In the execution cost problem, an investor wants to minimize the total expected cost and risk in the execution of a portfolio of risky assets in order to achieve desired positions. A major source of the execution cost is the price impact of both the investor's own trades and other concurrent institutional trades. Indeed, the price impact of large trades has been considered to be of the main reasons for fat tails of the short-term return's probability distribution function. However, current models in the literature on the execution cost problem typically assume normal distributions. This assumption fails to capture the characteristics of tail distributions due to institutional trades. In this paper we argue that compound jump-diffusion processes naturally model uncertain price impacts of other large trades. This jump diffusion model includes two compound Poisson processes where random jump amplitudes capture uncertain permanent price impacts of other large buy and sell trades. Using stochastic dynamic programming, we derive analytical solutions for minimizing the expected execution cost under discrete jump-diffusion models. Our results indicate that, when the expected market price change is nonzero, likely due to large trades, assumptions on the market price model and values of mean and covariance of the market price change can have a significant impact on the optimal execution strategy. Using simulations, we computationally illustrate minimum CVaR execution strategies under different models. Furthermore, we analyze qualitative and quantitative differences of the expected execution cost and risk between optimal execution strategies, determined under a multiplicative jump-diffusion model and an additive jump-diffusion model.