Gamma: a European option's delta against its stock's price¶

A European option's gamma, its delta's change with the stock's price, d2V/dS2 = phi(d1) / (S s), with s = sigma sqrt(T / P) the volatility over the option's life (Hull's S0 sigma sqrt(T)); a call's and a put's are one.

Per the money unit of S and K (the kernel is generic over it). 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_gamma.trait

namespace lusine.finance;

float<_^-1> option_gamma(float<_> spot, float<_> strike, float<year^-1> rate, float volatility, float<year> period,
                         float<year> years)
{
  let a = d1(spot, strike, rate, volatility, period, years);
  return normal_pdf(x = a) / (spot * (volatility * sqrt(years / period)));
}

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

Phi(d1) / (S sigma sqrt(T)), a call's and a put's alike.

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_gamma.trait'))

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

// lusine.finance.option_gamma: a European option's gamma, its delta's change with the stock's price, d2V/dS2 =
// phi(d1) / (S s), with s = sigma sqrt(T / P) the volatility over the option's life (Hull's S0 sigma sqrt(T)); a call's
// and a put's are one. Per the money unit of S and K (the kernel is generic over it).
// 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<_^-1> option_gamma(float<_> spot, float<_> strike, float<year^-1> rate, float volatility, float<year> period,
                         float<year> years)
{
  let a = d1(spot, strike, rate, volatility, period, years);
  return normal_pdf(x = a) / (spot * (volatility * sqrt(years / period)));
}

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 gamma is per euro. 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 = False
BS = """namespace bs;

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

struct option {
  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<EUR^-1> option_gamma;
};
lattice<option, 1> options(spacing = 1 m, boundary = wall);

void main()
{
  options.option_gamma = lusine.finance.option_gamma(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_gamma 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 per euro, 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_gamma.trait'], BS, {'bs.options': [n]})
    for field, v in (('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_gamma')])[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 norm.pdf(a) / (S * s)


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 = False BS = """namespace bs; lusine.version version = { { number = 1 } }; struct option { 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<EUR^-1> option_gamma; }; lattice<option, 1> options(spacing = 1 m, boundary = wall); void main() { options.option_gamma = lusine.finance.option_gamma(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_gamma 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 per euro, 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_gamma.trait'], BS, {'bs.options': [n]}) for field, v in (('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_gamma')])[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 norm.pdf(a) / (S * s) 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 float<EUR> spot; host float<EUR> strike; host float<year^-1> rate; host float volatility;
  host float<year> period; host float<year> years;
  float<EUR^-1> option_gamma;
};
lattice<option, 1> options(spacing = 1 m, boundary = wall);

void main()
{
  options.option_gamma = lusine.finance.option_gamma(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]:

g = kernel(None, 49.0, 50.0, 0.05, 0.20, 20 / 52)
assert abs(g - 0.066) <= 0.0005, g
print(f'ch. 19: gamma {g:.4f} (0.066)')

g = kernel(None, 49.0, 50.0, 0.05, 0.20, 20 / 52) assert abs(g - 0.066) <= 0.0005, g print(f'ch. 19: gamma {g:.4f} (0.066)')

ch. 19: gamma 0.0655 (0.066)

Reference: scipy's Black-Scholes¶

A thousand options: each gamma within 1e-5 per spot of the float64 account, and within 1e-5 of it relative where gamma S is above 1e-3.

In [5]:

g = kernel(None, SPOT, STRIKE, RATE, VOL, LIFE)
want = closed(True, SPOT, STRIKE, RATE, VOL, LIFE)
err = np.abs(g - want) * SPOT
assert np.max(err) <= 1e-5, np.max(err)
big = want * SPOT > 1e-3
assert np.max(np.abs(g[big] / want[big] - 1)) <= 1e-5, np.max(np.abs(g[big] / want[big] - 1))
print(f'largest error {np.max(err):.2e} per spot')

g = kernel(None, SPOT, STRIKE, RATE, VOL, LIFE) want = closed(True, SPOT, STRIKE, RATE, VOL, LIFE) err = np.abs(g - want) * SPOT assert np.max(err) <= 1e-5, np.max(err) big = want * SPOT > 1e-3 assert np.max(np.abs(g[big] / want[big] - 1)) <= 1e-5, np.max(np.abs(g[big] / want[big] - 1)) print(f'largest error {np.max(err):.2e} per spot')

largest error 1.13e-06 per spot

Invariant: where it peaks, and the life¶

Gamma is positive and highest where d1 = -sigma sqrt(T), at S = K e^(-(r + 3 sigma^2 / 2) T), to the grid's step; at the strike it grows as the life shortens, and far from it it falls.

In [6]:

S = np.linspace(20.0, 100.0, 401)
g1, g01 = kernel(None, S, 50.0, 0.05, 0.2, 1.0), kernel(None, S, 50.0, 0.05, 0.2, 0.1)
peak = 50.0 * np.exp(-(0.05 + 1.5 * 0.2 ** 2) * 1.0)
assert np.all(g1 > 0) and abs(S[np.argmax(g1)] - peak) <= S[1] - S[0], (S[np.argmax(g1)], peak)
at = np.argmin(np.abs(S - 50.0))
assert g01[at] > g1[at] and g01[0] < g1[0]
fig, ax = plt.subplots(figsize=(7, 3.5))
ax.plot(S, g1, label='a year left')
ax.plot(S, g01, label='a tenth of a year')
ax.set_xlabel('spot (EUR)')
ax.set_ylabel('gamma (per EUR)')
ax.legend()
plt.show()

S = np.linspace(20.0, 100.0, 401) g1, g01 = kernel(None, S, 50.0, 0.05, 0.2, 1.0), kernel(None, S, 50.0, 0.05, 0.2, 0.1) peak = 50.0 * np.exp(-(0.05 + 1.5 * 0.2 ** 2) * 1.0) assert np.all(g1 > 0) and abs(S[np.argmax(g1)] - peak) <= S[1] - S[0], (S[np.argmax(g1)], peak) at = np.argmin(np.abs(S - 50.0)) assert g01[at] > g1[at] and g01[0] < g1[0] fig, ax = plt.subplots(figsize=(7, 3.5)) ax.plot(S, g1, label='a year left') ax.plot(S, g01, label='a tenth of a year') ax.set_xlabel('spot (EUR)') ax.set_ylabel('gamma (per EUR)') ax.legend() plt.show()