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Add new tutorial for multi-currencies arbitrage using Bellman-Ford Algorithm #48
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Merge pull request #1 from leo-ai-for-trading/leo-ai-for-trading-patch-1
leo-ai-for-trading 59417e5
Fixed: Issue #47
leo-ai-for-trading b8eb4dc
Capital Structure Arbitrage
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Capital Structure Arbitrage
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Fixed: Issue #47
## Description Replying to this issue Related Issue How to extract Implied volatility from Quantconnect #47 Types of changes -[ ] Bug fix (non-breaking change which fixes an issue) -[ ] Refactor (non-breaking change which improves implementation) -[ ] New feature (non-breaking change which adds functionality) -[ x] Non-functional change (xml comments/documentation/etc) Checklist: -[ x] My code follows the code style of this project. -[ x] I have read the CONTRIBUTING [document](https://github.com/QuantConnect/Lean/blob/master/CONTRIBUTING.md). -[ x] My branch follows the naming convention bug-<issue#>-<description> or feature-<issue#>-<description>
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{ | ||
"nbformat": 4, | ||
"nbformat_minor": 0, | ||
"metadata": { | ||
"colab": { | ||
"provenance": [] | ||
}, | ||
"kernelspec": { | ||
"name": "python3", | ||
"display_name": "Python 3" | ||
}, | ||
"language_info": { | ||
"name": "python" | ||
} | ||
}, | ||
"cells": [ | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"![QuantConnect Logo](https://cdn.quantconnect.com/web/i/icon.png)\n", | ||
"<hr>" | ||
], | ||
"metadata": { | ||
"id": "6iZIAgxW8Ari" | ||
} | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"# Dupire model\n", | ||
"$C\\left(S_0, K, T\\right)=\\int_K^{\\infty} d S_T \\varphi\\left(S_T, T ; S_0\\right)\\left(S_T-K\\right)$\n", | ||
"### risk-neutral density function φ of the final spot $S_T$\n", | ||
"$\\varphi\\left(K, T ; S_0\\right)=\\frac{\\partial^2 C}{\\partial K^2}$\n", | ||
"## Drift\n", | ||
"$\\mu = r_t − D_t$\n", | ||
"## local volatility\n", | ||
"$\\sigma^2\\left(K, T, S_0\\right)=\\frac{\\frac{\\partial C}{\\partial T}}{\\frac{1}{2} K^2 \\frac{\\partial^2 C}{\\partial K^2}}$\n", | ||
"## undiscounted option price C in terms of the strike price K:\n", | ||
"### which is the Dupire equation when the underlying stock has risk-neutral drift μ.\n", | ||
"$\\frac{\\partial C}{\\partial T}=\\frac{\\sigma^2 K^2}{2} \\frac{\\partial^2 C}{\\partial K^2}+\\left(r_t-D_t\\right)\\left(C-K \\frac{\\partial C}{\\partial K}\\right)$\n", | ||
"# BSM model\n", | ||
"$Call=S N\\left(d_1\\right)-K e^{-r \\tau} N\\left(d_2\\right)$ \n", | ||
"\\\n", | ||
"$d_1=\\frac{\\ln (S / K)+\\left(r-y+\\sigma^2 / 2\\right) \\tau}{\\sigma \\sqrt{\\tau}}$\n", | ||
"\\\n", | ||
"$d_2=\\frac{\\ln (S / K)+\\left(r-y-\\sigma^2 / 2\\right) \\tau}{\\sigma \\sqrt{\\tau}}=d_1-\\sigma \\sqrt{\\tau}$\n", | ||
"\n", | ||
"## PDF of Normal \n", | ||
"$f(x)=\\frac{1}{\\sigma \\sqrt{2 \\pi}} e^{-\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^2}$\n", | ||
"\n", | ||
"## CDF of Normal\n", | ||
"$N(x)=\\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^x e^{-z^2 / 2} d z$\n" | ||
], | ||
"metadata": { | ||
"id": "fqocn6D58JDW" | ||
} | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"# Importing Libraries" | ||
], | ||
"metadata": { | ||
"id": "LeIimwLE8FOu" | ||
} | ||
}, | ||
{ | ||
"cell_type": "code", | ||
"source": [ | ||
"import sympy as smp \n", | ||
"import numpy as np\n", | ||
"from sympy.stats import P, E, variance, Die, Normal\n", | ||
"from sympy import simplify\n" | ||
], | ||
"metadata": { | ||
"id": "dk37ioq7wpBX" | ||
}, | ||
"execution_count": 2, | ||
"outputs": [] | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"\n", | ||
"# Risk-neutral density function φ of the final spot $S_T$\n", | ||
"$\\varphi\\left(K, T ; S_0\\right)=\\frac{\\partial^2 C}{\\partial K^2}$" | ||
], | ||
"metadata": { | ||
"id": "BOs4Ha4_9Go_" | ||
} | ||
}, | ||
{ | ||
"cell_type": "code", | ||
"source": [ | ||
"#define the function symbols\n", | ||
"Call,varphi,N = smp.symbols('Call varphi N', cls=smp.Function) \n", | ||
"S_0,K,T,S_T,D,r,sigma,tau ,d_1,d_2 = smp.symbols('S_0 K T S_T D r sigma tau d_1 d_2',real=True) \n", | ||
"\n", | ||
"\n", | ||
"Call = smp.symbols('Call', cls=smp.Function) \n", | ||
"f_d_1,f_d_2,F_d_1,F_d_2 = smp.symbols('f_d_1 f_d_2 F_d_1 F_d_2' , cls=smp.Function) \n", | ||
"mu_google,sigma_google= smp.symbols('mu_google sigma_google',real=True) \n", | ||
"f_d_1 = Normal('d_1', mu_google, sigma_google)\n", | ||
"f_d_2 = Normal('d_2', mu_google, sigma_google)\n", | ||
"\n", | ||
"\n", | ||
"F_d_1 = simplify(P(f_d_1>tau))\n", | ||
"F_d_2 = simplify(P(f_d_2>tau))\n", | ||
"Call = S_T * F_d_1 - K * smp.exp(-r*tau) * F_d_2\n", | ||
"\n", | ||
"\n", | ||
"varphi = smp.diff(Call,K,2)\n", | ||
"\n", | ||
"#varphi.subs([(Call,Call),(K,int(K))]).evalf()" | ||
], | ||
"metadata": { | ||
"id": "0G1VQsJF8M0M" | ||
}, | ||
"execution_count": 4, | ||
"outputs": [] | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"\n", | ||
"## Undiscounted option price C in terms of the strike price K:\n", | ||
"### which is the Dupire equation when the underlying stock has risk-neutral drift μ.\n", | ||
"$\\frac{\\partial C}{\\partial T}=\\frac{\\sigma^2 K^2}{2} \\frac{\\partial^2 C}{\\partial K^2}+\\left(r_t-D_t\\right)\\left(C-K \\frac{\\partial C}{\\partial K}\\right)$" | ||
], | ||
"metadata": { | ||
"id": "ORnbFO8A8XAJ" | ||
} | ||
}, | ||
{ | ||
"cell_type": "code", | ||
"source": [ | ||
"dCdT = sigma**2 * K / 2 * varphi + (r - D)* (Call - K * smp.diff(Call,K))\n", | ||
"dCdT" | ||
], | ||
"metadata": { | ||
"colab": { | ||
"base_uri": "https://localhost:8080/", | ||
"height": 137 | ||
}, | ||
"id": "Xf5AbHNS8XT6", | ||
"outputId": "fec23eba-66b6-4c57-8588-38d7d839f9ff" | ||
}, | ||
"execution_count": 5, | ||
"outputs": [ | ||
{ | ||
"output_type": "execute_result", | ||
"data": { | ||
"text/plain": [ | ||
"S_T*(-D + r)*Piecewise((erf(sqrt(2)*(mu_google - tau)/(2*sigma_google))/2 + 1/2, (Abs(arg(sigma_google)) < pi/4) | ((Abs(arg(sigma_google)) <= pi/4) & (Abs(2*arg(mu_google) - 4*arg(sigma_google) + 2*arg((mu_google - tau)/mu_google) + 2*pi) < pi))), (sqrt(2)*Integral(exp(-(_z - mu_google + tau)**2/(2*sigma_google**2)), (_z, 0, oo))/(2*sqrt(pi)*sigma_google), True))" | ||
], | ||
"text/latex": "$\\displaystyle S_{T} \\left(- D + r\\right) \\left(\\begin{cases} \\frac{\\operatorname{erf}{\\left(\\frac{\\sqrt{2} \\left(\\mu_{google} - \\tau\\right)}{2 \\sigma_{google}} \\right)}}{2} + \\frac{1}{2} & \\text{for}\\: \\left(\\left|{\\arg{\\left(\\sigma_{google} \\right)}}\\right| \\leq \\frac{\\pi}{4} \\wedge \\left|{2 \\arg{\\left(\\mu_{google} \\right)} - 4 \\arg{\\left(\\sigma_{google} \\right)} + 2 \\arg{\\left(\\frac{\\mu_{google} - \\tau}{\\mu_{google}} \\right)} + 2 \\pi}\\right| < \\pi\\right) \\vee \\left|{\\arg{\\left(\\sigma_{google} \\right)}}\\right| < \\frac{\\pi}{4} \\\\\\frac{\\sqrt{2} \\int\\limits_{0}^{\\infty} e^{- \\frac{\\left(z - \\mu_{google} + \\tau\\right)^{2}}{2 \\sigma_{google}^{2}}}\\, dz}{2 \\sqrt{\\pi} \\sigma_{google}} & \\text{otherwise} \\end{cases}\\right)$" | ||
}, | ||
"metadata": {}, | ||
"execution_count": 5 | ||
} | ||
] | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"## Local volatility\n", | ||
"$\\sigma^2\\left(K, T, S_0\\right)=\\frac{\\frac{\\partial C}{\\partial T}}{\\frac{1}{2} K^2 \\frac{\\partial^2 C}{\\partial K^2}}$\n", | ||
"\n", | ||
"The right-hand side of this equation can be computed from known European option prices. So, given a complete set of European option prices for all strikes and expirations, local volatilities are given uniquely by equation above.\n", | ||
"We can view this equation as a definition of the local volatility function regardless of what kind of process (stochastic volatility for example) actually governs the evolution of volatility." | ||
], | ||
"metadata": { | ||
"id": "tLMrOdpH8ci9" | ||
} | ||
}, | ||
{ | ||
"cell_type": "code", | ||
"source": [ | ||
"local_volatility = smp.sqrt( dCdT / (1/2*K**2 * varphi))\n", | ||
"local_volatility" | ||
], | ||
"metadata": { | ||
"colab": { | ||
"base_uri": "https://localhost:8080/", | ||
"height": 141 | ||
}, | ||
"id": "6-ut-zDY8aMU", | ||
"outputId": "7696d056-f450-4e39-e466-d3f270052379" | ||
}, | ||
"execution_count": 6, | ||
"outputs": [ | ||
{ | ||
"output_type": "execute_result", | ||
"data": { | ||
"text/plain": [ | ||
"sqrt(zoo*S_T*(-D + r)*Piecewise((erf(sqrt(2)*(mu_google - tau)/(2*sigma_google))/2 + 1/2, (Abs(arg(sigma_google)) < pi/4) | ((Abs(arg(sigma_google)) <= pi/4) & (Abs(2*arg(mu_google) - 4*arg(sigma_google) + 2*arg((mu_google - tau)/mu_google) + 2*pi) < pi))), (sqrt(2)*Integral(exp(-(_z - mu_google + tau)**2/(2*sigma_google**2)), (_z, 0, oo))/(2*sqrt(pi)*sigma_google), True)))" | ||
], | ||
"text/latex": "$\\displaystyle \\sqrt{\\tilde{\\infty} S_{T} \\left(- D + r\\right) \\left(\\begin{cases} \\frac{\\operatorname{erf}{\\left(\\frac{\\sqrt{2} \\left(\\mu_{google} - \\tau\\right)}{2 \\sigma_{google}} \\right)}}{2} + \\frac{1}{2} & \\text{for}\\: \\left(\\left|{\\arg{\\left(\\sigma_{google} \\right)}}\\right| \\leq \\frac{\\pi}{4} \\wedge \\left|{2 \\arg{\\left(\\mu_{google} \\right)} - 4 \\arg{\\left(\\sigma_{google} \\right)} + 2 \\arg{\\left(\\frac{\\mu_{google} - \\tau}{\\mu_{google}} \\right)} + 2 \\pi}\\right| < \\pi\\right) \\vee \\left|{\\arg{\\left(\\sigma_{google} \\right)}}\\right| < \\frac{\\pi}{4} \\\\\\frac{\\sqrt{2} \\int\\limits_{0}^{\\infty} e^{- \\frac{\\left(z - \\mu_{google} + \\tau\\right)^{2}}{2 \\sigma_{google}^{2}}}\\, dz}{2 \\sqrt{\\pi} \\sigma_{google}} & \\text{otherwise} \\end{cases}\\right)}$" | ||
}, | ||
"metadata": {}, | ||
"execution_count": 6 | ||
} | ||
] | ||
}, | ||
{ | ||
"cell_type": "markdown", | ||
"source": [ | ||
"# Using real values from QuantBook to calculate Local Volatility" | ||
], | ||
"metadata": { | ||
"id": "JSHTTZeU9ONk" | ||
} | ||
}, | ||
{ | ||
"cell_type": "code", | ||
"source": [ | ||
"# QuantBook Analysis Tool \n", | ||
"# For more information see [https://www.quantconnect.com/docs/v2/our-platform/research/getting-started]\n", | ||
"import datetime\n", | ||
"qb = QuantBook()\n", | ||
"option = qb.AddOption(\"GOOG\") \n", | ||
"option.SetFilter(-2, 2, 0, 90)\n", | ||
"history = qb.GetOptionHistory(option.Symbol, datetime(2023, 5, 7), datetime(2023, 5, 26))\n", | ||
"\n", | ||
"all_history = history.GetAllData()\n", | ||
"expiries = history.GetExpiryDates() \n", | ||
"strikes = history.GetStrikes()\n", | ||
"\n", | ||
"goog = qb.AddEquity(\"GOOG\")\n", | ||
"goog_df = qb.History(qb.Securities.Keys, datetime(2023, 5, 7), datetime(2023, 5, 26), Resolution.Daily)\n", | ||
"goog_df = goog_df['close'].unstack(level=0)\n", | ||
"S_T = goog_df.values[-1]\n", | ||
"all_history.head()" | ||
], | ||
"metadata": { | ||
"id": "pYG7bmQv88tI" | ||
}, | ||
"execution_count": null, | ||
"outputs": [] | ||
}, | ||
{ | ||
"cell_type": "code", | ||
"source": [ | ||
"#BSM model of Call option\n", | ||
"import math\n", | ||
"import sympy.stats\n", | ||
"\n", | ||
"####################### parameters #########################\n", | ||
"tau = 3/220 #expiry the 5/19\n", | ||
"r = 0.0341 \n", | ||
"D = 0.0\n", | ||
"K = strikes[-1]\n", | ||
"sigma = goog_df.std()\n", | ||
"S_T = float(goog_df.values[-1])\n", | ||
"mu_goog = goog_df.mean()\n", | ||
"#Calculate numerically the local volatility\n", | ||
"local_volatility.subs([(S_T,S_T),(K,int(K)),(tau,tau),(D,D),(sigma_google,sigma),(mu_google,mu_google)]).evalf()\n" | ||
], | ||
"metadata": { | ||
"id": "lcZxAl8K8msU" | ||
}, | ||
"execution_count": null, | ||
"outputs": [] | ||
} | ||
] | ||
} |
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Can use Lean's InterestRateProvider
and DividendYieldProvider