Superreplication of European multiasset derivatives with bounded stochastic volatility

Abstract

In this paper we analyze the superreplication approach in stochastic volatility models in the case of European multiasset derivatives. We prove that the Black-ScholesBarenblatt (BSB) equation gives a superhedging strategy even if its solution is not twice differentiable. This is done under convexity assumptions on the final payoff h that are verified in some applications presented here.

Approximations and asymptotics of upper hedging prices in multinomial models

“…We consider the following three values of a 4 as depicted in Figure 3 Next we consider the payoff f = sin(10S n ), which has a lot of changes from convexity to concavity, and similarly calculate the upper hedging prices. The results are shown in Figures 7,8,9. Also in this case the upper hedging prices under the quadnomial model equal those under the trinomial model with increasing N, provided that a 4 = 1.5.…”

On properties of solutions to Black–Scholes–Barenblatt equations

“…where C > 0 and m ∈ N depending on ϕ. We have the following solvability result for the viscosity solution of BSB equation (1.6) (see Theorem 7 in Gozzi and Vargiolu [4]).…”

A uniform asymptotic expansion for stochastic volatility model in pricing multi‐asset European options

“…where 1 , 2 are expected return rates of S 1 , S 2 respectively, W (1) and W (2) are standard Brownian motions. As in [7], in this paper we assume that the volatilities 1 (t, ) = f (Y 1 ) and 2 (t, ) = g(Y 2 ) are functions of diffusion processes on the circles S 1 and S 2 in a 'fast' time scale t/ε and t/ with…”

“…As in [4] we assume the stochastic volatilities to be fast mean-reverting and the driving diffusion process are on circles. Multi-asset option models are more natural in real world, which are widely studied, see for example [6,7]. To avoid risk, investors would like to choose portfolio which consists of stocks, bonds and so on.…”