Why is the discounted stock price modeled as an exponential? (reading "Stochastic Calculus for Finan

[Hugolp]

I am reading "Stochastic Calculus for Finance II" by Shreve to learn about Black-Scholes and derivatives. Until now, coming from an engineering background, Im getting everything, but I have what I think is probably a very simple (read stupid) doubt. From page 155:

We shall often consider the discounted stock price e^(- rt) s(t) and the discounted portfolio value of an agent, e^(- rt) X(t).​

Where s(t) is the stochastic ecuation modeling the nominal price of a stock and X(t) the same but for the nominal price of the portfolio.

I understand why it can be useful to adjust the price by the (assumed) risk free interest rate since it can give you a better baseline than the nominal price, but what I dont get is why is it an exponential?. Im pretty sure is something very stupid/simple Im missing but since Im studying on my own I dont want to just give it a pass and assume/believe things without understanding them.
So can anyone explain me why the discounted price is an exponential? Thanks in advance.

 

[babinu]

I am reading "Stochastic Calculus for Finance II" by Shreve to learn about Black-Scholes and derivatives. Until now, coming from an engineering background, Im getting everything, but I have what I think is probably a very simple (read stupid) doubt. From page 155:

We shall often consider the discounted stock price e^(- rt) s(t) and the discounted portfolio value of an agent, e^(- rt) X(t).​

Where s(t) is the stochastic ecuation modeling the nominal price of a stock and X(t) the same but for the nominal price of the portfolio.

I understand why it can be useful to adjust the price by the (assumed) risk free interest rate since it can give you a better baseline than the nominal price, but what I dont get is why is it an exponential?. Im pretty sure is something very stupid/simple Im missing but since Im studying on my own I dont want to just give it a pass and assume/believe things without understanding them.
So can anyone explain me why the discounted price is an exponential? Thanks in advance.

This is happening because we are using continuous compounding. In general if we have n compounding periods, the rate would be (1+r/n)^nt, where r is the rate for n periods. As n goes to infitinity , this comes to e^rt. Let me know if you are clear with this.

Thanks
Babinu

 

[Jeffrey K.W. Lo]

If it was annual rate then it'd be s(t) / (1+r)
If it was semi annual then it'd be s(t) / (1+(r/2))^2

In the case of e^-rt, it's continuous compounding, google/wiki for more info

Jeffrey

 

[Hugolp]

This is happening because we are using continuous compounding. In general if we have n compounding periods, the rate would be (1+r/n)^nt, where r is the rate for n periods. As n goes to infitinity , this comes to e^rt. Let me know if you are clear with this.

Thanks
Babinu

Yes, it makes sense. Thanks.