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"cell_type": "markdown",
"metadata": {},
"source": [
"![](\"https://raw.githubusercontent.com/israeldi/quantlab/master/assets/images/Program-Logo.png\")
\n",
"\n",
"# QuantLab: Black Scholes Volatility Surface\n",
"### [(Go to Quant Lab)](https://israeldi.github.io/quantlab/)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Table of Contents\n",
"\n",
"1. [Random Numbers](#1.-Random-Numbers)\n",
"2. [Plotting Random Samples](#2.-Plotting-Random-Samples)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 4. Valuation\n",
"#### ([Back to Top](#Table-of-Contents))\n",
"\n",
"### 4.1 Valuation of European call options in Black-Scholes-Merton model "
]
},
{
"cell_type": "code",
"execution_count": 280,
"metadata": {},
"outputs": [],
"source": [
"from math import log, sqrt, exp\n",
"from scipy import stats\n",
"\n",
"\n",
"def bsm_call_value(S0, K, T, r, sigma):\n",
" ''' Valuation of European call option in BSM model.\n",
" Analytical formula.\n",
"\n",
" Parameters\n",
" ==========\n",
" S0: float\n",
" initial stock/index level\n",
" K: float\n",
" strike price\n",
" T: float\n",
" maturity date (in year fractions)\n",
" r: float\n",
" constant risk-free short rate\n",
" sigma: float\n",
" volatility factor in diffusion term\n",
"\n",
" Returns\n",
" =======\n",
" value: float\n",
" present value of the European call option\n",
" '''\n",
"\n",
" S0 = float(S0)\n",
" d1 = (log(S0 / K) + (r + 0.5 * (sigma ** 2)) * T) / (sigma * sqrt(T))\n",
" d2 = (log(S0 / K) + (r - 0.5 * (sigma ** 2)) * T) / (sigma * sqrt(T))\n",
" # stats.norm.cdf --> cumulative distribution function\n",
" # for normal distribution\n",
" value = (S0 * stats.norm.cdf(d1, 0.0, 1.0) -\n",
" K * exp(-r * T) * stats.norm.cdf(d2, 0.0, 1.0))\n",
" return value"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 4.2 Vega function and implied volatility estimation"
]
},
{
"cell_type": "code",
"execution_count": 346,
"metadata": {},
"outputs": [],
"source": [
"def bsm_vega(S0, K, T, r, sigma):\n",
" ''' Vega of European option in BSM model.\n",
"\n",
" Parameters\n",
" ==========\n",
" S0: float\n",
" initial stock/index level\n",
" K: float\n",
" strike price\n",
" T: float\n",
" maturity date (in year fractions)\n",
" r: float\n",
" constant risk-free short rate\n",
" sigma: float\n",
" volatility factor in diffusion term\n",
"\n",
" Returns\n",
" =======\n",
" vega: float\n",
" partial derivative of BSM formula with respect\n",
" to sigma, i.e. Vega\n",
"\n",
" '''\n",
" from math import log, sqrt\n",
" from scipy import stats\n",
" \n",
" '''\n",
" \n",
" YOUR CODE HERE\n",
" \n",
" '''\n",
" return\n",
"\n",
"def bsm_call_imp_vol(S0, K, T, r, C0, sigma_est, it=100):\n",
" ''' Implied volatility of European call option in BSM model.\n",
"\n",
" Parameters\n",
" ==========\n",
" S0: float\n",
" initial stock/index level\n",
" K: float\n",
" strike price\n",
" T: float\n",
" maturity date (in year fractions)\n",
" r: float\n",
" constant risk-free short rate\n",
" sigma_est: float\n",
" estimate of impl. volatility\n",
" it: integer\n",
" number of iterations\n",
"\n",
" Returns\n",
" =======\n",
" simga_est: float\n",
" numerically estimated implied volatility\n",
" '''\n",
" \n",
" \n",
" '''\n",
" \n",
" YOUR CODE HERE\n",
" \n",
" '''\n",
" \n",
" return"
]
},
{
"cell_type": "code",
"execution_count": 377,
"metadata": {},
"outputs": [],
"source": [
"# Parameters\n",
"\n",
"S0= 500\n",
"K = 500\n",
"T = 2\n",
"r = 0\n",
"sigma_est = 0.9\n",
"C0= 50\n",
"it = 100\n",
"\n",
"imp_vol = bsm_call_imp_vol(S0, K, T, r, C0, sigma_est, it)"
]
},
{
"cell_type": "code",
"execution_count": 381,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 1.3 Plot Surface"
]
},
{
"cell_type": "code",
"execution_count": 382,
"metadata": {},
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