Convergence analysis and optimal strike choice for static hedges of general path-independent pay-offs

Abstract

In this paper we propose a new algorithm to find the optimal static replicating portfolios for general path-independent nonlinear payoff functions and give an estimate for the rate of convergence that is absent in the literature. We choose the static replication by designing an adaptation function arising in the error bound between the nonlinear payoff function and the linear spline approximation and derive the equidistribution equation for selecting the optimal strikes. The numerical tests for variance swaps, swaptions, static quadratic hedges, and also for a jump diffusion process allowing for the default of the underlying asset, show that the proposed iterative equidistribution equation algorithm is simple, fast and accurate. The paper generalizes and improves the results of the static replication and approximation in the literature.JEL classification. C, C6, C63, G, G1, G12

Explicit Pricing Formulas for European Option with Asset Exposed to Double Defaults Risk

“…Figure 1 also shows that the value of call option at time 0 under this model is the same as the one at time 0 with the stock exposed to counterparty risk (refer to Ma et al [8]); i.e., the double defaults risk model becomes counterparty default risk model when 2 = 0. This is because the default intensity 2 = 0 means that endogenous default risk has not occurred during the life of the option.…”

“…We use the conditional density approach of default, which is particularly suitable to study what goes on after the default and was adopted by Jiao and Pham [6] for the optimal investment problem, to derive the explicit distribution of the stock price at expire time and then obtain the analytic formulas for valuation of the European call and put options. We also compare the pricing results of double defaults risk model with Black and Scholes 2 Discrete Dynamics in Nature and Society [1] default-free option model and Ma et al [8] counterparty default risk option model. The rest of this paper is organized as follows.…”

“…The explicit valuation of European options with assets exposed to exogenous counterparty default risk was partly given by Ma et al [8]. Yan derived analytical formulas for lookback and barrier options on underlying assets that are subject to an exogenous counterparty risk (see Yan [9]).…”

Pricing Formula for Exotic Options with Assets Exposed to Counterparty Risk

“…≥ 0. According to [12], the joint density function under̃of (̂, ) involved in and ( ) (see (10)) iŝ,̂(…”

“…The explicit valuation of vanilla European options with this counterparty default risk was partly given by Ma et al [10]. However, the derivation of the analytic formula for pricing lookback and barrier options with this default risk model has not been done in the previous literature.…”