Deep combinatorial optimisation for optimal stopping time problems and stochastic impulse control. Application to swing options pricing and fixed transaction costs options hedging

American-style options are used not only by traditional asset managers but also by energy companies to hedge "optimised assets" by finding optimal decisions to optimise their P&L and find their value. A common modelling of a power plant unit P&L is done using swing options which are American options allowing to exercise at most l times the option with possibly a constraint on the delay between two exercise dates (see Carmona and Touzi (2008) or Warin (2012) for gas storage modelling). Formally, for T 0, we are given a stochastic processes ( X t) t 0 defined on a probability space (Ω, F, F ( F t) t 0, P) and one wants to find an increasing sequence of F stopping times τ ( τ 1,τ 2,...,τ l) that maximises the expectation of some objective function f E Pnull l null i 1f ( τ i,X τ i) 1 τ i Tnull . Numerical methods to solve the optimal stopping problem when l 1,f ( x,t) e rt g (x) and X is Markovian include: - Dynamic programming equation: the option price P 0 is computed using the following backward discrete scheme over a grid t 0 0 t 1 ... t N T: P t N g ( X T), P t i max( g ( X t i),e r (t i 1 t i) E P( P t i 1 F t i)), i 0,...,N 1 . One then needs to perform regression to compute the conditional expectations, see Longstaff and Schwartz (2001) or Bouchard and Warin (2012).