Heston Model: Hedging Under Stochastic Volatility

13. Heston Model: Hedging Under Stochastic Volatility#

Extends the BS hedging analysis to Heston. The main questions:

What happens when you hedge with BS deltas but reality follows Heston?
Does using the “correct” Heston delta actually help?
What risk can’t you hedge away with stock alone?

13.1. Heston Model Recap#

Under Heston [Heston, 1993], stock and variance follow:

\[dS_t = r S_t \, dt + \sqrt{v_t} S_t \, dW_t^S\]

\[dv_t = \kappa(\theta - v_t) \, dt + \xi \sqrt{v_t} \, dW_t^v\]

with \(\text{Corr}(dW^S, dW^v) = \rho\).

The parameters: \(v_0\) is initial variance, \(\theta\) is long-run variance, \(\kappa\) controls mean-reversion speed, \(\xi\) is vol-of-vol, and \(\rho\) is typically negative (the leverage effect—when stock drops, vol tends to rise).

For hedging, the key difference from BS is that volatility is now a second source of randomness. Option prices depend on the current vol level, and that level moves around.

Model Parameters
========================================
Spot S0:          100
Strike K:         100
Time T:           0.25 years
Risk-free r:      0.05
Initial vol:      20.0%
Long-run vol:     20.0%
Vol-of-vol xi:    0.3
Correlation rho:  -0.7

13.2. Heston Greeks via COS Method#

Under Heston, Greeks need to be computed numerically. We use the COS method [Fang and Oosterlee, 2008]—analytical derivatives for delta/gamma, finite differences for vega and theta.

The important difference from BS: Heston delta depends on the current variance \(v_t\), not just spot. Even if spot doesn’t move, delta changes as vol moves around.

Greeks Comparison: BS vs Heston
==================================================
Greek                  BS       Heston       Diff %
--------------------------------------------------
Price            4.614997     4.587070       -0.61%
Delta            0.569460     0.614251        7.87%
Gamma            0.039288     0.038115       -2.99%
Theta          -10.474151   -10.361430        1.08%
Vega            19.644000    15.375729      -21.73%

../_images/c91dec74dd1756f6bb5e4fd3932dc5b574d8e78f1724ff1b32fcf1ed6817d501.png

13.3. Simulating Heston Paths#

We use Euler discretization:

\[S_{t+\Delta t} = S_t \exp\left[(r - \tfrac{1}{2}v_t)\Delta t + \sqrt{v_t}\sqrt{\Delta t}\, Z_1\right]\]

\[v_{t+\Delta t} = v_t + \kappa(\theta - v_t)\Delta t + \xi\sqrt{v_t}\sqrt{\Delta t}\, Z_2\]

where \(Z_1\) and \(Z_2\) are correlated normals with \(\text{Corr}(Z_1, Z_2) = \rho\).

../_images/0c3a8c9af14f461db81d5ba09d6b6d150c24688fe81728b5e71ad9b64b512e39.png

13.4. BS Delta vs Heston Delta#

Here’s the experiment: reality follows Heston, we sell an option and delta-hedge. We compare hedging with BS delta (using \(\sigma = \sqrt{v_t}\)) vs the full Heston delta.

Both methods observe the true variance at each hedge time. The question is whether knowing the full Heston dynamics—mean-reversion, vol-of-vol, correlation—actually improves the hedge.

We expect the unhedgeable vol risk to dominate the delta choice — let’s see.

Show code cell source

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def hedge_under_heston(S0, K, T, r, v0, kappa, theta, xi, rho, n_hedge, hedge_model='heston'):
    """ Delta-hedge a short call under Heston dynamics.
    hedge_model: 'heston' uses Heston delta, 'bs' uses BS delta with sqrt(v_t)
    """
    dt = T / n_hedge
    
    # Simulate Heston path
    times, S, v = simulate_heston_path(S0, v0, r, kappa, theta, xi, rho, T, n_hedge)
    
    # Initial option price (Heston price - this is what we receive)
    V0 = Heston_price(S0, K, T, r, kappa, v0, theta, xi, rho)
    
    # Get initial delta
    if hedge_model == 'heston':
        g = cos_greeks_full(S0, K, T, r, v0, model='heston', N=256, L=12,
                            heston_cf=Heston_CF, kappa=kappa, theta=theta, xi=xi, rho=rho)
        delta = g['delta']
    else:  # BS
        g = bs_greeks_analytical(S0, K, T, r, np.sqrt(v0), 'call')
        delta = g['delta']
    
    shares = delta
    cash = V0 - delta * S0
    
    portfolio_values = [V0]
    deltas_used = [delta]
    
    for i in range(1, n_hedge + 1):

        tau = T - times[i]
        cash *= np.exp(r * dt)
        
        portfolio_value = shares * S[i] + cash
        
        # Get new delta
        if tau > 1e-6:
            if hedge_model == 'heston':
                g = cos_greeks_full(S[i], K, tau, r, v[i], model='heston', N=256, L=12,
                                    heston_cf=Heston_CF, kappa=kappa, theta=theta, xi=xi, rho=rho)
                new_delta = g['delta']
            else:  # BS with current vol
                g = bs_greeks_analytical(S[i], K, tau, r, np.sqrt(v[i]), 'call')
                new_delta = g['delta']
        else:
            new_delta = 1.0 if S[i] > K else 0.0
        
        # Rebalance
        shares_traded = new_delta - shares
        cash -= shares_traded * S[i]
        shares = new_delta
        
        portfolio_values.append(portfolio_value)
        deltas_used.append(new_delta)
    
    # Final PnL
    payoff = max(S[-1] - K, 0)
    final_portfolio = shares * S[-1] + cash
    hedge_pnl = final_portfolio - payoff
    
    return {
        'times': times,
        'spot': S,
        'vol': v,
        'portfolio_values': np.array(portfolio_values),
        'deltas': np.array(deltas_used),
        'hedge_pnl': hedge_pnl,
        'payoff': payoff,
        'final_portfolio': final_portfolio
    }

Running hedge comparison...

Hedge PnL Statistics
============================================================
Metric                         BS Delta       Heston Delta
------------------------------------------------------------
Mean PnL                        -0.0424            -0.0433
Std PnL                          0.8005             0.8826
5th Percentile                  -1.5596            -1.6255
95th Percentile                  1.1424             1.1698

../_images/a6ad7567de13b37bef47931c55bb1beb5e79f3db20f0a637cca6ebce0bccf107.png

Key Finding:
  BS Delta std:     0.8005
  Heston Delta std: 0.8826
  Correlation:      0.9530

The high correlation shows both methods face the SAME underlying risk: vol moves.

PnL from both methods is highly correlated — the unhedgeable vol risk dominates the delta choice. Model choice matters more for exotics with strong path-dependence.

13.5. The Unhedgeable Risk: Vol-of-Vol#

Even with the correct Heston delta, risk remains. The option value depends on two state variables, \(V = V(S_t, v_t, t)\), and the delta-hedged PnL is:

\[d\Pi = \frac{\partial V}{\partial v} dv + \frac{1}{2}\frac{\partial^2 V}{\partial S^2}(dS)^2 + \frac{\partial V}{\partial t}dt\]

Stock hedges the \(dS\) risk. But the \(dv\) term—exposure to variance moves—requires other instruments: variance swaps, VIX options, or other vanilla options.

With only stock, the \(dv\) exposure remains. And this exposure scales with \(\xi\) (vol-of-vol): higher \(\xi\) means more variance in variance, means more unhedgeable risk.

Testing vol-of-vol impact...

  xi=0.1: std=0.5416

  xi=0.3: std=0.9037

  xi=0.5: std=1.3142

  xi=0.8: std=1.9244

../_images/d08c54da58b33068e45c90784e4d98bb732d0b2b28f6c81e2349f29d2cfd8e78.png

13.6. The Leverage Effect#

The correlation \(\rho\) between stock returns and variance creates the leverage effect: when stock drops, vol tends to rise (\(\rho < 0\)). This creates negatively-skewed returns—the left tail is fatter.

For hedging, this is bad news. Large down moves coincide with vol spikes, so your hedge deteriorates exactly when you need it most. The PnL distribution becomes negatively skewed.

Testing correlation impact...

  rho=-0.9: mean=-0.0420, std=0.8852

  rho=-0.5: mean=-0.0362, std=0.9147

  rho=+0.0: mean=-0.0349, std=0.9318

  rho=+0.5: mean=-0.0146, std=0.9282

../_images/d7f8029dec053b8adf9cb9c05db87ffbe9ebda55c08343296303938fd81934a5.png

For vanilla hedging, BS delta with current vol works nearly as well as Heston delta. The dominant risk is vol moves, which stock can’t hedge — need variance swaps or other options for that.

TODO: add vega hedging with a second option to quantify how much residual risk drops.