Delta: a European option's value against its stock's price¶

A European option's delta, its value's change with the stock's price, dV/dS: N(d1) for a call, N(d1) - 1 for a put; dimensionless (money over money).

The parameters are lusine.finance.option_value's. Source: Hull, Options, Futures, and Other Derivatives, 9th ed. (Pearson, 2014), ch. 19. Checked against a reference in numpy, which has its own tests.

Source

Hull, Options, Futures, and Other Derivatives, 9th ed. (2014), ch. 19.

Proof

reference, invariant.

option_delta.trait

namespace lusine.finance;

float option_delta(bool call, float<_> spot, float<_> strike, float<year^-1> rate, float volatility,
                   float<year> period, float<year> years)
{
  let n = normal_cdf(x = d1(spot, strike, rate, volatility, period, years));
  return call ? n : n - 1.0;
}

Uses, without proving it here: finance.d1, finance.normal_cdf.

N(d1) for a call, N(d1) - 1 for a put.

In [1]:

import os, sys
root = os.getcwd()
while not os.path.isfile(os.path.join(root, 'docsite.json')):
    root = os.path.dirname(root)
sys.path.insert(0, os.path.join(root, 'tools'))
import numpy as np
import matplotlib.pyplot as plt
from scipy.special import ndtr
from scipy.stats import norm
from lusine import trait_run
EPS = float(np.finfo(np.float32).eps)   # Trait's float is binary32
YEAR = 31557600.0                        # lusine's year row, the Julian year in seconds: the host speaks base units

import os, sys root = os.getcwd() while not os.path.isfile(os.path.join(root, 'docsite.json')): root = os.path.dirname(root) sys.path.insert(0, os.path.join(root, 'tools')) import numpy as np import matplotlib.pyplot as plt from scipy.special import ndtr from scipy.stats import norm from lusine import trait_run EPS = float(np.finfo(np.float32).eps) # Trait's float is binary32 YEAR = 31557600.0 # lusine's year row, the Julian year in seconds: the host speaks base units

The kernel¶

The law as the library states it, and as this page runs it.

In [2]:

print(trait_run.source('finance/option_delta.trait'))

print(trait_run.source('finance/option_delta.trait'))

// lusine.finance.option_delta: a European option's delta, its value's change with the stock's price, dV/dS: N(d1) for a
// call, N(d1) - 1 for a put; dimensionless (money over money). The parameters are lusine.finance.option_value's.
// Source: Hull, Options, Futures, and Other Derivatives, 9th ed. (Pearson, 2014), ch. 19.
// Checked against a reference in numpy, which has its own tests.

namespace lusine.finance;

float option_delta(bool call, float<_> spot, float<_> strike, float<year^-1> rate, float volatility,
                   float<year> period, float<year> years)
{
  let n = normal_cdf(x = d1(spot, strike, rate, volatility, period, years));
  return call ? n : n - 1.0;
}

The program¶

A lattice of options, one an item, the kernel bound over them: the prices in euros, the rate a year, the volatility over a period (a year here, Hull's), the life in years. The host speaks the base units, so a time is stored in seconds and a rate per second, and delta, money over money, is dimensionless. kernel below is that call; closed is the independent account, Black-Scholes in float64 with scipy's exact normal distribution, at the parameters as the device holds them. held rounds them so.

In [3]:

CALL = True
BS = """namespace bs;

lusine.version version = { { number = 1 } };

struct option {
  host bool call; host float<EUR> spot; host float<EUR> strike; host float<year^-1> rate; host float volatility;
  host float<year> period; host float<year> years;
  float option_delta;
};
lattice<option, 1> options(spacing = 1 m, boundary = wall);

void main()
{
  options.option_delta = lusine.finance.option_delta(options.call, options.spot, options.strike, options.rate, options.volatility,
                                    options.period, options.years);
}
"""
print(BS)


def held(call, spot, strike, rate, volatility, years, period=1.0):
    """The parameters as the device holds them, in the host's units (euros, a year, years): each rounded to binary32 in
    the base units it is stored in."""
    f = lambda v, scale=1.0: np.float32(np.asarray(v, dtype=float) * scale).astype(float) / scale
    return (np.asarray(call, dtype=bool), f(spot), f(strike), f(rate, 1 / YEAR), f(volatility), f(years, YEAR),
            f(period, YEAR))


def kernel(call, spot, strike, rate, volatility, years, period=1.0):
    """The kernel's option_delta for each option: prices in euros, the rate a year, the volatility over `period` years and the
    life in years, stored in the base units (seconds); the result back in dimensionless, one an option (a
    single option's alone)."""
    n = max(np.size(a) for a in (call, spot, strike, rate, volatility, years, period))
    run = trait_run.Program(['finance/option_delta.trait'], BS, {'bs.options': [n]})
    for field, v in (('call', call), ('spot', spot), ('strike', strike), ('rate', np.asarray(rate) / YEAR),
                     ('volatility', volatility), ('period', np.asarray(period) * YEAR),
                     ('years', np.asarray(years) * YEAR)):
        run.store('bs.options', field, np.broadcast_to(np.asarray(v), (n,)))
    out = run.steps('bs.main', 1, [('bs.options', 'option_delta')])[0][0]
    return out if n > 1 else float(out[0])


def closed(call, spot, strike, rate, volatility, years, period=1.0):
    """The independent account: Black-Scholes in float64 with scipy's exact normal distribution, at the parameters
    the device holds."""
    c, S, K, r, v, T, P = held(call, spot, strike, rate, volatility, years, period)
    s = v * np.sqrt(T / P)
    a = (np.log(S / K) + (r + 0.5 * (v * v / P)) * T) / s
    b = a - s
    df = K * np.exp(-r * T)
    return np.where(c, ndtr(a), ndtr(a) - 1.0)


rng = np.random.default_rng(391)
N = 1000
SPOT, STRIKE = rng.uniform(20, 200, N), rng.uniform(20, 200, N)
RATE, VOL, LIFE = rng.uniform(-0.02, 0.15, N), rng.uniform(0.05, 0.9, N), rng.uniform(0.02, 5.0, N)
CALLS = rng.random(N) < 0.5

CALL = True BS = """namespace bs; lusine.version version = { { number = 1 } }; struct option { host bool call; host float<EUR> spot; host float<EUR> strike; host float<year^-1> rate; host float volatility; host float<year> period; host float<year> years; float option_delta; }; lattice<option, 1> options(spacing = 1 m, boundary = wall); void main() { options.option_delta = lusine.finance.option_delta(options.call, options.spot, options.strike, options.rate, options.volatility, options.period, options.years); } """ print(BS) def held(call, spot, strike, rate, volatility, years, period=1.0): """The parameters as the device holds them, in the host's units (euros, a year, years): each rounded to binary32 in the base units it is stored in.""" f = lambda v, scale=1.0: np.float32(np.asarray(v, dtype=float) * scale).astype(float) / scale return (np.asarray(call, dtype=bool), f(spot), f(strike), f(rate, 1 / YEAR), f(volatility), f(years, YEAR), f(period, YEAR)) def kernel(call, spot, strike, rate, volatility, years, period=1.0): """The kernel's option_delta for each option: prices in euros, the rate a year, the volatility over `period` years and the life in years, stored in the base units (seconds); the result back in dimensionless, one an option (a single option's alone).""" n = max(np.size(a) for a in (call, spot, strike, rate, volatility, years, period)) run = trait_run.Program(['finance/option_delta.trait'], BS, {'bs.options': [n]}) for field, v in (('call', call), ('spot', spot), ('strike', strike), ('rate', np.asarray(rate) / YEAR), ('volatility', volatility), ('period', np.asarray(period) * YEAR), ('years', np.asarray(years) * YEAR)): run.store('bs.options', field, np.broadcast_to(np.asarray(v), (n,))) out = run.steps('bs.main', 1, [('bs.options', 'option_delta')])[0][0] return out if n > 1 else float(out[0]) def closed(call, spot, strike, rate, volatility, years, period=1.0): """The independent account: Black-Scholes in float64 with scipy's exact normal distribution, at the parameters the device holds.""" c, S, K, r, v, T, P = held(call, spot, strike, rate, volatility, years, period) s = v * np.sqrt(T / P) a = (np.log(S / K) + (r + 0.5 * (v * v / P)) * T) / s b = a - s df = K * np.exp(-r * T) return np.where(c, ndtr(a), ndtr(a) - 1.0) rng = np.random.default_rng(391) N = 1000 SPOT, STRIKE = rng.uniform(20, 200, N), rng.uniform(20, 200, N) RATE, VOL, LIFE = rng.uniform(-0.02, 0.15, N), rng.uniform(0.05, 0.9, N), rng.uniform(0.02, 5.0, N) CALLS = rng.random(N) < 0.5

namespace bs;

lusine.version version = { { number = 1 } };

struct option {
  host bool call; host float<EUR> spot; host float<EUR> strike; host float<year^-1> rate; host float volatility;
  host float<year> period; host float<year> years;
  float option_delta;
};
lattice<option, 1> options(spacing = 1 m, boundary = wall);

void main()
{
  options.option_delta = lusine.finance.option_delta(options.call, options.spot, options.strike, options.rate, options.volatility,
                                    options.period, options.years);
}

Reference: Hull's example¶

Hull's printed values (Options, Futures, and Other Derivatives, 9th ed., 2014) are quoted from sources that were opened, the book itself not: Example 15.6 (S0 = 42, K = 40, r = 10%, sigma = 20%, six months) as QFRM's BS_Simple documentation cites it, and ch. 19's running example (S0 = 49, K = 50, r = 5%, sigma = 20%, 20 weeks) as the black-scholes-greeks README quotes it. Each is held to half a unit in the last digit printed.

In [4]:

d = kernel(True, 49.0, 50.0, 0.05, 0.20, 20 / 52)
assert abs(d - 0.522) <= 0.0005, d
print(f'ch. 19: delta {d:.4f} (0.522)')

d = kernel(True, 49.0, 50.0, 0.05, 0.20, 20 / 52) assert abs(d - 0.522) <= 0.0005, d print(f'ch. 19: delta {d:.4f} (0.522)')

ch. 19: delta 0.5216 (0.522)

Reference: scipy's Black-Scholes¶

A thousand options, calls and puts: each delta within 1e-6 of the float64 account.

In [5]:

d = kernel(CALLS, SPOT, STRIKE, RATE, VOL, LIFE)
err = np.abs(d - closed(CALLS, SPOT, STRIKE, RATE, VOL, LIFE))
assert np.max(err) <= 1e-6, np.max(err)
print(f'largest error {np.max(err):.2e}')

d = kernel(CALLS, SPOT, STRIKE, RATE, VOL, LIFE) err = np.abs(d - closed(CALLS, SPOT, STRIKE, RATE, VOL, LIFE)) assert np.max(err) <= 1e-6, np.max(err) print(f'largest error {np.max(err):.2e}')

largest error 2.18e-07

Invariant: a call's and a put's are one apart¶

A call's delta is in [0, 1], a put's in [-1, 0], and they differ by one, to binary32's rounding of one; delta rises with the spot.

In [6]:

c = kernel(True, SPOT, STRIKE, RATE, VOL, LIFE)
p = kernel(False, SPOT, STRIKE, RATE, VOL, LIFE)
assert np.all((c >= 0) & (c <= 1) & (p >= -1) & (p <= 0))
assert np.max(np.abs(c - p - 1.0)) <= EPS, np.max(np.abs(c - p - 1.0))
S = np.linspace(10.0, 150.0, 300)
up = kernel(True, S, 50.0, 0.05, 0.2, 1.0)
assert np.all(np.diff(up) >= 0)
fig, ax = plt.subplots(figsize=(7, 3.5))
ax.plot(S, up, label='call')
ax.plot(S, kernel(False, S, 50.0, 0.05, 0.2, 1.0), label='put')
ax.set_xlabel('spot (EUR)')
ax.set_ylabel('delta')
ax.legend()
plt.show()

c = kernel(True, SPOT, STRIKE, RATE, VOL, LIFE) p = kernel(False, SPOT, STRIKE, RATE, VOL, LIFE) assert np.all((c >= 0) & (c <= 1) & (p >= -1) & (p <= 0)) assert np.max(np.abs(c - p - 1.0)) <= EPS, np.max(np.abs(c - p - 1.0)) S = np.linspace(10.0, 150.0, 300) up = kernel(True, S, 50.0, 0.05, 0.2, 1.0) assert np.all(np.diff(up) >= 0) fig, ax = plt.subplots(figsize=(7, 3.5)) ax.plot(S, up, label='call') ax.plot(S, kernel(False, S, 50.0, 0.05, 0.2, 1.0), label='put') ax.set_xlabel('spot (EUR)') ax.set_ylabel('delta') ax.legend() plt.show()