 EUR,
    rate = -0.01776675074159802 year^-1, volatility = 0.6865046104255538, years = 4.1593741745940465 year,
    reference = -1.6289120516197095 EUR/year, rest = 1.6289120516197098 EUR/year },
  { call = true, spot = 183.22855401336517 EUR, strike = 180.44889574953785 EUR,
    rate = 0.022967575251798763 year^-1, volatility = 0.44718485541057645, years = 4.795045054343254 year,
    reference = -7.499703857319932 EUR/year, rest = 7.499703857319931 EUR/year },
  { call = false, spot = 55.846083764705995 EUR, strike = 53.78166280056927 EUR,
    rate = 0.12564516571602508 year^-1, volatility = 0.47603203171217895, years = 0.9746002449187416 year,
    reference = -1.8136730871319493 EUR/year, rest = 1.8136730871319493 EUR/year },
  { call = false, spot = 49.43203182526905 EUR, strike = 181.98461219516167 EUR,
    rate = 0.12752664626939325 year^-1, volatility = 0.10648053312598907, years = 0.6180760422632282 year,
    reference = 21.448850278088127 EUR/year, rest = -21.448850278088127 EUR/year },
  { call = true, spot = 155.62101568917203 EUR, strike = 162.04319920333035 EUR,
    rate = 0.10691574785743298 year^-1, volatility = 0.60801310807133, years = 2.030221913104255 year,
    reference = -16.52128963509906 EUR/year, rest = 16.521289635099055 EUR/year },
  { call = false, spot = 143.56758434849291 EUR, strike = 148.75367003537065 EUR,
    rate = 0.12858898784657127 year^-1, volatility = 0.7002982559148344, years = 4.144197661866689 year,
    reference = 1.6007896588490453 EUR/year, rest = -1.600789658849048 EUR/year },
  { call = false, spot = 197.30127196208014 EUR, strike = 87.76729450260797 EUR,
    rate = 0.13782776344187664 year^-1, volatility = 0.284284922323631, years = 4.1099414619067645 year,
    reference = -0.030960439660711006 EUR/year, rest = 0.030960439660711395 EUR/year },
  { call = true, spot = 176.93386638351492 EUR, strike = 24.762009772194894 EUR,
    rate = 0.002562164252424219 year^-1, volatility = 0.5123741136193225, years = 0.8223266264332635 year,
    reference = -0.06422631768016202 EUR/year, rest = 0.06422631768016202 EUR/year },
  { call = true, spot = 22.383447260917606 EUR, strike = 87.65377119357558 EUR,
    rate = 0.12941660432027086 year^-1, volatility = 0.8920939592482549, years = 4.887267819355272 year,
    reference = -2.0177554246823908 EUR/year, rest = 2.0177554246823908 EUR/year },
  { call = false, spot = 198.92285038640267 EUR, strike = 28.77192047369769 EUR,
    rate = 0.07350521936785097 year^-1, volatility = 0.259816172475044, years = 4.990816046864923 year,
    reference = -0.000368371137469891 EUR/year, rest = 0.00036837113747056096 EUR/year },
  { call = true, spot = 29.333652993672846 EUR, strike = 155.9661717776625 EUR,
    rate = 0.13608761873657335 year^-1, volatility = 0.7921456835739259, years = 0.5821716906310608 year,
    reference = -0.4338674752351436 EUR/year, rest = 0.4338674752351436 EUR/year },
  { call = true, spot = 96.2014567940601 EUR, strike = 187.00927996286674 EUR,
    rate = 0.028803522459524823 year^-1, volatility = 0.3952282877503409, years = 4.334615968395207 year,
    reference = -4.214634223649003 EUR/year, rest = 4.214634223649003 EUR/year },
  { call = false, spot = 84.30148792212937 EUR, strike = 34.445458985634986 EUR,
    rate = 0.07128430993582072 year^-1, volatility = 0.24547475493628396, years = 4.191866317697686 year,
    reference = -0.03336844822854679 EUR/year, rest = 0.03336844822854709 EUR/year },
  { call = true, spot = 159.96599103543278 EUR, strike = 135.11847347833998 EUR,
    rate = 0.09059069229451537 year^-1, volatility = 0.11230768482870263, years = 0.8190242915046948 year,
    reference = -11.456528824633825 EUR/year, rest = 11.456528824633823 EUR/year },
};

// The sweep: the step, from 0 to 140 and round again; the spot, from 10 EUR to 150 EUR; and the call's theta and the
// put's, a year of calendar time's.
struct { integer step = -1; float<EUR> spot; float<EUR/year> called; float<EUR/year> put; } sweep;

// The checks: Hull's chapter 19 example's theta -4.31 a year to half a unit in the last digit printed; the thirty-two
// options each within 1e-6 of the spot a year of the float64 theta; and with the float64 delta, gamma and value, theta
// + r S delta + sigma^2 S^2 gamma / 2 = r V for each, within the same.
struct { bool hull = true; bool reference = true; bool equation = 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; }

bool near(float<EUR/year> theta, float<EUR/year> reference, float<EUR> spot)
{
  return abs(theta - reference) * 1.0 year <= 0.000001 * spot;
}

bool solves(float<EUR/year> theta, float<EUR/year> rest, float<EUR> spot)
{
  return abs(theta + rest) * 1.0 year <= 0.000001 * spot;
}

output table<lusine.widget> panels = {
  lusine.series { { panel = "the sweep", chart = "theta", quantity = sweep.called, steps = 141, against = sweep.spot,
                    label = "call" } },
  lusine.series { { panel = "the sweep", chart = "theta", quantity = sweep.put, steps = 141, against = sweep.spot,
                    label = "put" } },
  lusine.value { { panel = "the sweep", quantity = sweep.spot, label = "spot" } },
  lusine.value { { panel = "the sweep", quantity = sweep.called, label = "call" } },
  lusine.value { { panel = "the sweep", quantity = sweep.put, label = "put" } },
  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.equation, label = "the Black-Scholes equation" } },
};

void main()
{
  sweep.step = sweep.step >= 140 ? 0 : sweep.step + 1;
  sweep.spot = settings.first + float(sweep.step) * settings.stride;
  sweep.called = lusine.finance.option_theta(call = true, sweep.spot, settings.strike, settings.rate,
                                             settings.volatility, settings.period, settings.years);
  sweep.put = lusine.finance.option_theta(call = false, sweep.spot, settings.strike, settings.rate,
                                          settings.volatility, settings.period, settings.years);
  options.theta = lusine.finance.option_theta(options.call, options.spot, options.strike, options.rate,
                                              options.volatility, settings.period, options.years);
  options.holds = near(options.theta, options.reference, options.spot);
  options.equation = solves(options.theta, options.rest, options.spot);
  checks.hull = checks.hull && abs(lusine.finance.option_theta(call = true, running.spot, running.strike,
                                                              running.rate, running.volatility, settings.period,
                                                              running.years)
                                   + 4.31 EUR/year) <= 0.005 EUR/year;
  checks.reference = checks.reference && failing(holds = options.holds) == 0;
  checks.equation = checks.equation && failing(holds = options.equation) == 0;
  lamps.colour = lamp_colour(holds = checks.hull && checks.reference && checks.equation);
}

The kernel's other proof (a thousand options against Black-Scholes in float64 and its equation) is on its proof page.