I can not find Transition Probability Matrix that adds up to 1 in the first row. Am I missing something ?
Here is the definition of problem:
An investor bought shares of a company at $30 per share. Currently the value of the
share is $31. The investor will sell the shares either as soon as he makes 10% or more,
or when the stock drops to $30. At the end of each trading day the stock gains $2 with
probability (w.p.) 0.1, $1 w.p. 0.3, remains the same w.p. 0.3 and loses $1 w.p. 0.3.
What is the probability of making at least 10% ?
Can someone point me the right direction ? Thank you in advance.
You should define your states more carefully.
Let there be 4 states in the state-space S = {0, 1, 2, 3}. Let us introduce the idea of “unrealised” profit/loss (URPL), which is the amount you would have gained/lost, supposing you close all positions (i.e. sell all stocks) immediately. For example, in your problem, initially the stock is at 31 while the investor bought it at 30, so the initial URPL is 31 - 30 = 1.
Now note that you have set up a “limit order” which caps your URPL at 3 (or 4). You also have a “stop loss order” which caps your URPL at 0. After the URPL reaches these values, it stops moving because the investor will sell the stocks, realising his profits/losses. In terms of Markov Chains, these are absorbing states.
Now let’s describe the finite state-space S:
0: URPL = 0
1: URPL = 1
2: URPL = 2
3: URPL = 3 or 4
Now, can you set up the problem?