{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Derivatives framework example\n",
"This example is the accompanied code to https://ssrn.com/abstract=4217884. It illustrates how to use the relative market values v parameter in MeanCVaR and MeanVariance."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import pandas as pd\n",
"import seaborn as sns\n",
"import fortitudo.tech as ft\n",
"import matplotlib.pyplot as plt"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" Mean | \n",
" Volatility | \n",
" Skewness | \n",
" Kurtosis | \n",
"
\n",
" \n",
" \n",
" \n",
" | Gov & MBS | \n",
" -0.007 | \n",
" 0.032 | \n",
" 0.096 | \n",
" 3.023 | \n",
"
\n",
" \n",
" | Corp IG | \n",
" -0.004 | \n",
" 0.034 | \n",
" 0.107 | \n",
" 3.109 | \n",
"
\n",
" \n",
" | Corp HY | \n",
" 0.019 | \n",
" 0.061 | \n",
" 0.173 | \n",
" 2.971 | \n",
"
\n",
" \n",
" | EM Debt | \n",
" 0.027 | \n",
" 0.075 | \n",
" 0.217 | \n",
" 3.057 | \n",
"
\n",
" \n",
" | DM Equity | \n",
" 0.064 | \n",
" 0.149 | \n",
" 0.396 | \n",
" 3.148 | \n",
"
\n",
" \n",
" | EM Equity | \n",
" 0.080 | \n",
" 0.269 | \n",
" 0.766 | \n",
" 4.099 | \n",
"
\n",
" \n",
" | Private Equity | \n",
" 0.137 | \n",
" 0.278 | \n",
" 0.716 | \n",
" 3.758 | \n",
"
\n",
" \n",
" | Infrastructure | \n",
" 0.059 | \n",
" 0.108 | \n",
" 0.311 | \n",
" 3.193 | \n",
"
\n",
" \n",
" | Real Estate | \n",
" 0.043 | \n",
" 0.081 | \n",
" 0.234 | \n",
" 3.092 | \n",
"
\n",
" \n",
" | Hedge Funds | \n",
" 0.048 | \n",
" 0.072 | \n",
" 0.204 | \n",
" 3.050 | \n",
"
\n",
" \n",
"
\n",
"
"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Plot ATMF P&L vs underlying for illustration\n",
"plt.scatter(pnl['DM Equity'], call_atmf_pnl, marker='.')\n",
"plt.scatter(pnl['DM Equity'], put_atmf_pnl, marker='.')\n",
"plt.title('ATMF options P&L')\n",
"plt.legend(['ATMF Call', 'ATMF Put'])\n",
"plt.ylabel('Option return')\n",
"plt.xlabel('DM Equity return')\n",
"plt.show()"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" Mean | \n",
" Volatility | \n",
" Skewness | \n",
" Kurtosis | \n",
"
\n",
" \n",
" \n",
" \n",
" | Gov & MBS | \n",
" -0.007 | \n",
" 0.032 | \n",
" 0.096 | \n",
" 3.023 | \n",
"
\n",
" \n",
" | Corp IG | \n",
" -0.004 | \n",
" 0.034 | \n",
" 0.107 | \n",
" 3.109 | \n",
"
\n",
" \n",
" | Corp HY | \n",
" 0.019 | \n",
" 0.061 | \n",
" 0.173 | \n",
" 2.971 | \n",
"
\n",
" \n",
" | EM Debt | \n",
" 0.027 | \n",
" 0.075 | \n",
" 0.217 | \n",
" 3.057 | \n",
"
\n",
" \n",
" | DM Equity | \n",
" 0.064 | \n",
" 0.149 | \n",
" 0.396 | \n",
" 3.148 | \n",
"
\n",
" \n",
" | EM Equity | \n",
" 0.080 | \n",
" 0.269 | \n",
" 0.766 | \n",
" 4.099 | \n",
"
\n",
" \n",
" | Private Equity | \n",
" 0.137 | \n",
" 0.278 | \n",
" 0.716 | \n",
" 3.758 | \n",
"
\n",
" \n",
" | Infrastructure | \n",
" 0.059 | \n",
" 0.108 | \n",
" 0.311 | \n",
" 3.193 | \n",
"
\n",
" \n",
" | Real Estate | \n",
" 0.043 | \n",
" 0.081 | \n",
" 0.234 | \n",
" 3.092 | \n",
"
\n",
" \n",
" | Hedge Funds | \n",
" 0.048 | \n",
" 0.072 | \n",
" 0.204 | \n",
" 3.050 | \n",
"
\n",
" \n",
" | Put 90 | \n",
" -0.016 | \n",
" 0.026 | \n",
" 4.316 | \n",
" 23.824 | \n",
"
\n",
" \n",
" | Put 95 | \n",
" -0.022 | \n",
" 0.040 | \n",
" 2.966 | \n",
" 12.238 | \n",
"
\n",
" \n",
" | Put ATMF | \n",
" -0.029 | \n",
" 0.057 | \n",
" 2.069 | \n",
" 6.960 | \n",
"
\n",
" \n",
" | Call ATMF | \n",
" 0.035 | \n",
" 0.114 | \n",
" 1.327 | \n",
" 4.539 | \n",
"
\n",
" \n",
" | Call 105 | \n",
" 0.029 | \n",
" 0.098 | \n",
" 1.764 | \n",
" 6.149 | \n",
"
\n",
" \n",
" | Call 110 | \n",
" 0.022 | \n",
" 0.081 | \n",
" 2.328 | \n",
" 8.940 | \n",
"
\n",
" \n",
"
\n",
"
"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Plot option P&L distributions\n",
"pnl.iloc[:, -6:].plot(kind='density')\n",
"plt.xlim([-0.1, 0.1])\n",
"plt.xlabel('Relative P&L')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Prior portfolio optimization"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"# Optimization constraints\n",
"v = np.hstack((np.ones(I), [put_90, put_95, put_atmf, call_atmf, call_105, call_110]))\n",
"G = np.vstack((np.eye(len(v)), -np.eye(len(v))))\n",
"options_bounds = 0.5 * np.ones(6)\n",
"h = np.hstack((0.25 * np.ones(I), options_bounds, np.zeros(I), options_bounds))"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" 90%-CVaR optimized | \n",
" Variance optimized | \n",
"
\n",
" \n",
" \n",
" \n",
" | Gov & MBS | \n",
" -0.00 | \n",
" 0.00 | \n",
"
\n",
" \n",
" | Corp IG | \n",
" 0.00 | \n",
" 0.00 | \n",
"
\n",
" \n",
" | Corp HY | \n",
" 0.00 | \n",
" 0.00 | \n",
"
\n",
" \n",
" | EM Debt | \n",
" 15.05 | \n",
" 17.76 | \n",
"
\n",
" \n",
" | DM Equity | \n",
" 16.81 | \n",
" 11.98 | \n",
"
\n",
" \n",
" | EM Equity | \n",
" 0.00 | \n",
" 0.00 | \n",
"
\n",
" \n",
" | Private Equity | \n",
" 4.81 | \n",
" 3.83 | \n",
"
\n",
" \n",
" | Infrastructure | \n",
" 18.60 | \n",
" 22.82 | \n",
"
\n",
" \n",
" | Real Estate | \n",
" 18.44 | \n",
" 21.94 | \n",
"
\n",
" \n",
" | Hedge Funds | \n",
" 25.00 | \n",
" 25.00 | \n",
"
\n",
" \n",
" | Put 90 | \n",
" -50.00 | \n",
" -50.00 | \n",
"
\n",
" \n",
" | Put 95 | \n",
" -0.31 | \n",
" -50.00 | \n",
"
\n",
" \n",
" | Put ATMF | \n",
" 50.00 | \n",
" 35.61 | \n",
"
\n",
" \n",
" | Call ATMF | \n",
" -50.00 | \n",
" -49.99 | \n",
"
\n",
" \n",
" | Call 105 | \n",
" 36.50 | \n",
" 1.69 | \n",
"
\n",
" \n",
" | Call 110 | \n",
" 50.00 | \n",
" 25.71 | \n",
"
\n",
" \n",
"
\n",
"
"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Plot portfolio P&L distributions\n",
"pfs_pnl = pd.DataFrame(pnl.values @ prior_results, columns=portfolio_names)\n",
"\n",
"pfs_pnl.plot(kind='density')\n",
"plt.title('Prior optimization')\n",
"plt.xlim([-0.2, 0.5])\n",
"plt.xlabel('Portfolio return')\n",
"plt.show()"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" 90%-CVaR optimized | \n",
" Variance optimized | \n",
"
\n",
" \n",
" \n",
" \n",
" | Vol | \n",
" 8.2 | \n",
" 5.9 | \n",
"
\n",
" \n",
" | VaR | \n",
" 4.7 | \n",
" 5.2 | \n",
"
\n",
" \n",
" | CVaR | \n",
" 6.7 | \n",
" 10.4 | \n",
"
\n",
" \n",
"
\n",
"
\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" Mean | \n",
" Volatility | \n",
" Skewness | \n",
" Kurtosis | \n",
"
\n",
" \n",
" \n",
" \n",
" | Gov & MBS | \n",
" -0.007 | \n",
" 0.032 | \n",
" 0.043 | \n",
" 2.999 | \n",
"
\n",
" \n",
" | Corp IG | \n",
" -0.004 | \n",
" 0.033 | \n",
" 0.152 | \n",
" 3.140 | \n",
"
\n",
" \n",
" | Corp HY | \n",
" 0.017 | \n",
" 0.071 | \n",
" -0.010 | \n",
" 2.792 | \n",
"
\n",
" \n",
" | EM Debt | \n",
" 0.025 | \n",
" 0.082 | \n",
" 0.083 | \n",
" 3.104 | \n",
"
\n",
" \n",
" | DM Equity | \n",
" 0.064 | \n",
" 0.200 | \n",
" -0.100 | \n",
" 2.750 | \n",
"
\n",
" \n",
" | EM Equity | \n",
" 0.080 | \n",
" 0.326 | \n",
" 0.571 | \n",
" 4.035 | \n",
"
\n",
" \n",
" | Private Equity | \n",
" 0.142 | \n",
" 0.339 | \n",
" 0.441 | \n",
" 3.247 | \n",
"
\n",
" \n",
" | Infrastructure | \n",
" 0.058 | \n",
" 0.114 | \n",
" 0.303 | \n",
" 3.082 | \n",
"
\n",
" \n",
" | Real Estate | \n",
" 0.040 | \n",
" 0.090 | \n",
" 0.114 | \n",
" 2.962 | \n",
"
\n",
" \n",
" | Hedge Funds | \n",
" 0.045 | \n",
" 0.093 | \n",
" -0.220 | \n",
" 3.012 | \n",
"
\n",
" \n",
" | Put 90 | \n",
" 0.004 | \n",
" 0.063 | \n",
" 2.348 | \n",
" 7.224 | \n",
"
\n",
" \n",
" | Put 95 | \n",
" 0.000 | \n",
" 0.080 | \n",
" 2.036 | \n",
" 5.830 | \n",
"
\n",
" \n",
" | Put ATMF | \n",
" -0.005 | \n",
" 0.098 | \n",
" 1.750 | \n",
" 4.756 | \n",
"
\n",
" \n",
" | Call ATMF | \n",
" 0.059 | \n",
" 0.132 | \n",
" 1.164 | \n",
" 4.189 | \n",
"
\n",
" \n",
" | Call 105 | \n",
" 0.051 | \n",
" 0.116 | \n",
" 1.543 | \n",
" 5.482 | \n",
"
\n",
" \n",
" | Call 110 | \n",
" 0.041 | \n",
" 0.099 | \n",
" 2.026 | \n",
" 7.630 | \n",
"
\n",
" \n",
"
\n",
"
"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Plot DM Equity P&L distribution\n",
"sns.kdeplot(x=pnl['DM Equity'])\n",
"sns.kdeplot(x=pnl['DM Equity'], weights=q[:, 0])\n",
"plt.title('DM Equity marginal distributions')\n",
"plt.legend(['Prior', 'Posterior'])\n",
"plt.xlabel('DM Equity return')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Posterior portfolio optimization\n"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [],
"source": [
"cvar_opt_post = ft.MeanCVaR(R, G, h, v=v, p=q, alpha=alpha)\n",
"\n",
"mean_post = stats_post['Mean'].values\n",
"cov_post = ft.covariance_matrix(R, q).values\n",
"var_opt_post = ft.MeanVariance(mean_post, cov_post, G, h, v=v)\n",
"\n",
"port_cvar_post = cvar_opt_post.efficient_portfolio(0.05)\n",
"port_var_post = var_opt_post.efficient_portfolio(0.05)\n",
"post_results = np.hstack((port_cvar_post, port_var_post))"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" 90%-CVaR optimized | \n",
" Variance optimized | \n",
"
\n",
" \n",
" \n",
" \n",
" | Gov & MBS | \n",
" 18.56 | \n",
" 0.00 | \n",
"
\n",
" \n",
" | Corp IG | \n",
" 0.00 | \n",
" 0.00 | \n",
"
\n",
" \n",
" | Corp HY | \n",
" 0.00 | \n",
" 0.00 | \n",
"
\n",
" \n",
" | EM Debt | \n",
" 7.05 | \n",
" 17.92 | \n",
"
\n",
" \n",
" | DM Equity | \n",
" 25.00 | \n",
" 1.39 | \n",
"
\n",
" \n",
" | EM Equity | \n",
" 0.00 | \n",
" 0.00 | \n",
"
\n",
" \n",
" | Private Equity | \n",
" 2.70 | \n",
" 5.03 | \n",
"
\n",
" \n",
" | Infrastructure | \n",
" 9.94 | \n",
" 25.00 | \n",
"
\n",
" \n",
" | Real Estate | \n",
" 9.37 | \n",
" 22.68 | \n",
"
\n",
" \n",
" | Hedge Funds | \n",
" 25.00 | \n",
" 25.00 | \n",
"
\n",
" \n",
" | Put 90 | \n",
" 14.27 | \n",
" 50.00 | \n",
"
\n",
" \n",
" | Put 95 | \n",
" -9.41 | \n",
" 49.93 | \n",
"
\n",
" \n",
" | Put ATMF | \n",
" 47.40 | \n",
" -10.30 | \n",
"
\n",
" \n",
" | Call ATMF | \n",
" -50.00 | \n",
" 4.60 | \n",
"
\n",
" \n",
" | Call 105 | \n",
" 39.37 | \n",
" 35.25 | \n",
"
\n",
" \n",
" | Call 110 | \n",
" 50.00 | \n",
" -50.00 | \n",
"
\n",
" \n",
"
\n",
"
"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"pfs_pnl_post = pnl @ post_results\n",
"pfs_pnl_post.columns = portfolio_names\n",
"\n",
"sns.kdeplot(x=pfs_pnl_post.values[:, 0], weights=q[:, 0])\n",
"sns.kdeplot(x=pfs_pnl_post.values[:, 1], weights=q[:, 0])\n",
"plt.title('Posterior optimization')\n",
"plt.legend(portfolio_names)\n",
"plt.xlabel('Portfolio return')\n",
"plt.xlim([-0.2, 0.5])\n",
"plt.show()"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" 90%-CVaR optimized | \n",
" Variance optimized | \n",
"
\n",
" \n",
" \n",
" \n",
" | Vol | \n",
" 8.9 | \n",
" 6.3 | \n",
"
\n",
" \n",
" | VaR | \n",
" 3.6 | \n",
" 5.2 | \n",
"
\n",
" \n",
" | CVaR | \n",
" 4.6 | \n",
" 7.3 | \n",
"
\n",
" \n",
"
\n",
"