When several state variables move together, the FE-610 notes replace the scalar Itô formula with its multidimensional version. The Hessian term contains both variances and cross-variations, so correlation enters the drift of a function of two stocks.
\nFor a two-dimensional process driven by correlated Brownian motions, the covariance matrix is part of the model. A portfolio or derivative can therefore depend on both individual volatilities and the covariance term. The notes use a two-dimensional Itô process and Lévy's characterization to keep track of these cross terms.
\nThis is the bridge from one-stock Black–Scholes to a market model with several assets: the same calculus works, but the matrix of quadratic covariations must be carried through every derivative.
","contentMarkdown":"When several state variables move together, the FE-610 notes replace the scalar Itô formula with its multidimensional version. The Hessian term contains both variances and cross-variations, so correlation enters the drift of a function of two stocks.\n\nFor a two-dimensional process driven by correlated Brownian motions, the covariance matrix is part of the model. A portfolio or derivative can therefore depend on both individual volatilities and the covariance term. The notes use a two-dimensional Itô process and Lévy's characterization to keep track of these cross terms.\n\nThis is the bridge from one-stock Black–Scholes to a market model with several assets: the same calculus works, but the matrix of quadratic covariations must be carried through every derivative.","dataUrl":"https://sharifhsn.dev/api/posts/multidimensional-ito-calculus.json","date":"2024-10-24","datePublished":"2024-10-24","description":"When several state variables move together, the FE-610 notes replace the scalar Itô formula with its multidimensional version. The Hessian term contains both variances and cross-va…","site":"https://sharifhsn.dev","slug":"multidimensional-ito-calculus","source":"FE-610 | Stochastic Calculus","sourceUrl":null,"tags":["Stochastic Calculus","Multidimensional Models","Correlation"],"title":"Multidimensional Itô Calculus","url":"https://sharifhsn.dev/blog/multidimensional-ito-calculus/","version":"1","wordCount":116}