II. Module 5 Homework [practice]: Arbitrage Bounds
- Due No due date
- Points 12
- Questions 4
- Time Limit None
- Allowed Attempts 2
Instructions
I always found arbitrage bounds annoying because they seemed to require you to be clever -- oh, payoff combination so and so is always greater than payoff combination such an such, so the price must be greater too. In any sort of real situation, with maybe 5-10 assets, I can't count on being clever enough to see all the possibilities. (Suppose this question were, given put and call options at 10 different strikes, find the arbitrage bounds on a given call. Fun, eh?) Nor, can I be sure I haven't left something out. Wouldn't it be nice if there were a constructive way to find arbitrage bounds?
There is. Arbitrage bounds are a linear program. And linear programs have constructive solutions.
To see the point, state the problem as a search for the discount factor: We want to look among all strictly positive (hence, arbitrage bound) discount factors which price the stock and bond to find the discount factor which generates the largest/smallest price for the option,
s.t.
meaning, really, that we are looking for the vector that maximizes or minimizes
subject to constraints expressed similarly. Since always enter together, risk vs risk aversion is irrelevant, and we can choose instead the risk neutral probabilities as a function
Writing the problem this way
s.t.
With continuous states, we are looking for the function that
s.t.
Now look hard at what we have. You're choosing the vector/function the objective is a linear objective, and the constraints are linear constraints. This is a linear program!
This approach also gives a useful reexpression of the problem. We can look at the risk neutral probabilities that generate arbitrage bounds. If we have a set of real probabilities (like a lognormal), we can divide the real from the risk neutral and find the discount factor, and we can at least think about whether that is a reasonable discount factor for the problem. Often, it is not -- it rises as declines, it concentrates all mass on a few points, it's all ziggy zaggy, its variance implies a Sharpe ratio in the hundreds and so on. That may suggest further refinements in pricing, when you don't want to jump all the way to continuous time no-arbitrage arguments.
Ok, this is supposed to be a quiz, not a lecture.