{"componentChunkName":"component---src-templates-blog-post-js","path":"/bs-merton-polymarket/","result":{"data":{"site":{"siteMetadata":{"title":"/dev/yukarinoki"}},"markdownRemark":{"id":"83216695-eaac-529e-8777-6878f45fead8","excerpt":"日本語要約: Black-ScholesモデルとMertonジャンプ拡散モデルを用いて、Polymarketの5分BTCバイナリオプションの理論価格を算出し、実際の市場価格と比較します。Monte Carlo…","html":"
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日本語要約: Black-ScholesモデルとMertonジャンプ拡散モデルを用いて、Polymarketの5分BTCバイナリオプションの理論価格を算出し、実際の市場価格と比較します。Monte Carloシミュレーションのコード例を交えながら、伝統的金融モデルが暗号通貨の超短期市場でどこまで通用するかを検証します。

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I’ve been obsessed with Polymarket’s ultra-short-term BTC binary options lately. These are 5-minute contracts where you’re betting whether BTC/USDT will be above or below a strike price at expiry. The tick-by-tick nature of these contracts makes them a perfect playground for testing option pricing models against reality.

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The question I wanted to answer: can classical Black-Scholes or the more sophisticated Merton jump-diffusion model actually predict what Polymarket traders are pricing in? Spoiler: the answer is nuanced and interesting.

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Why Crypto Binary Options Are a Weird Beast

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Binary options are the simplest derivative you can imagine. You get paid $1 if BTC is above the strike at expiry, $0 otherwise. The “fair” price is just $P(S_T > K)$ - the probability that the spot price exceeds the strike at expiration.

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But crypto makes this deceptively hard to model:

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Polymarket runs these contracts with strikes spaced around the current spot price (e.g., if BTC is at $68,450, you’ll see strikes at $68,400, $68,500, $68,600). I pulled about 1,200 trades across multiple contract cycles to compare with model predictions.

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Black-Scholes for Binary Options

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The standard BS framework assumes BTC follows geometric Brownian motion (GBM):

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$$dS = \\mu S , dt + \\sigma S , dW$$

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where $\\mu$ is the drift, $\\sigma$ is the volatility, and $dW$ is a Wiener process.

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For a cash-or-nothing binary call (pays $1 if $S_T > K$), the analytical price is:

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$$C{binary} = e^{-rT} \\cdot N(d2)$$

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where:

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$$d_2 = \\frac{\\ln(S/K) + (r - \\frac{1}{2}\\sigma^2)T}{\\sigma\\sqrt{T}}$$

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and $N(\\cdot)$ is the standard normal CDF.

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For 5-minute expiries, $T$ is tiny (about $9.5 \\times 10^{-6}$ years), $r$ is basically zero, and the whole thing reduces to:

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$$C_{binary} \\approx N\\left(\\frac{\\ln(S/K) - \\frac{1}{2}\\sigma^2 T}{\\sigma\\sqrt{T}}\\right)$$

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The critical input is $\\sigma$. I estimated it using rolling 1-hour realized volatility from Binance tick data, annualized. Typical values during my observation period were 45-65% annualized.

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The BS dashboard above shows the model tracking real prices. It works… okay. The shape is right - prices near 0.5 when the spot is at the strike, approaching 1 or 0 as you move away. But there’s consistent mispricing at the tails.

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Where Black-Scholes Breaks Down

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BS assumes returns are log-normally distributed. In a 5-minute window for BTC, the actual return distribution has:

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  1. Fat tails: the kurtosis of 5-minute BTC returns is typically 8-15 (normal would be 3). Extreme moves are way more likely than BS predicts.
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  2. Jumps: BTC doesn’t move continuously. Liquidation cascades, large market orders, and news events create discrete jumps. In my dataset, about 3-4% of 5-minute windows had moves exceeding 3 standard deviations.
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  3. Asymmetry: negative jumps (crashes) are slightly more common and larger than positive jumps, giving negative skewness.
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  4. Volatility clustering: a big move in the last 5 minutes makes a big move in the next 5 minutes much more likely. BS assumes constant vol.
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The practical effect: BS underprices deep out-of-the-money binaries. If the strike is $200 above spot and you have 5 minutes left, BS says “basically zero probability” but the market says “small but non-trivial” because traders know jumps happen.

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Merton Jump-Diffusion Model

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Merton (1976) extended GBM by adding a compound Poisson jump process:

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$$dS = (\\mu - \\lambda k)S , dt + \\sigma S , dW + S , dJ$$

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where:

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The key insight: between jumps, the price follows GBM with reduced drift (to compensate for the expected jump contribution). When a jump arrives, the price instantaneously moves by a log-normal factor.

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For binary option pricing under Merton, the analytical formula is a series expansion:

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$$C{binary}^{Merton} = \\sum{n=0}^{\\infty} \\frac{e^{-\\lambda’ T}(\\lambda’ T)^n}{n!} \\cdot N(d_2^{(n)})$$

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where $\\lambda’ = \\lambda(1+k)$ and:

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$$d2^{(n)} = \\frac{\\ln(S/K) + (r - \\frac{1}{2}\\sigman^2 - \\lambda k)T + n\\muJ}{\\sigman\\sqrt{T}}$$

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$$\\sigman = \\sqrt{\\sigma^2 + n\\sigmaJ^2 / T}$$

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I calibrated the jump parameters from the empirical 5-minute return distribution:

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Interactive Simulator

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Before diving into the code, try the interactive simulator below. You can adjust the model parameters (volatility, jump intensity, etc.) and run Monte Carlo simulations to see how BS and Merton pricing curves diverge in real-time.

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Monte Carlo Simulation Approach

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While the analytical Merton formula works, I also ran Monte Carlo simulations for both models. This lets me sanity-check the analytics and easily extend to more complex dynamics later.

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Here’s the simulation code:

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import numpy as np\nfrom scipy.stats import norm\n\ndef simulate_bs_binary(S0, K, T, sigma, r=0.0, n_sims=100000):\n    \"\"\"\n    Monte Carlo price for a binary call under GBM.\n    Returns P(S_T > K).\n    \"\"\"\n    Z = np.random.standard_normal(n_sims)\n    S_T = S0 * np.exp((r - 0.5 * sigma**2) * T + sigma * np.sqrt(T) * Z)\n    payoff = (S_T > K).astype(float)\n    return np.exp(-r * T) * payoff.mean()\n\n\ndef simulate_merton_binary(S0, K, T, sigma, lam, mu_j, sigma_j,\n                           r=0.0, n_sims=100000):\n    \"\"\"\n    Monte Carlo price for a binary call under Merton jump-diffusion.\n    Returns P(S_T > K) with Poisson jumps.\n    \"\"\"\n    # Number of jumps in [0, T] for each path\n    N_jumps = np.random.poisson(lam * T, n_sims)\n\n    # Compensator for jump drift\n    k = np.exp(mu_j + 0.5 * sigma_j**2) - 1\n    drift = (r - 0.5 * sigma**2 - lam * k) * T\n\n    # Diffusion component\n    Z = np.random.standard_normal(n_sims)\n    diffusion = sigma * np.sqrt(T) * Z\n\n    # Jump component: sum of N_jumps log-normal jumps\n    jump_component = np.zeros(n_sims)\n    for i in range(n_sims):\n        if N_jumps[i] > 0:\n            jumps = np.random.normal(mu_j, sigma_j, N_jumps[i])\n            jump_component[i] = jumps.sum()\n\n    # Terminal price\n    S_T = S0 * np.exp(drift + diffusion + jump_component)\n    payoff = (S_T > K).astype(float)\n    return np.exp(-r * T) * payoff.mean()\n\n\n# Example: BTC at $68,450, strike at $68,500, 5 min to expiry\nS0 = 68450.0\nK = 68500.0\nT = 5.0 / (365.25 * 24 * 60)  # 5 minutes in years\nsigma = 0.55  # 55% annualized vol\n\n# BS price\nbs_price = simulate_bs_binary(S0, K, T, sigma, n_sims=500000)\nprint(f\"BS Binary Price: {bs_price:.4f}\")\n\n# Merton price\nlam = 500      # ~1 jump per 10 five-min intervals\nmu_j = -0.0002\nsigma_j = 0.003\n\nmerton_price = simulate_merton_binary(S0, K, T, sigma, lam, mu_j, sigma_j,\n                                       n_sims=500000)\nprint(f\"Merton Binary Price: {merton_price:.4f}\")\n\n# Analytical BS for comparison\nd2 = (np.log(S0/K) + (0 - 0.5*sigma**2)*T) / (sigma*np.sqrt(T))\nbs_analytical = norm.cdf(d2)\nprint(f\"BS Analytical: {bs_analytical:.4f}\")
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The loop in simulate_merton_binary is intentionally naive for clarity. In practice you’d vectorize it (pre-allocate max jumps and mask), which gets you ~10x speedup. For 500k paths on a 5-minute contract, runtime is about 2 seconds on my machine.

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Comparison Against Real Polymarket Data

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This is where it gets interesting. I collected trade data from Polymarket’s BTC 5-minute binary contracts over several hours, tracking:

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Then I computed BS and Merton model prices for each observed trade and plotted them together.

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