Problem
Suppose you bought a stock at $100. You have placed a limit order to sell the stock if it reaches $150. This is called the profit taking price. Now you want to limit your downside and need a stop loss order. Assume that stock moves up or down by $1. In the traders view a bull market scenario occurs with the probability of an up move of p = 0.55 and that of a down move of q = 0.45. Also, in the traders view a bear market scenario occurs with the probability of an up move of p = 0.45 and that of a down move of q = 0.55. What is the best stop loss to set? Use the Gamblers Ruin methodology.
Now suppose that bull market (and hence the bear market) probabilities change to p= 0.60 and q = 0.4. How does the stop loss change? Plot a few such probabilities (p) versus their best stop losses. Does the plot make sense? Explain the relationship intuitively.