# Pricing a vanilla call option with a fixed dividend

I have started a finance course few months ago and am looking for a way to compute the price of a 1-year call option with a fixed dividend paid after 6 months. Using Black and Scholes I know how to compute the price of the option with a continuous dividend q but not for a fixed dividend D. My initial guess was to use a binomial tree, but I am not sure it is the optimal approach. If some of you are familiar with the topic, I would gladly read what they have to say. Thank you! :D

Sophie

## 1 Answer

There are a couple of options that you can use to account for dollar amount dividends. Firstly, if dividends are expected to increase or decrease in proportion to the stock price, you can convert the dividends into a percentage by dividing the latest dividend by the last stock price on the day the dividend was declared and multiply by the number of dividends in a given year. For example, if the stock price is \$100 and the company issues a \$1 dividend quarterly, then the annualized yield is 4% (1/100*4). The continuously compounded yield for Black Scholes would be $ln(1+0.04)$.

Alternatively, the reason that dividends are important in Black Scholes is because dividends would not be paid to the holders of the options. If you think about it, then, you could remove the present value of all dividends in the term of the option (discounted at a risk adjusted cost of equity) from the present value of the stock price. This would account for the future dividends paid and treat the company as a company that does not pay dividends.

Because you are looking at one fixed dividend in 6 months, I would use the latter method because you do not know what the stock price will be at that time.