How much value is there in attempting to out-model your counterparty?

fair_option_pricing.ipynb

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   "source": [
    "import math\n",
    "import QuantLib as ql # 1.35\n",
    "import polars as pl # 0.20.13\n",
    "from lets_plot import * # 4.3.3\n",
    "LetsPlot.setup_html()"
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   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
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   "source": [
    "def black_scholes_euro_price(spot: float,\n",
    "                             strike: float,\n",
    "                             rfr: float,\n",
    "                             vol: float,\n",
    "                             maturity_days: int,\n",
    "                             tenor: int = 365,\n",
    "                             option_type: str = 'call') -> float:\n",
    "    \"\"\"\n",
    "    Calculate the Black-Scholes price for a European call or put option.\n",
    "    \n",
    "    Parameters:\n",
    "    - spot: Current stock price (S)\n",
    "    - strike: Strike price (K)\n",
    "    - rfr: Risk-free interest rate (r) as a decimal (e.g., 0.05 for 5%)\n",
    "    - vol: Volatility of the underlying stock (σ) as a decimal (e.g., 0.2 for 20%)\n",
    "    - maturity_days: Time to maturity in days (T)\n",
    "    - tenor: Number of days in a year (e.g., 365 for daily)\n",
    "    - option_type: 'call' or 'put'\n",
    "    \n",
    "    Returns:\n",
    "    - Option price\n",
    "    \"\"\"\n",
    "    # Convert maturity from days to years\n",
    "    maturity = maturity_days / tenor\n",
    "\n",
    "    # Calculate d1 and d2\n",
    "    d1 = (math.log(spot / strike) + (rfr + 0.5 * vol ** 2) * maturity) / (vol * math.sqrt(maturity))\n",
    "    d2 = d1 - vol * math.sqrt(maturity)\n",
    "\n",
    "    # Calculate N(d1) and N(d2) for call, or N(-d1) and N(-d2) for put\n",
    "    cdf = ql.CumulativeNormalDistribution()\n",
    "    if option_type.lower() == 'call':\n",
    "        N_d1 = cdf(d1)\n",
    "        N_d2 = cdf(d2)\n",
    "        # Call option price\n",
    "        price = spot * N_d1 - strike * math.exp(-rfr * maturity) * N_d2\n",
    "    elif option_type.lower() == 'put':\n",
    "        N_minus_d1 = cdf(-d1)\n",
    "        N_minus_d2 = cdf(-d2)\n",
    "        # Put option price\n",
    "        price = strike * math.exp(-rfr * maturity) * N_minus_d2 - spot * N_minus_d1\n",
    "    else:\n",
    "        raise ValueError(\"option_type must be 'call' or 'put'\")\n",
    "\n",
    "    return price\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Let's plot the Black-Scholes Euro price for the URA\n",
    "# call option in Jan'26 against a varying implied volatility\n",
    "# value to better understand what the market is offering\n",
    "# us right now (well not technically right now, technically\n",
    "# as of the close on Fri Sep 6th, 2024, but you know what\n",
    "# I mean)."
   ]
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  {
   "cell_type": "code",
   "execution_count": 5,
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   "source": [
    "spot = 23.18\n",
    "strike = 20\n",
    "risk_free_rate = 0.0533\n",
    "maturity_days = 496\n",
    "tenor = 365\n",
    "option_type = 'call'\n",
    "implied_vols = [\n",
    "    0.25,\n",
    "    0.26,\n",
    "    0.27,\n",
    "    0.28,\n",
    "    0.29,\n",
    "    0.30,\n",
    "    0.31,\n",
    "    0.32,\n",
    "    0.33,\n",
    "    0.34,\n",
    "    0.35,\n",
    "    0.36,\n",
    "    0.37,\n",
    "    0.38,\n",
    "    0.39,\n",
    "    0.40\n",
    "]"
   ]
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   "execution_count": 6,
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       "           var plotSpec={\n",
       "\"data\":{\n",
       "\"implied_vol\":[0.25,0.26,0.27,0.28,0.29,0.3,0.31,0.32,0.33,0.34,0.35,0.36,0.37,0.38,0.39,0.4],\n",
       "\"price\":[5.35970318421672,5.432328142434251,5.506376153646071,5.581708642535213,5.658201834535841,5.735744952652185,5.814238637813466,5.893593569811271,5.973729265687588,6.054573033650895,6.136059062472778,6.218127628438625,6.300724404055801,6.383799854734672,6.467308711492144,6.551209509362076]\n",
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       "\"text\":\"BS Euro Prices vs. Implied Volatility\"\n",
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   "source": [
    "bs_euro_option_prices = []\n",
    "for implied_vol in implied_vols:\n",
    "    price = black_scholes_euro_price(spot,\n",
    "                                     strike,\n",
    "                                     risk_free_rate,\n",
    "                                     implied_vol,\n",
    "                                     maturity_days,\n",
    "                                     tenor,\n",
    "                                     option_type)\n",
    "    bs_euro_option_prices.append(price)\n",
    "\n",
    "data = {\n",
    "    'implied_vol': implied_vols,\n",
    "    'price': bs_euro_option_prices\n",
    "}\n",
    "df = pl.DataFrame(data)\n",
    "\n",
    "ggplot(df, aes(x='implied_vol', y='price')) + \\\n",
    "    geom_line(color='blue') + \\\n",
    "    ggtitle('BS Euro Prices vs. Implied Volatility') + \\\n",
    "    labs(x='Implied Volatility', y='Option Price') + \\\n",
    "    theme(plot_title=element_text(size=18, hjust=0.5), legend_position='bottom')"
   ]
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  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
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   "source": [
    "# The market pricing is 6.00x6.40 for this option with\n",
    "# non-zero volume (13 contracts traded on Fri Sep 6th)\n",
    "# so we are certain that we can buy the 20C for 6.40\n",
    "# and sell the 20C for 6.00.\n",
    "#\n",
    "# Using the interactive chart above, we see that at\n",
    "# 0.34 implied vol, the BS Euro price is 6.05, close enough\n",
    "# to the 6.00 bid, and at 0.38 implied vol, the BS Euro\n",
    "# price is 6.38, close enough to the 6.40 ask.\n",
    "#\n",
    "# From this we can infer that your counterparty's model\n",
    "# is predicting that 0.36 implied vol is roughly \"fair\"\n",
    "# and in order to trade profitably against you, they will\n",
    "# buy the option from you at a 0.34 implied vol or they will\n",
    "# sell the option to you at a 0.38 implied vol, a 2 vol point\n",
    "# edge in their favor in either direction.\n",
    "#\n",
    "# Remember that your counterparty is not a directional trader\n",
    "# and this volatility edge is how they stay in business. Your\n",
    "# trade will be delta-hedged on their book, while your trade\n",
    "# is likely a levered directional bet (i.e. a delta bet).\n",
    "#\n",
    "# Let's assume you decide to take the trade and fill at 6.40.\n",
    "# Suppose that 0.34 implied vol is \"fair\" and that the market\n",
    "# will continue to demand a 2 vol point edge in either direction:\n",
    "# What will the projected price of the BS Euro option be at\n",
    "# a variety of underlying prices 90 calendar days from now, and\n",
    "# how much difference in that price exists between \"fair\"\n",
    "# implied volatility at 0.34 and our counterparty's 2 vol point edge?"
   ]
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   "cell_type": "code",
   "execution_count": 8,
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   "source": [
    "spots = [\n",
    "    15,\n",
    "    16,\n",
    "    17,\n",
    "    18,\n",
    "    19,\n",
    "    20,\n",
    "    21,\n",
    "    22,\n",
    "    23,\n",
    "    24,\n",
    "    25,\n",
    "    26,\n",
    "    27,\n",
    "    28,\n",
    "    29,\n",
    "    30\n",
    "]\n",
    "strike = 20\n",
    "risk_free_rate = 0.0533\n",
    "maturity_days = 406 # 90 calendar days forward from now\n",
    "tenor = 365\n",
    "option_type = 'call'\n",
    "fair_implied_vol = 0.34\n",
    "edge_implied_vol = 0.32 # 2 vol pts below 0.34 \"fair\" implied vol"
   ]
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   "execution_count": 9,
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    "bs_fair_euro_option_prices = []\n",
    "bs_edge_euro_option_prices = []\n",
    "for spot in spots:\n",
    "    fair_price = black_scholes_euro_price(\n",
    "        spot,\n",
    "        strike,\n",
    "        risk_free_rate,\n",
    "        fair_implied_vol,\n",
    "        maturity_days,\n",
    "        tenor,\n",
    "        option_type\n",
    "    )\n",
    "    bs_fair_euro_option_prices.append(fair_price)\n",
    "\n",
    "    edge_price = black_scholes_euro_price(\n",
    "        spot,\n",
    "        strike,\n",
    "        risk_free_rate,\n",
    "        edge_implied_vol,\n",
    "        maturity_days,\n",
    "        tenor,\n",
    "        option_type\n",
    "    )\n",
    "    bs_edge_euro_option_prices.append(edge_price)\n",
    "\n",
    "fair_data = {\n",
    "    'spot': spots,\n",
    "    'price': bs_fair_euro_option_prices,\n",
    "    'dataset': ['fair'] * len(spots)\n",
    "}\n",
    "fair_df = pl.DataFrame(fair_data)\n",
    "\n",
    "edge_data = {\n",
    "    'spot': spots,\n",
    "    'price': bs_edge_euro_option_prices,\n",
    "    'dataset': ['edge'] * len(spots)\n",
    "}\n",
    "edge_df = pl.DataFrame(edge_data)\n",
    "\n",
    "combined_df = pl.concat([fair_df, edge_df])\n",
    "\n",
    "ggplot(combined_df) + \\\n",
    "    geom_line(aes(x='spot', y='price', color='dataset')) + \\\n",
    "    scale_color_manual(values=['gray', 'magenta']) + \\\n",
    "    ggtitle('BS Euro Prices vs. Spot Price at T+90') + \\\n",
    "    labs(x='Spot Price', y='Option Price') + \\\n",
    "    theme(plot_title=element_text(size=18, hjust=0.5), legend_position='bottom')\n"
   ]
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  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# So how much damage does this 2 vol point edge really do?\n",
    "# At 90 days, almost none lol. You would be better off spending\n",
    "# your time improving your forecasts for price than hand-wringing\n",
    "# over the \"fair\" price of an option as a retail trader.\n",
    "#\n",
    "# If you're a professional working on a volatility trading\n",
    "# desk, that's a different story. This edge can be meaningful\n",
    "# as you trade large quantities of contracts (e.g. 1000+) and \n",
    "# have the capital necessary to hedge your deltas effectively.\n",
    "# Time spent building a faster, more accurate volatility model\n",
    "# for the surface of a particular security can be lucrative."
   ]
  }
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