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Black–Scholes and delta hedging in production

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Start with the idea

Black–Scholes prices a replicable payoff under a set of idealised assumptions. The drift of the real-world stock forecast disappears from the price because replication replaces a directional forecast with no-arbitrage financing.

Symbols, units & horizon

C: European call value with no dividends in the displayed BS equation
S or S₀: spot
K: strike
r: continuously compounded annual risk-free rate
T: years remaining
σ: annualised return volatility
E_Q: expectation under the risk-neutral probability measure, not physical expected return
Z,Z₁,Z₂: standard normal draws
(x)⁺: max(x,0)
N or Φ: standard normal cumulative probability
φ: standard normal density
d₁,d₂: dimensionless normal thresholds
μ: price drift in the physical-measure Heston model
v: instantaneous return variance per year
κ: variance mean-reversion speed per year
θ: long-run variance
ξ: volatility of variance parameter
W_S,W_v: correlated Brownian motions
ρ: shock correlation
⟨ ⟩: quadratic covariation
v⁺: max(v,0) in the numerical scheme
Δ: call delta, sensitivity to spot
Γ: gamma, second spot derivative
Vega: option value change per unit decimal volatility
m(t): expected variance at time t in the ODE
v₀: initial variance

When and why to use this

Use the formula as a baseline for implied volatility, scenario sensitivities and option-price checks. The Heston extension explains how allowing variance to move introduces skew and additional risks.

C = S N (d_{1}) - K e^{- r T} N (d_{2}), d_{1,2} = \frac{\ln (\frac{S}{K}) + (r \pm \frac{1}{2} σ^{2}) T}{σ \sqrt{T}}

Risk-neutral valuation + Gaussian integral

Integrate the risk-neutral call payoff

For a European call without dividends, risk-neutral pricing gives $C = e^{- r T} E_{Q} [(S_{T} - K)^{+}]$ , with $S_{T} = S e^{(r - \frac{σ^{2}}{2}) T + σ \sqrt{T} Z}$ .
The exercise condition Sₜ>K is Z>−d₂, where $d_{2} = \frac{[\ln (\frac{S}{K}) + (r - \frac{σ^{2}}{2}) T]}{(σ \sqrt{T})}$ . Split the payoff expectation into a stock term and K times the exercise probability.
Use $e^{a Z} φ (Z) = e^{\frac{a^{2}}{2}} φ (Z - a)$ (complete the square). The stock term becomes $S N (d_{2} + σ \sqrt{T}) = S N (d_{1})$ ; the strike term is $K e^{- r T} N (d_{2})$ . Subtract.
Differentiate the result, using the normal-density cancellation, to get $Δ = N (d_{1})$ and $Γ = \frac{φ (d_{1})}{(S σ \sqrt{T})}$ . Vega per 1.00 volatility is $S φ (d_{1}) \sqrt{T}$ .

Work it by hand

S=K=100, r=.05, T=1, σ=.20: d₁=.35, d₂=.15; N≈.63683,.55962; C≈10.4506. These are risk-neutral pricing quantities.

$N (d_{2})$ is the risk-neutral probability of finishing in the money; $N (d_{1})$ is the delta. The formula is the solution of the PDE you get by plugging Itô's lemma into a hedged portfolio and insisting it earn the risk-free rate — which is why $μ$ does not appear: the hedge removed it.

Delta hedging uses the option model's local stock sensitivity to choose the hedge. With discrete rebalancing there is residual error. Under suitable diffusion, regularity and fine-grid assumptions, the standard deviation of the discretisation component can scale approximately with $\sqrt{rehedge interval}$ ; in that regime, a quarter of the interval gives about half that component's standard deviation. This is not a universal scaling law for total trading P&L. Jumps, near-expiry payoff kinks, changing parameters and transaction costs can materially change the result.

Greek	definition	what it tells the desk
Δ delta	$\frac{\partial V}{\partial S}$	shares to hold against one option
Γ gamma	$\frac{\partial^{2} V}{\partial S^{2}}$	how fast delta changes; convexity; peaks at the money near expiry
Θ theta	$\frac{\partial V}{\partial t}$	daily decay; the bill for gamma
ν vega	$\frac{\partial V}{\partial σ}$	exposure to implied vol; largest at the money, long-dated

At-the-money call, T → 0. Gamma goes…

$Γ = \frac{N^{'} (d_{1})}{(S σ \sqrt{T})}$ . As $T \to 0$ at the money, the denominator vanishes. Delta flips from 0 to 1 across the strike in an instant; hedging it is impossible. This is "pin risk", and it is why expiry days are dangerous for short-gamma books.

Heston: volatility is itself random

d S = μ S d t + \sqrt{v} S d W_{S}, d v = κ (θ - v) d t + ξ \sqrt{v} d W_{v}, ⟨ d W_{S}, d W_{v} ⟩ = ρ d t

Model specification + discretisation + ODE

Discretise stochastic variance and correlated shocks

Heston specifies variance rather than deriving it from price. Over Δt, a full-truncation Euler update is $v_{n e x t} = v + κ (θ - v^{+}) Δ t + ξ \sqrt{v^{+} Δ t} Z_{v}$ , with $v^{+} = \max (v, 0)$ .
For independent unit-normal Z₁,Z₂, set $Z_{S} = Z_{1}$ and $Z_{v} = ρ Z_{1} + \sqrt{1 - ρ^{2}} Z_{2}$ . Variance is ρ²+(1−ρ²)=1 and covariance with Zₛ is ρ.
Taking expectation of the variance equation removes the shock. Solving $\frac{d m}{d t} = κ (θ - m)$ gives $E [v_{T}] = θ + (v_{0} - θ) e^{- κ T}$ .

Work it by hand

ρ=−.7, Z₁=1,Z₂=0 gives Zᵥ=−.7. With κ=2, θ=.04, v₀=.09, T=1, expected variance=.04+.05e⁻²≈.04677.

Black–Scholes prices every strike with one σ; the market does not. Plot implied vol against strike and you get a smile (or, in equities, a skew — puts are dearer). Heston generates the smile from two extra ingredients: vol-of-vol $ξ$ fattens both tails, and correlation $ρ$ between price and vol tilts it. $ρ < 0$ (crashes come with vol spikes) produces the equity skew.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

from math import log, exp, sqrt
from statistics import NormalDist

def black_scholes_call(spot, strike, rate, volatility, years):
    if min(spot,strike,volatility,years) <= 0:
        raise ValueError("Positive S, K, sigma and T required")
    d1 = (log(spot/strike)+(rate+.5*volatility**2)*years)/(volatility*sqrt(years))
    d2 = d1-volatility*sqrt(years)
    normal = NormalDist()
    return spot*normal.cdf(d1)-strike*exp(-rate*years)*normal.cdf(d2)

def heston_step(spot, raw_variance, mu, kappa, theta, xi, rho, dt, z1, z2):
    """Full-truncation Euler variance; independent standard-normal z1, z2.
    Preserve raw variance for the next step; positive part is used in coefficients.
    Spot uses a log step. Discretisation has bias; this is not an exact sampler.
    """
    if not -1 <= rho <= 1 or dt < 0:
        raise ValueError("Invalid correlation or time step")
    v = max(raw_variance, 0)
    shock_s = sqrt(dt)*z1
    shock_v = sqrt(dt)*(rho*z1+sqrt(1-rho*rho)*z2)
    next_v = raw_variance+kappa*(theta-v)*dt+xi*sqrt(v)*shock_v
    next_s = spot*exp((mu-.5*v)*dt+sqrt(v)*shock_s)
    return next_s, next_v

print(black_scholes_call(100,100,.05,.2,1))
def bs_spot_vol_greeks(spot, strike, rate, volatility, years):
    d1 = (log(spot/strike)+(rate+.5*volatility**2)*years)/(volatility*sqrt(years))
    density = NormalDist().pdf(d1)
    return {"delta":NormalDist().cdf(d1),
            "gamma":density/(spot*volatility*sqrt(years)),
            "vega_per_unit":spot*density*sqrt(years)}

def heston_expected_variance(initial_variance, long_run_variance, speed, years):
    return long_run_variance+(initial_variance-long_run_variance)*exp(-speed*years)

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