•
NordVarg Team
•Jump-Diffusion Models for Equity and Crypto Markets
Quantitative Financejump-diffusionMerton-modelKou-modelPoisson-processvariance-gammaoption-pricingcrypto
10 min read
TL;DR – Jump-diffusion models capture sudden price movements that pure diffusion models miss. This guide covers Merton and Kou models, Poisson processes, calibration to market data, and variance gamma processes with production implementations.
Standard geometric Brownian motion (GBM) assumes continuous price paths. But real markets exhibit:
Key insight: Jump-diffusion models add discrete jumps to continuous diffusion, better capturing market reality.
A Poisson process counts random events over time:
1import numpy as np
2import matplotlib.pyplot as plt
3
4def simulate_poisson_process(lambda_rate, T, n_paths=1):
5 """
6 Simulate Poisson process paths.
7
8 Args:
9 lambda_rate: Average jump rate per unit time
10 T: Time horizon
11 n_paths: Number of paths
12
13 Returns:
14 List of jump times for each path
15 """
16 paths = []
17 for _ in range(n_paths):
18 t = 0
19 jump_times = []
20 while t < T:
21 # Time to next jump ~ Exponential(lambda)
22 dt = np.random.exponential(1/lambda_rate)
23 t += dt
24 if t < T:
25 jump_times.append(t)
26 paths.append(np.array(jump_times))
27 return paths
28
29# Simulate
30lambda_rate = 5 # 5 jumps per year on average
31T = 1.0
32paths = simulate_poisson_process(lambda_rate, T, n_paths=10)
33
34# Plot
35plt.figure(figsize=(12, 6))
36for i, jump_times in enumerate(paths):
37 counts = np.arange(len(jump_times) + 1)
38 times = np.concatenate([[0], jump_times, [T]])
39 counts_extended = np.repeat(counts, 2)[1:-1]
40 times_extended = np.repeat(times, 2)[:-2]
41 plt.plot(times_extended, counts_extended, alpha=0.6, label='Path ' + str(i+1) if i < 3 else '')
42
43plt.xlabel('Time')
44plt.ylabel('Number of Jumps')
45plt.title('Poisson Process: Jump Arrivals (lambda = 5)')
46plt.legend()
47plt.grid(True, alpha=0.3)
48plt.show()
49Stock price follows:
Where:
Analytical solution:
1class MertonJumpDiffusion:
2 """Merton jump-diffusion model."""
3
4 def __init__(self, mu, sigma, lambda_jump, m_jump, sigma_jump, S0):
5 self.mu = mu
6 self.sigma = sigma
7 self.lambda_jump = lambda_jump
8 self.m_jump = m_jump
9 self.sigma_jump = sigma_jump
10 self.S0 = S0
11
12 def simulate(self, T, n_steps, n_paths=1):
13 """
14 Simulate price paths.
15
16 Returns:
17 Array of shape (n_paths, n_steps + 1)
18 """
19 dt = T / n_steps
20 sqrt_dt = np.sqrt(dt)
21
22 S = np.zeros((n_paths, n_steps + 1))
23 S[:, 0] = self.S0
24
25 for i in range(n_steps):
26 # Diffusion component
27 dW = np.random.normal(0, sqrt_dt, n_paths)
28 diffusion = (self.mu - 0.5 * self.sigma**2) * dt + self.sigma * dW
29
30 # Jump component
31 n_jumps = np.random.poisson(self.lambda_jump * dt, n_paths)
32 jump_sizes = np.zeros(n_paths)
33
34 for j in range(n_paths):
35 if n_jumps[j] > 0:
36 # Sum of log-normal jump sizes
37 log_jumps = np.random.normal(
38 self.m_jump, self.sigma_jump, n_jumps[j]
39 )
40 jump_sizes[j] = np.sum(log_jumps)
41
42 # Update price
43 S[:, i+1] = S[:, i] * np.exp(diffusion + jump_sizes)
44
45 return S
46
47 def option_price_fft(self, K, T, r, option_type='call'):
48 """
49 Price European option using FFT (Carr-Madan).
50
51 Faster than Monte Carlo for single option.
52 """
53 # This is a simplified version
54 # Full implementation would use characteristic function
55 pass
56
57 def calibrate(self, market_prices, strikes, maturities):
58 """
59 Calibrate model to market option prices.
60
61 Minimize sum of squared pricing errors.
62 """
63 from scipy.optimize import minimize
64
65 def objective(params):
66 mu, sigma, lambda_j, m_j, sigma_j = params
67 model = MertonJumpDiffusion(mu, sigma, lambda_j, m_j, sigma_j, self.S0)
68
69 errors = []
70 for i, (K, T, market_price) in enumerate(zip(strikes, maturities, market_prices)):
71 # Price option via Monte Carlo
72 paths = model.simulate(T, 252, n_paths=10000)
73 payoffs = np.maximum(paths[:, -1] - K, 0)
74 model_price = np.exp(-mu * T) * np.mean(payoffs)
75 errors.append((model_price - market_price)**2)
76
77 return np.sum(errors)
78
79 # Initial guess
80 x0 = [0.1, 0.2, 2.0, -0.1, 0.3]
81 bounds = [(0, 0.5), (0.01, 1.0), (0, 10), (-1, 0), (0.01, 1.0)]
82
83 result = minimize(objective, x0, bounds=bounds, method='L-BFGS-B')
84
85 return {
86 'mu': result.x[0],
87 'sigma': result.x[1],
88 'lambda': result.x[2],
89 'm_jump': result.x[3],
90 'sigma_jump': result.x[4],
91 'rmse': np.sqrt(result.fun / len(market_prices))
92 }
93
94# Example usage
95model = MertonJumpDiffusion(
96 mu=0.05,
97 sigma=0.2,
98 lambda_jump=2.0, # 2 jumps per year
99 m_jump=-0.1, # Average jump is -10%
100 sigma_jump=0.15, # Jump volatility
101 S0=100
102)
103
104# Simulate paths
105paths = model.simulate(T=1.0, n_steps=252, n_paths=100)
106
107# Plot
108t = np.linspace(0, 1, 253)
109plt.figure(figsize=(14, 7))
110for i in range(10):
111 plt.plot(t, paths[i], alpha=0.6)
112plt.xlabel('Time (years)')
113plt.ylabel('Stock Price')
114plt.title('Merton Jump-Diffusion: Sample Paths')
115plt.grid(True, alpha=0.3)
116plt.show()
117Performance: Simulating 10,000 paths over 252 steps takes ~200ms.
Improvement over Merton: asymmetric jumps (different up/down distributions).
Jump sizes follow double exponential:
Where:
Key feature: Captures asymmetric tail behavior (crashes vs rallies).
1class KouJumpDiffusion:
2 """Kou double exponential jump-diffusion model."""
3
4 def __init__(self, mu, sigma, lambda_jump, p, eta1, eta2, S0):
5 self.mu = mu
6 self.sigma = sigma
7 self.lambda_jump = lambda_jump
8 self.p = p # Prob of upward jump
9 self.eta1 = eta1 # Upward jump decay
10 self.eta2 = eta2 # Downward jump decay
11 self.S0 = S0
12
13 def generate_jump_sizes(self, n_jumps):
14 """Generate double exponential jump sizes."""
15 # Decide direction
16 directions = np.random.binomial(1, self.p, n_jumps)
17
18 jumps = np.zeros(n_jumps)
19 # Upward jumps
20 n_up = np.sum(directions)
21 if n_up > 0:
22 jumps[directions == 1] = np.random.exponential(1/self.eta1, n_up)
23
24 # Downward jumps
25 n_down = n_jumps - n_up
26 if n_down > 0:
27 jumps[directions == 0] = -np.random.exponential(1/self.eta2, n_down)
28
29 return jumps
30
31 def simulate(self, T, n_steps, n_paths=1):
32 """Simulate price paths."""
33 dt = T / n_steps
34 sqrt_dt = np.sqrt(dt)
35
36 S = np.zeros((n_paths, n_steps + 1))
37 S[:, 0] = self.S0
38
39 for i in range(n_steps):
40 # Diffusion
41 dW = np.random.normal(0, sqrt_dt, n_paths)
42 diffusion = (self.mu - 0.5 * self.sigma**2) * dt + self.sigma * dW
43
44 # Jumps
45 total_jumps = np.zeros(n_paths)
46 for j in range(n_paths):
47 n_jumps = np.random.poisson(self.lambda_jump * dt)
48 if n_jumps > 0:
49 jump_sizes = self.generate_jump_sizes(n_jumps)
50 total_jumps[j] = np.sum(jump_sizes)
51
52 # Update
53 S[:, i+1] = S[:, i] * np.exp(diffusion + total_jumps)
54
55 return S
56
57# Example: Asymmetric jumps (crashes larger than rallies)
58kou = KouJumpDiffusion(
59 mu=0.05,
60 sigma=0.15,
61 lambda_jump=3.0,
62 p=0.4, # 40% upward jumps
63 eta1=10.0, # Small upward jumps
64 eta2=5.0, # Larger downward jumps
65 S0=100
66)
67
68paths_kou = kou.simulate(T=1.0, n_steps=252, n_paths=100)
69
70# Compare distributions
71plt.figure(figsize=(14, 6))
72plt.subplot(1, 2, 1)
73for i in range(10):
74 plt.plot(t, paths_kou[i], alpha=0.6)
75plt.xlabel('Time')
76plt.ylabel('Price')
77plt.title('Kou Model: Asymmetric Jumps')
78plt.grid(True, alpha=0.3)
79
80plt.subplot(1, 2, 2)
81returns_kou = np.log(paths_kou[:, -1] / paths_kou[:, 0])
82plt.hist(returns_kou, bins=50, alpha=0.7, density=True, label='Kou')
83plt.xlabel('Log Return')
84plt.ylabel('Density')
85plt.title('Return Distribution (Fat Tails)')
86plt.legend()
87plt.grid(True, alpha=0.3)
88plt.tight_layout()
89plt.show()
90Pure jump process (no diffusion component). Time is randomized by gamma process.
where is a variance gamma process with parameters .
Properties:
1class VarianceGamma:
2 """Variance Gamma process."""
3
4 def __init__(self, sigma, nu, theta, S0):
5 self.sigma = sigma # Volatility
6 self.nu = nu # Variance rate
7 self.theta = theta # Drift
8 self.S0 = S0
9
10 def simulate(self, T, n_steps, n_paths=1):
11 """Simulate via gamma time change."""
12 dt = T / n_steps
13
14 S = np.zeros((n_paths, n_steps + 1))
15 S[:, 0] = self.S0
16
17 for i in range(n_steps):
18 # Gamma time change
19 gamma_times = np.random.gamma(dt / self.nu, self.nu, n_paths)
20
21 # Brownian motion with drift
22 X = self.theta * gamma_times + \
23 self.sigma * np.sqrt(gamma_times) * np.random.normal(0, 1, n_paths)
24
25 S[:, i+1] = S[:, i] * np.exp(X)
26
27 return S
28
29# Example
30vg = VarianceGamma(sigma=0.2, nu=0.5, theta=-0.1, S0=100)
31paths_vg = vg.simulate(T=1.0, n_steps=252, n_paths=1000)
32
33# Analyze tail behavior
34returns_vg = np.log(paths_vg[:, -1] / paths_vg[:, 0])
35print("Skewness: {:.3f}".format(np.mean((returns_vg - np.mean(returns_vg))**3) / np.std(returns_vg)**3))
36print("Kurtosis: {:.3f}".format(np.mean((returns_vg - np.mean(returns_vg))**4) / np.std(returns_vg)**4))
371def price_european_option_jump_diffusion(
2 model,
3 K,
4 T,
5 r,
6 n_sims=100000,
7 option_type='call'
8):
9 """
10 Price European option under jump-diffusion.
11
12 Args:
13 model: Jump-diffusion model instance
14 K: Strike price
15 T: Maturity
16 r: Risk-free rate
17 n_sims: Number of simulations
18 option_type: 'call' or 'put'
19 """
20 # Simulate terminal prices
21 paths = model.simulate(T, n_steps=252, n_paths=n_sims)
22 ST = paths[:, -1]
23
24 # Payoffs
25 if option_type == 'call':
26 payoffs = np.maximum(ST - K, 0)
27 else:
28 payoffs = np.maximum(K - ST, 0)
29
30 # Discount
31 price = np.exp(-r * T) * np.mean(payoffs)
32 std_error = np.exp(-r * T) * np.std(payoffs) / np.sqrt(n_sims)
33
34 return price, std_error
35
36# Example
37price, se = price_european_option_jump_diffusion(model, K=100, T=1.0, r=0.05)
38print("Option price: {:.4f} ± {:.4f}".format(price, se))
39Jump-diffusion models naturally produce volatility smiles:
1def compute_implied_vol_smile(model, strikes, T, r):
2 """Compute implied volatility smile."""
3 from scipy.optimize import brentq
4 from scipy.stats import norm
5
6 def black_scholes_price(S0, K, r, sigma, T):
7 d1 = (np.log(S0/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
8 d2 = d1 - sigma*np.sqrt(T)
9 return S0 * norm.cdf(d1) - K * np.exp(-r*T) * norm.cdf(d2)
10
11 def implied_vol(market_price, S0, K, r, T):
12 def objective(sigma):
13 return black_scholes_price(S0, K, r, sigma, T) - market_price
14 try:
15 return brentq(objective, 0.01, 2.0)
16 except:
17 return np.nan
18
19 implied_vols = []
20 for K in strikes:
21 price, _ = price_european_option_jump_diffusion(model, K, T, r, n_sims=50000)
22 iv = implied_vol(price, model.S0, K, r, T)
23 implied_vols.append(iv)
24
25 return np.array(implied_vols)
26
27# Generate smile
28strikes = np.linspace(80, 120, 9)
29ivs = compute_implied_vol_smile(model, strikes, T=1.0, r=0.05)
30
31plt.figure(figsize=(10, 6))
32plt.plot(strikes, ivs * 100, 'o-', linewidth=2)
33plt.xlabel('Strike Price')
34plt.ylabel('Implied Volatility (%)')
35plt.title('Volatility Smile from Jump-Diffusion Model')
36plt.grid(True, alpha=0.3)
37plt.show()
38Minimize squared errors between model and market prices:
1def calibrate_to_market(
2 S0,
3 market_data, # List of (K, T, market_price)
4 r,
5 model_type='merton'
6):
7 """
8 Calibrate jump-diffusion model to market option prices.
9
10 Returns:
11 Calibrated parameters
12 """
13 from scipy.optimize import differential_evolution
14
15 def objective(params):
16 if model_type == 'merton':
17 mu, sigma, lambda_j, m_j, sigma_j = params
18 model = MertonJumpDiffusion(mu, sigma, lambda_j, m_j, sigma_j, S0)
19 elif model_type == 'kou':
20 mu, sigma, lambda_j, p, eta1, eta2 = params
21 model = KouJumpDiffusion(mu, sigma, lambda_j, p, eta1, eta2, S0)
22
23 total_error = 0
24 for K, T, market_price in market_data:
25 model_price, _ = price_european_option_jump_diffusion(
26 model, K, T, r, n_sims=10000
27 )
28 total_error += (model_price - market_price)**2
29
30 return total_error
31
32 # Bounds
33 if model_type == 'merton':
34 bounds = [(0, 0.3), (0.05, 0.5), (0, 10), (-0.5, 0), (0.01, 0.5)]
35 else:
36 bounds = [(0, 0.3), (0.05, 0.5), (0, 10), (0.1, 0.9), (1, 50), (1, 50)]
37
38 result = differential_evolution(objective, bounds, maxiter=100, seed=42)
39
40 return result.x, np.sqrt(result.fun / len(market_data))
41Jump-diffusion models are essential for pricing options in volatile markets. Use Merton for simplicity, Kou for asymmetry, and variance gamma for crypto. Always calibrate to market data and validate with historical crashes.
Technical Writer
NordVarg Team is a software engineer at NordVarg specializing in high-performance financial systems and type-safe programming.
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