Options with discontinuous payoffs are called Binary options. An example is the cash-or-nothing callwhich pays nothing if the stock price at the maturity of the option is below the strike price and pays a fixed amount Q if the stock price is above the strike price. Another type of binary option is an asset-or-nothing call. This call pays nothing if the stock price is below the strike price at the maturity of the option and pays the stock price (ST) if the stock price is above the strike price. A regular European call is equivalent to a long position in an asset-or-nothing call and a short position in cash-or-nothing call.
The stock price is $50, the volatility of the stock is 30%, and the interest rate is 6%. Consider a cash-or-nothing call with a strike price of $52 and maturity of 6 months and the fixed amount it pays, when the stock price is above the strike price, is the strike price itself. Also consider a asset-or-nothing call with the same strike price and maturity as the earlier call. Use simulation to arrive the price of these two calls. Also, verify the relationship between cash-or-nothing call, asset-or-nothing call, and a regular call option. Do 1000 simulations.