![\"Transposition](\"images/Xlogo.png\" "\\"Vector")
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" \n", " Stochastic calculus and control - tutorial
\n", "Stochastic Calculus
\n", "Eduardo Abi Jaber
\n", "\n",
" \n", " Stochastic calculus and control - tutorial
\n", "Stochastic Calculus
\n", "Eduardo Abi Jaber
\n", "Who am I? ๐
\n", "![\"Transposition](\"images/rock3.png\" "\\"Vector")
\n",
"\n",
"\n",
"---\n",
"\n",
"# Bachelier model\n",
"\n",
"\n",
"First stochastic model for the stock price has been introduced in 1903 by `Louis Bachelier` in his PhD thesis under the supervision of the celebrated mathematician Henri Poincarรฉ. \n",
"\t\n",
"\n",
"![\"Transposition](\"images/Bachelier.jpg\" "\\"Vector")
\n",
" \n", " Louis Bachelier (1903)
\n", "$\\bullet$ Bachelier is considered as the forefather of mathematical finance and a pioneer in the study of stochastic processes.
\n", "$\\bullet$ PhD not well received because it attempted to apply mathematics to an unfamiliar area for mathematicians\n", "
\n", "$\\bullet$ Although Bachelier's work on random walks predated Einstein's celebrated study of Brownian motion by five years, the pioneering nature of his work was recognized only after several decades, first by Andrey Kolmogorov\n", "
\n", " \n", " In the Bachelier model, the stock price is stochastic and follows a Brownian motion \n", "$$S_t = S_0 + \\sigma W_t, \\quad t \\leq T, $$\n", "where $S_0>0$ is the initial price of the asset, $T$ is the investement horizon, and $\\sigma>0$.\n", "\n", "\n", "> What is a `Brownian motion` ?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "![\"Transposition](\"images/BS.jpg\" "\\"Vector")
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"\n",
"\n",
"Scholes received the 1997 Nobel Memorial Prize in Economic Sciences, Black wasn't awarded because he died in 1995.\n",
"\n",
"\n"
]
},
{
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"Let $W$ be a scalar Brownian motion and define\n",
"$$\n",
"S_t = S_0 \\exp\\!\\left(\\int_0^t \\Big(r(s)-\\tfrac12\\sigma^2(s)\\Big)\\,ds + \\int_0^t \\sigma(s)\\,dW_s\\right),\n",
"$$\n",
"for some constant $S_0>0$ and some deterministic functions $r$ and $\\sigma$.\n",
"\n",
"1. Provide an explicit expression of $\\mathbb{P}(S_T\\ge K)$ for some $K>0$ in terms of the cdf $\\mathbf{N}$ of the standard $\\mathcal{N}(0,1)$ distribution.\n",
"\n",
"2. Provide an explicit expression of\n",
"$$\n",
"\\mathbb{E}\\!\\left[e^{-\\int_0^T r(s)\\,ds}(S_T-K)^+\\right].\n",
"$$\n",
"\n",
"3. Prove that $S$ solves a stochastic differential equation.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"toto = 2 "
]
},
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"cell_type": "markdown",
"metadata": {},
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"\n",
"\n",
"