European Option Pricing with Stochastic Volatility Models Under Parameter Uncertainty

Abstract

We consider stochastic volatility models under parameter uncertainty and investigate how model derived prices of European options are affected. We let the pricing parameters evolve dynamically in time within a specified region, and formalise the problem as a control problem where the control acts on the parameters to maximise/minimise the option value. Through a dual representation with backward stochastic differential equations, we obtain explicit equations for Heston's model and investigate several numerical solutions thereof. In an empirical study, we apply our results to market data from the S&P 500 index where the model is estimated to historical asset prices. We find that the conservative model-prices cover 98% of the considered market-prices for a set of European call options.

Deep Signature FBSDE Algorithm

“…From Figure 6, we can see that our method provides a better pricing bound over the recursive MARS degree 2 with variance reduction method (denoted as "MARS" in the figure) in [10], by providing a slightly wider bound for the optimally controlled value process. The closeup plots are in Figure 7.…”

“…In this section, we study the European call option pricing problem with stochastic volatility model under parameter uncertainty. The general model setup follows from [10], where the asset price S and the variance process V in the Heston model are given by…”

“…where η t ∈ R 3×1 can be computed explicitly. We implement the example in [10] with the same experiment set up: S 0 = 100, V 0 = 0.0457, r = 0.05, κ = 5.070, θ = 0.0457, σ = 0.4800, ρ = −0.767, K = 100, T = 1, and covariance matrix Σ = Diag(2.5e − 05, 0.25, 1e − 04) ∈ R 3×3 .…”

“…European call options in the Heston model under parameter uncertainty. Recall that, under the risk neutral measure Q, we consider the following Heston model in [10] for a European call option pricing problem with stochastic volatility model under parameter uncertainty. For t ∈ [0, T ],…”

“…On the other hand, the high frequency and path-dependent features show up naturally in option pricing problems and non-linear expectations within various financial contexts, e.g. limit order book [5,6,9,16,23,25], nonlinear pricing [29,37], Asian option pricing [28], model ambiguity [3,4,10,15], stochastic games and mean field games [13,31], etc.…”