![]() They are really the same model just expressed differently. It gives you a future price model that allows you to price a stock as if the risk-free rate can be used to discount the future value to get the right present value. The closed form solution does not give you the actual price model. But the future value of the stock must look like a smaller number (say, perhaps, 94) so that the price today, $S_0$, is maybe 89 or some such. So, for example, if $B_0=100$ and $r=5%$ the future value of the bond in one year is 105, and its present value is 100. This is a bit of mathematical sleight of hand but it all works out the same. ![]() Simply, if you 'roll forward' a simulation the stock will outperform the bond on average, but if you see a price model under risk-neutrality the path must be such that when you discount future values to today they must give you a fair value today for the stock. ![]() As a result the future value of the stock at the same time must be below $B_t$ so that it discounts back to a lower value at $t=0$ using $r$ as the discount rate to earn a return that compensates for the risk. So if we start with $S_0=B_0$ the bond trajectory of price $B_t$ must discount back to $B_0$ when the risk-free rate is used. The closed-form solution does everything in risk-neutral space. Calculate Sum of price increment and stock price and this gives the simulated stock price value. In essence, the stock is priced today at a discount to the bond. Multiplicate this with the stock price, this gives the price increment. I simulated the values with the following formula: I started with the famous geometric brownian motion. I want to simulate stock price paths with different stochastic processes. ![]()
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