Option Strategies for Trading on Bounded Stock Moves

# What is the strategy for this piece of information


# What is the strategy for this piece of information












Heavy Math background, very light finance background:

Suppose I have a stock $S$ whose price is measured by the market once on times $t_0$ $t_1$ $t_2$.

Now the market has some opinion for how the stock behaves and it has priced the stock and options derived on it accordingly: (say S(t+1) - S(t)) is normally distributed with mean $0$ and standard deviation $\sigma$.

Now suppose a oracle (or insider?) approaches you and says that $a_0 \sigma > |S(t+1) - S(t) | + |S(t+2) - S(t+1)| > a_1 \sigma$.

For some constants $a_0, a_1, a_0 - a_1 < \sigma $ which is a much tighter bound than what the market can have any reasonable opinion about.

What sort of portfolio can you construct to profit off of of this? Using just going long and short, calls and puts, as well as long and short the underlying asset I can't seem to cook up any portfolio and am wondering if there is an algorithmic way to make a portfolio, or a systematic way to prove its not possible.

### Additional Information:

Say I posed the question as the oracle lets you know the $S(t+1) > S(t) + \sigma$. Then a portfolio would be to buy call options with strike price equal to the current price.

Say I posed the question as knowing that $|S(t+1) - S(t)| > \sigma$ then a portfolio, would be to go and buy a call and a put option with strike price equal to the current price. (Since either way the stock is high enough or low enough that one of the options can be executed to cover the cost of the initial and yield some profit, independent of which direction the stock moves).

In that sense, here I have another inequality, and I want to construct a corresponding portfolio for this inequality.

## Answer by Attack68 (score 2, accepted)

https://quant.stackexchange.com/a/41293

I just want to attempt to clarify something about your question:

> Say I posed the question as the oracle lets you know the S(t+1)>S(t)+σ. Then a portfolio would be to buy call options with strike price equal to the current price.

Well actually, no, I wouldn't do this. In this case the precise information possessed is that the stock price at expiry will be greater than the current price by $\sigma$, but you do not know by how much. If you buy a call option with a strike price equal to the current price you have unnecessarily introduced market risk into your profit, i.e. the higher the price goes then the more money you make, but your profit is undetermined.

However, if you go long the market (+1 delta) and sell a call option with a strike at $S(t)+\sigma$ then with certainty you will accrue a profit of the option premium plus $\sigma$. The value of the information is then a fixed deterministic amount. This is a covered call, but equivalently you could also just sell a put option with strike $S(t)+\sigma$, (put-call parity).

> Say I posed the question as knowing that |S(t+1)−S(t)|>σ then a portfolio, would be to go and buy a call and a put option with strike price equal to the current price.

Well for the same reason the information gives certainty about a range so, instead of doing a straddle as your comment suggests, I would do a reverse iron butterfly. The value of that information can also be determined as a fixed amount, in that case, with the same profit for all outcomes conditional upon your insider information.

## With regards to the question

You will observe that as at time, $t+1$, $|S(t+1)-S(t)|=k$ is a known parameter. Therefore as a trader with flexibility of timing of execution, one particular choice (I am not saying I can prove this is optimal) would be to wait until time $t+1$ and then you have the information about the price at time $S(t+2)$: i.e.

$$a_0 \sigma - k > |S(t+2)−S(t+1)| > a_1 \sigma - k$$

Now you are reduced to some specific cases:

- Case 1: $a_1 \sigma - k > 0$ then the reverse iron butterfly can be applied as above (since you have a lower bound of market movement). You also have an upper bound of market movements to further refine your trading window from which you profit by selling a strangle. (executed at time $t+1$ dependent on $k$).

- Case 2: $a_1 \sigma - k < 0$ and $a_0 \sigma - k > 0$ then you only have an upper bound, the lower bound is replaced by 0 (so the reverse iron butterfly no longer works) but the upper bound strategy in case 1 will still generate fixed profit by selling a strangle dependent upon $k$.

- Case 3: $a_0 \sigma - k < 0$. Well this actually contradicts your piece of information since by definition of modulus $|S(t+2)−S(t+1)| \geq 0 $ hence you can conclude that $k = |S(t+1) - S(t)| \leq a_0 \sigma$, and therefore this opens a trade option at time $t$ with expiry $t+1$ again playing on the fact that you know the price range. So again you sell a strangle here with strikes dependent upon $a_0 \sigma$.

If I was assessing the value of the piece of information I would assess the value of the strangle in case 3 as a fixed amount, and plus the values of cases 1 and 2 integrated over all legitimate values of $k$, which represents the expectation of the informational value from time $t+1$ to $t+2$.

Have only pondered on this for a brief period so I welcome peer review and criticism of these ideas.

## Answer by amdopt (score 2)

https://quant.stackexchange.com/a/41283

> What sort of portfolio can you construct to profit off of this? Using just going long and short, calls and puts, as well as long and short the underlying asset I can't seem to cook up any portfolio

This is by no means a comprehensive list, but three strategies come to mind when looking for a long vega position.

- Long straddle/strangle

- Short butterfly/Short Iron Condor--both function the same way. You profit from a move beyond of predetermined limits that depend on the strikes that you choose.

- Calendar/Reverse Calendar spreads--You could do a lot with these including be vega AND theta positive at the same time. You may also find these called Time spreads or Horizontal spreads.

> In that sense, here I have another inequality, and I want to construct a corresponding portfolio for this inequality.

Here you seem to be looking for an option on one of the options strategies (or combination of them) I mentioned above. I'm not sure why this would be necessary when (according to the scenario you describe) you already have a stock that can be bought or shorted and vanilla options derived from that stock which you can do pretty much anything with. That being said I'm sure you could find a counterparty to trade with but, why reinvent the wheel with a more complex wheel that functions the same way and yields the same result?

I'm happy to elaborate on anything above but a basic Google search for those options strategy names will produce tons of results. Good luck!