Explicit formulae for the valuation of European options with price impacts

In this work, we examine the consequences of trading a large position in vanilla European options within a multi-period binomial model framework for the underlying asset price, S. Given the significant size of the transaction, we expect both the derivative's price and the underlying asset's price to be affected by market impacts. Consequently, derivative valuation should incorporate these effects. To address this, we not only utilize a multi-period binomial model to represent the price process S but also incorporate trading impacts in a multiplicative manner. Moreover, we conduct our analysis in discrete time to better capture the influence of price impacts. Our findings suggest, for instance, that the strike price should be determined by both the trade's magnitude and parameterized market impacts. We present explicit formulas for European option prices under market impacts and offer numerical examples to elucidate our findings. Upon request, we can provide code implemented in the statistical package R. 91B24, 91B70, 91G15, 91G30.