Diffusion equations: convergence of the functional scheme derived from the Binomial tree with local volatility for non smooth payoff functions.

Julien Baptiste (1, 2) , Emmanuel Lépinette (2)

1 PSL - Université Paris Sciences et Lettres
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

The function solution to the functional scheme derived from the Binomial tree financial model with local volatility converges to the solution of a diffusion equation of type ht(t, x)+ x2σ2(t,x) hxx(t, x) = 0 as the number of discrete dates n → ∞. Contrarily to classical numerical methods, in particular finite difference methods, the principle is only based on a discretization in time. We establish the uniform convergence in time of the scheme and provide the rate of convergence when the payoff function is not necessarily smooth as in finance. We illustrate the convergence result and compare its performance to the finite difference and finite element methods by numerical examples.

Mots clés: and phrases: Binomial tree model; Liquidation value; Transaction costs; Diffusion partial differential equations; Finite difference scheme; European option pricing; Financial market models; European options; finite element scheme

Domaines: Equations aux dérivées partielles [math.AP]

Julien Baptiste, Emmanuel Lépinette. Diffusion equations: convergence of the functional scheme derived from the Binomial tree with local volatility for non smooth payoff functions.. Applied Mathematical Finance, 2018, 25 (5-6), ⟨10.1080/1350486X.2018.1513806⟩. ⟨hal-01507267v3⟩