We suppose that the price of a firm follows a nonlinear stochastic delay differential equation. We also assume that any claim whose value depends on firm value and time follows a nonlinear stochastic delay differential equation. Using self-financed strategy and replication we are able to derive a random partial differential equation (RPDE) satisfied by any corporate claim whose value is a function of firm value and time. Under specific final and boundary conditions, we solve the RPDE for the debt value and loan guarantees within a single period and homogeneous class of debt. We then analyze the risk structure of a levered firm. We also evaluate loan guarantees in the presence of more than one debt. Furthermore, we perform numerical simulations for specific companies and compare our results with existing models.

This thesis investigated Stochastic Runge–Kutta Lawson (SRKL) schemes and their application to the Heston model. Two distinct SRKL discretization methods were used to simulate a single asset’s dynamics under the Heston model, notably the Euler–Maruyama and Midpoint schemes. Additionally, standard Monte Carlo and variance reduction techniques were implemented. European and Asian option prices were estimated and compared with a benchmark value regarding accuracy, effectiveness, and computational complexity. Findings showed that the SRKL Euler–Maruyama schemes exhibited promise in enhancing the price for simple and path-dependent options. Consequently, integrating SRKL numerical methods into option valuation provides notable advantages by addressing challenges posed by the Heston model’s SDEs. Given the limited scope of this research topic, it is imperative to conduct further studies to understand the use of SRKL schemes within other models.