Construction of Itos integral

Question

I am trying to understand the below:

Question 1 how can [W(t1) - W(t0)] = [W(t1) - W(0)] =[W(t1) - 0] =some positive number be a profit or loss? In this calculation the purchase price is not taken into consideration. The above positive number make sense if it called portfolio value at the end of time t1.

Below is the text extract from Stochastic Calculus Bk2: Continuous Time Models Also below is construction of the topic leading to the above question.

Ito integral for simple integrands

Assume that $\Delta(t) $ is constant in t on each subinterval [tj,tj+1). Such a process $\Delta(t) $ is a simple process.

Here we choose a single path of a simple process $\Delta(t) $

Values of $\Delta(t) $ depends up on a particular path $\omega $ belonging sample space $ \Omega $. If we choose a different $\omega $ then we will have a different $\Delta(t) $ for each time interval.

$\Delta(t) $ depends only upon the information available till time t.

Since there is no information available at time t = 0, $\Delta(0) $ will be the same for all the paths. Hence the first piece of $\Delta(t) $ for $ 0\leqq t \leqq t1 $ does not really depend on $\omega $

The value of the $\Delta(t) $ on the second interval [t1,t2) can depend on observations made during the first time interval [0,t1)

We shall think of the interplay between the simple process $\Delta $ and the Brownian Motion W(t) in the following way.

Regard W(t) as the price per share of an asset at time t. For here we assume W(t) can take only positive values.

Think of t0, t1,. . . .tn-1 as the trading dates in the asset, and think of ∆(t0), ∆(t1), . . . .,∆(tn-1) as the position (number of shares) taken in the asset at each trading date and held to the next trading date.

The gain from trading at each time t is given by:

I(t) = ∆(t0)[W(t) - W(t0)] = ∆(0)W(t), 0 ≤ t ≤ t1,

∆(t0) is similar to ∆(0), this is some positive quantity. Here W(t0) is similar to W(0), W(0)=0

For W(t) to be a Brownian motion, W(0) = 0 is the requirement.

I understand that Itos integral will give profit or loss at any point in time. But without taking into consideration the purchase price/cost, how can we call the value at t1 (at the the end of first time interval) as profit or loss. I can understand if it called Portfolio value.

Kindly clarify/guide.

Thank you

Answer 1

First, let me address different aspects of your question. The way the Ito-integral is built in this book is via a typical measure-theoretic argument. First the integral is defined over a simple (or step function), then using convergence arguments, the general integral is derived. I think that the book of Shreve is very short on these derivations. However, I also think this is not the main purpose of his writings.

Furthermore, one of the main starting assumptions often made in financial mathematics is that of frictionless markets. Within frictionless markets, all costs are omitted. This makes it easier to price derivatives. Note that this assumption can be easily relaxed in a later stage, such that you can obtain more realistic models.

However, for this passage from the book, you don't need frictionless markets. As Shreve is talking about a gain and not about a profit nor a loss. Just the difference between an earlier position in the asset and the current one. Due to the price change, the asset becomes more worth (or maybe less, if a negative change is considered) and therefore there is a gain. In particular, the gain will be equal to the increment of the Brownian motion. As a Brownian motion is considered, it is known (by definition) that the initial value of the process is equal to 0 and hence the gain will be $W(t)$.

I hope you find it useful.

Answer 2

I would like add another answer for completeness, should someone stumble upon this in the future.

If $(X_t:t\geq 0)$ is simple adapted process, square-integrable defined by:

$$X_t = \sum_{j=0}^{n-1}Y_{t_j}\mathbf{1}_{t\in(t_j,t_{j+1}]}$$

then the Ito integral $I_t(X)$ is a random variable in $L^2$, defined as:

$$I_t(X) = \int_{0}^{t}X_s dB_s = \sum_{j=0}^{n-1}Y_{t_j}(B_{t_{j+1}} - B_{t_j})$$

If you vary the end-point $t$, $(I_t(X):t\leq T)$ becomes a stochastic process. One can simply think of it as some investment strategy, where the stock price follows a brownian motion; $B_t$ is the price of the stock at time $t$; and you buy $Y_{t_j}$ amount of stocks, based on the information available upto time $t_j$ and sell them at time $t_{j+1}$. Then, the cumulative gain/loss is $I_t(X)$.

The above integral is a martingale transform. Martingale transforms are to Ito integral, as Riemann sums are to the Riemann integral.