Stochatic process (sample qs 70)

[cygnewhite]

70. Assume the Black-Scholes framework.
You are given:

(i) S(t) is the time-t price of a stock.

(ii) The stock pays dividends continuously at a rate proportional to its price.
The dividend yield is 2%.

(iii) S(t) satisfies
d (St)/St 0.1d 0.2d (Zt)
where {Z(t)} is a standard Brownian motion.

(iv) An investor employs a proportional investment strategy. At every point of
time, 80% of her assets are invested in the stock and 20% in a risk-free
asset earning the risk-free rate.

(v) The continuously compounded risk-free interest rate is 5%.
Let W(t) be the value of the investor’s assets at time t, t >= 0.
Determine W(t).

The solution is
For t ≥ 0, the rate of return over the time interval from t to t + dt is


d (Wt)/Wt= 0.8[d(St)/St + δdt ] + 0.2rdt

Why do we have to add δdt here? Isn't the rate of return for stock a-δ = 0.1 the same for the stock portion of this investment?

[cygnewhite]

Sorry for such a long qs. Stochastic process is a very hard topic for me because I often can't understand the solution. Any reccommend supplemental book about this topic?

[d45]

This also bothered be for a while . Then i assumed a solution and stopped thinking about it.
I assumed that dividends from the stock are reinvested in it and this solved the problem :P . I am not saying i am correct, so i would also like to know the definite solution to this.

[WhatsGolden]

I'm certainly not an expert on this, but I believe it is due to how the question is phrased. This asks for total assets rather than stock price. The stock price is modeled by using a-δ, but if we are considering total assets, this would leave out the dividend that is gained. Hence why it is added back in.

[LifeIsAPoissonProcess]

Originally Posted by WhatsGolden

I'm certainly not an expert on this, but I believe it is due to how the question is phrased. This asks for total assets rather than stock price. The stock price is modeled by using a-δ, but if we are considering total assets, this would leave out the dividend that is gained. Hence why it is added back in.

Let's say there are two identical companies, one with continuous dividend-paying stock, and one with non-dividend paying stock. The cumulative assets of having $1 in either one are exactly the same assuming dividends are re-invested. The only difference is that the stock price for the dividend-paying company will be lower, but since those dividends were reinvested, you have more shares of stock than when you started.