Monte Carlo Exotic Options

This example demonstrates pricing exotic options using Monte Carlo simulation with various variance reduction techniques.

Setup

julia

using QuantNova

Basic Monte Carlo Pricing

European Options (Benchmark)

Start with European options to verify against Black-Scholes:

julia

S0 = 100.0   # Initial spot
K = 100.0    # Strike
T = 1.0      # Time to maturity
r = 0.05     # Risk-free rate
σ = 0.20     # Volatility

dynamics = GBMDynamics(r, σ)

# Monte Carlo price
result = mc_price(S0, T, EuropeanCall(K), dynamics;
    npaths = 100000,
    nsteps = 252,
    antithetic = true
)

# Black-Scholes benchmark
bs_price = black_scholes(S0, K, T, r, σ, :call)

println("European Call (K=100):")
println("  Monte Carlo: \$$(round(result.price, digits=4)) ± $(round(result.stderr, digits=4))")
println("  Black-Scholes: \$$(round(bs_price, digits=4))")
println("  Error: $(round(abs(result.price - bs_price), digits=4))")
println("  95% CI: [\$$(round(result.ci_lower, digits=4)), \$$(round(result.ci_upper, digits=4))]")

Output:

European Call (K=100):
  Monte Carlo: 10.4477 +/- 0.0329
  Black-Scholes: 10.4506
  Error: 0.0029
  95% CI: [10.3832, 10.5122]

Asian Options

Asian options depend on the average price over the life of the option:

julia

# Asian call: max(avg(S) - K, 0)
asian_call = AsianCall(100.0)
asian_put = AsianPut(100.0)

result_call = mc_price(S0, T, asian_call, dynamics;
    npaths = 100000,
    nsteps = 252  # Daily averaging
)

result_put = mc_price(S0, T, asian_put, dynamics;
    npaths = 100000,
    nsteps = 252
)

println("\nAsian Options (K=100, daily averaging):")
println("  Call: \$$(round(result_call.price, digits=4)) ± $(round(result_call.stderr, digits=4))")
println("  Put:  \$$(round(result_put.price, digits=4)) ± $(round(result_put.stderr, digits=4))")

# Compare to European
eu_call = mc_price(S0, T, EuropeanCall(100.0), dynamics; npaths=100000)
println("\n  Asian call is $(round((1 - result_call.price/eu_call.price)*100, digits=1))% cheaper than European")
println("  (Averaging reduces volatility exposure)")

Output:

Asian Options (K=100, daily averaging):
  Call: 5.7374 +/- 0.0246
  Put:  3.2851 +/- 0.0171

  Asian call is 45.1% cheaper than European
  (Averaging reduces volatility exposure)

Averaging Frequency Impact

julia

println("\nImpact of Averaging Frequency:")
for nsteps in [12, 52, 252]
    freq = nsteps == 12 ? "Monthly" : (nsteps == 52 ? "Weekly" : "Daily")
    result = mc_price(S0, T, AsianCall(100.0), dynamics;
        npaths=50000, nsteps=nsteps)
    println("  $freq: \$$(round(result.price, digits=4))")
end

Output:

Impact of Averaging Frequency:
  Monthly: 5.6694
  Weekly: 5.7589
  Daily: 5.7062

Barrier Options

Barrier options are knocked out (or in) when the price crosses a barrier:

julia

# Up-and-out call: knocked out if S ≥ barrier
K = 100.0
B = 130.0  # Barrier

up_out_call = UpAndOutCall(K, B)

result = mc_price(S0, T, up_out_call, dynamics;
    npaths = 100000,
    nsteps = 252
)

println("\nUp-and-Out Call (K=100, B=130):")
println("  Price: \$$(round(result.price, digits=4)) ± $(round(result.stderr, digits=4))")

# Compare to vanilla
vanilla = mc_price(S0, T, EuropeanCall(K), dynamics; npaths=100000)
println("  European Call: \$$(round(vanilla.price, digits=4))")
println("  Knockout discount: $(round((1 - result.price/vanilla.price)*100, digits=1))%")

Output:

Up-and-Out Call (K=100, B=130):
  Price: 3.5972 +/- 0.0289
  European Call: 10.4506
  Knockout discount: 65.6%

Down-and-Out Put

julia

# Down-and-out put: knocked out if S ≤ barrier
K = 100.0
B = 80.0  # Barrier

down_out_put = DownAndOutPut(K, B)

result = mc_price(S0, T, down_out_put, dynamics;
    npaths = 100000,
    nsteps = 252
)

println("\nDown-and-Out Put (K=100, B=80):")
println("  Price: \$$(round(result.price, digits=4))")

Output:

Down-and-Out Put (K=100, B=80):
  Price: 1.7633

Barrier Monitoring Frequency

julia

println("\nBarrier Monitoring Frequency Impact:")
println("(More monitoring = more chances to hit barrier = lower price)")
for nsteps in [12, 52, 252, 504]
    result = mc_price(S0, T, UpAndOutCall(100.0, 120.0), dynamics;
        npaths = 50000,
        nsteps = nsteps
    )
    println("  $nsteps steps/year: \$$(round(result.price, digits=4))")
end

Output:

Barrier Monitoring Frequency Impact:
(More monitoring = more chances to hit barrier = lower price)
  12 steps/year: 1.8603
  52 steps/year: 1.4968
  252 steps/year: 1.3657
  504 steps/year: 1.3002

American Options (Longstaff-Schwartz)

American options can be exercised early. The Longstaff-Schwartz algorithm uses regression to estimate continuation values:

julia

# American put (early exercise is valuable when ITM)
am_put = AmericanPut(100.0)

result = lsm_price(S0, T, am_put, dynamics;
    npaths = 50000,
    nsteps = 50  # Exercise dates
)

println("\nAmerican Put (K=100):")
println("  Price: \$$(round(result.price, digits=4))")

# European put for comparison
eu_put = mc_price(S0, T, EuropeanPut(100.0), dynamics; npaths=50000)
premium = result.price - eu_put.price
println("  European Put: \$$(round(eu_put.price, digits=4))")
println("  Early Exercise Premium: \$$(round(premium, digits=4)) ($(round(premium/eu_put.price*100, digits=1))%)")

Output:

American Put (K=100):
  Price: 6.0647
  European Put: 5.5873
  Early Exercise Premium: 0.4774 (8.5%)

American vs European by Moneyness

julia

println("\nEarly Exercise Premium by Moneyness:")
println("Strike | American | European | Premium")
println("-"^45)
for K in [80.0, 90.0, 100.0, 110.0, 120.0]
    am = lsm_price(S0, T, AmericanPut(K), dynamics; npaths=30000, nsteps=50)
    eu = mc_price(S0, T, EuropeanPut(K), dynamics; npaths=30000)
    premium = am.price - eu.price
    pct = eu.price > 0.01 ? premium / eu.price * 100 : 0.0
    println("$(Int(K))     | $(round(am.price, digits=2))      | $(round(eu.price, digits=2))      | $(round(premium, digits=2)) ($(round(pct, digits=1))%)")
end

Variance Reduction Techniques

Antithetic Variates

Uses negatively correlated path pairs to reduce variance:

julia

# Without antithetic
result_no_anti = mc_price(S0, T, EuropeanCall(100.0), dynamics;
    npaths = 50000,
    antithetic = false
)

# With antithetic
result_anti = mc_price(S0, T, EuropeanCall(100.0), dynamics;
    npaths = 50000,  # Same number of paths
    antithetic = true
)

println("\nAntithetic Variates Impact:")
println("  Without: \$$(round(result_no_anti.price, digits=4)) ± $(round(result_no_anti.stderr, digits=4))")
println("  With:    \$$(round(result_anti.price, digits=4)) ± $(round(result_anti.stderr, digits=4))")
println("  Variance reduction: $(round((1 - (result_anti.stderr/result_no_anti.stderr)^2)*100, digits=1))%")

Quasi-Monte Carlo (Sobol Sequences)

QMC uses low-discrepancy sequences for better convergence:

julia

# Standard MC
result_mc = mc_price(S0, T, EuropeanCall(100.0), dynamics;
    npaths = 10000,
    nsteps = 50
)

# Quasi-Monte Carlo
result_qmc = mc_price_qmc(S0, T, EuropeanCall(100.0), dynamics;
    npaths = 10000,
    nsteps = 50
)

println("\nQMC vs Standard MC (10,000 paths):")
println("  Standard MC: \$$(round(result_mc.price, digits=4)) ± $(round(result_mc.stderr, digits=4))")
println("  QMC:         \$$(round(result_qmc.price, digits=4))")

Convergence Comparison

julia

println("\nConvergence Analysis:")
println("Paths  | MC Error  | QMC Error | MC Stderr")
println("-"^50)

bs_price = black_scholes(S0, K, T, r, σ, :call)

for npaths in [1000, 5000, 10000, 50000]
    mc = mc_price(S0, T, EuropeanCall(K), dynamics; npaths=npaths, nsteps=50)
    qmc = mc_price_qmc(S0, T, EuropeanCall(K), dynamics; npaths=npaths, nsteps=50)

    mc_err = abs(mc.price - bs_price)
    qmc_err = abs(qmc.price - bs_price)

    println("$(lpad(npaths, 6)) | $(round(mc_err, digits=4))    | $(round(qmc_err, digits=4))    | $(round(mc.stderr, digits=4))")
end

Stochastic Volatility (Heston)

The Heston model has stochastic variance:

julia

# Heston parameters
heston = HestonDynamics(
    0.05,   # r - risk-free rate
    0.04,   # v0 - initial variance
    2.0,    # κ - mean reversion speed
    0.04,   # θ - long-term variance
    0.3,    # ξ - vol of vol
    -0.7    # ρ - correlation (negative = leverage effect)
)

# MC price under Heston
result = mc_price(S0, T, EuropeanCall(100.0), heston;
    npaths = 50000,
    nsteps = 252
)

# Compare to semi-analytical Heston
params = HestonParams(0.04, 0.04, 2.0, 0.3, -0.7)
analytical = heston_price(S0, 100.0, T, 0.05, params, :call)

println("\nHeston Model European Call:")
println("  Monte Carlo: \$$(round(result.price, digits=4)) ± $(round(result.stderr, digits=4))")
println("  Semi-analytical: \$$(round(analytical, digits=4))")

Heston Smile

julia

println("\nHeston Implied Volatility Smile:")
println("Strike | MC Price | Implied Vol")
println("-"^40)

for K in [80.0, 90.0, 100.0, 110.0, 120.0]
    result = mc_price(S0, T, EuropeanCall(K), heston; npaths=30000, nsteps=100)

    # Back out implied vol
    iv = implied_vol_search(result.price, S0, K, T, r)
    println("$(Int(K))     | \$$(round(result.price, digits=3))   | $(round(iv*100, digits=2))%")
end

# Helper function
function implied_vol_search(price, S, K, T, r; tol=1e-6)
    σ_low, σ_high = 0.01, 2.0
    for _ in 1:100
        σ_mid = (σ_low + σ_high) / 2
        model_price = black_scholes(S, K, T, r, σ_mid, :call)
        if abs(model_price - price) < tol
            return σ_mid
        elseif model_price > price
            σ_high = σ_mid
        else
            σ_low = σ_mid
        end
    end
    return (σ_low + σ_high) / 2
end

Monte Carlo Greeks

Compute sensitivities via pathwise differentiation:

julia

using Enzyme  # Or ForwardDiff

dynamics = GBMDynamics(0.05, 0.20)
payoff = EuropeanCall(100.0)

# Delta only
delta = mc_delta(S0, T, payoff, dynamics;
    npaths = 10000,
    nsteps = 50,
    backend = EnzymeBackend()
)

# Delta and Vega together
greeks = mc_greeks(S0, T, payoff, dynamics;
    npaths = 10000,
    nsteps = 50,
    backend = EnzymeBackend()
)

println("\nMonte Carlo Greeks:")
println("  Delta: $(round(delta, digits=4))")
println("  Vega:  $(round(greeks.vega, digits=4))")

# Compare to analytical
bs_delta, _, bs_vega, _, _ = black_scholes_greeks(S0, K, T, r, σ, :call)
println("\nBlack-Scholes Greeks:")
println("  Delta: $(round(bs_delta, digits=4))")
println("  Vega:  $(round(bs_vega, digits=4))")

Path Simulation

For custom payoffs or analysis, simulate paths directly:

julia

# Simulate a single GBM path
dynamics = GBMDynamics(0.05, 0.20)
path = QuantNova.MonteCarlo.simulate_gbm(S0, T, 252, dynamics)

println("\nSimulated GBM Path:")
println("  Initial: \$$(round(path[1], digits=2))")
println("  Final:   \$$(round(path[end], digits=2))")
println("  Min:     \$$(round(minimum(path), digits=2))")
println("  Max:     \$$(round(maximum(path), digits=2))")

# Antithetic pair
path1, path2 = QuantNova.MonteCarlo.simulate_gbm_antithetic(S0, T, 252, dynamics)
println("\nAntithetic Pair:")
println("  Path 1 final: \$$(round(path1[end], digits=2))")
println("  Path 2 final: \$$(round(path2[end], digits=2))")

# QMC path (deterministic)
Z = sobol_normals(252, 1)
path_qmc = QuantNova.MonteCarlo.simulate_gbm_qmc(S0, T, 252, dynamics, Z[1, :])
println("\nQMC Path (deterministic):")
println("  Final: \$$(round(path_qmc[end], digits=2))")

Custom Payoffs

Implement custom exotic payoffs:

julia

# Example: Lookback call (payoff = S_max - K)
function price_lookback_call(S0, T, K, dynamics; npaths=50000, nsteps=252)
    total_payoff = 0.0

    for _ in 1:npaths
        path = QuantNova.MonteCarlo.simulate_gbm(S0, T, nsteps, dynamics)
        S_max = maximum(path)
        payoff = max(S_max - K, 0)
        total_payoff += payoff
    end

    r = dynamics.r
    return exp(-r * T) * total_payoff / npaths
end

lookback_price = price_lookback_call(S0, T, 100.0, dynamics)
println("\nLookback Call (K=100):")
println("  Price: \$$(round(lookback_price, digits=4))")

# Compare to vanilla
vanilla = black_scholes(S0, 100.0, T, 0.05, 0.20, :call)
println("  European Call: \$$(round(vanilla, digits=4))")
println("  Lookback premium: $(round((lookback_price/vanilla - 1)*100, digits=1))%")

Next Steps

Option Pricing - Black-Scholes basics
Portfolio Risk - Risk management
Monte Carlo Manual - Full reference

Monte Carlo Exotic Options ​

Setup ​

Basic Monte Carlo Pricing ​

European Options (Benchmark) ​

Asian Options ​

Averaging Frequency Impact ​

Barrier Options ​

Down-and-Out Put ​

Barrier Monitoring Frequency ​

American Options (Longstaff-Schwartz) ​

American vs European by Moneyness ​

Variance Reduction Techniques ​

Antithetic Variates ​

Quasi-Monte Carlo (Sobol Sequences) ​

Convergence Comparison ​

Stochastic Volatility (Heston) ​

Heston Smile ​

Monte Carlo Greeks ​

Path Simulation ​

Custom Payoffs ​

Next Steps ​

Monte Carlo Exotic Options

Setup

Basic Monte Carlo Pricing

European Options (Benchmark)

Asian Options

Averaging Frequency Impact

Barrier Options

Down-and-Out Put

Barrier Monitoring Frequency

American Options (Longstaff-Schwartz)

American vs European by Moneyness

Variance Reduction Techniques

Antithetic Variates

Quasi-Monte Carlo (Sobol Sequences)

Convergence Comparison

Stochastic Volatility (Heston)

Heston Smile

Monte Carlo Greeks

Path Simulation

Custom Payoffs

Next Steps