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# Insurance Maths

Discussion in 'Options' started by UMU, Mar 7, 2009.

1. ### UMU

Let's say an event with 2 possible outcomes (for example up/down) has a 50% probability for each.
If you 'invest' (ie. bet) \$100 into this event and want insure your bet
against a loss, then how much should the insurance cost?
Can this be calculated?

You need to add the distribution of outcomes.

3. ### UMU

Oops sorry I forgot that.

Let's say it is the same amount.
Ie. \$100 brings either \$100 profit, or one loses the invested \$100.

Now the next question is: is the outcome known instantaneously? If not how long does it take to know it? In addition could you get out of the insurance or not if the result is known after some time, and you can watch the object that determines the outcome.

5. ### JJacksET4

UMU,

If you have a 50% shot of either losing the entire \$100 or gaining \$100 and the investment is \$100, why would someone insure that for a small price? Think about it - if every "bet" would be insured for just \$25, you would make a small fortune quickly by doing the bet over and over, making \$100 (minus \$25) 50% of the time and losing only \$25 50% of the time.

So, in that case, I would have to imagine it would cost around \$50 to insure, and thus wipe out any profits.

However, it strikes me that your example is much more like Blackjack then the stock market. The stock market is not usually a bet where you then lose 100% or double your money, and the final outcome often isn't decided for a long time.

For the stock market, you have to decide:
1. How long do you want the insurance to last for?
2. How much of the bet do you want insured?

For example, if you buy a stock at \$50, and you want insurance if it falls below \$50 for 2 years, that insurance will cost alot more then if you want insurance if it falls below \$40 and that insurance is only good for 3 months. That of course wouldn't be complete insurance as you would eat the \$10/share in losses before the insurance from the puts kicks in.

There may or may not be a perfect formula for what the insurance should cost, but the BS formulas basically will tell you what the insurance will cost you once you know the numbers to input into the formula.

JJacksET4

6. ### UMU

Yes, the outcome is known immediately after the event.
Just think of a coin flipping game...
I just wonder if this can be insured and how much would/should the insurance cost.
Of course this is just theory, but it could be helpful in some price model developments.

the expectancy of your win, which is \$50? So you can not arb it.

If you were to insure a sum of wins, then you need to do a bit more work.

PS: The questions you are posing suggests to me that you are posing questions similar to what Louis Bachelier posed 100 years ago! This is flattering to you, because Bachelier is regarded at the father of modern finance. Since you are posing them independently, you have a brain between your shoulders. Congrats!

8. ### spindr0

These guys can ruin in circles all they want but if they're dealing with options, there's no what the insurance "should cost" but only what the insurance "will cost" since the marketplace determines the premiums.

9. ### dmo

Here's how you calculate the fair value of any bet. You have to know each possible outcome, the probability of each outcome, and the payout of each outcome. Then for each outcome you multiply the probability times the payout. Then you add all the products - and that is the fair value of that bet.

Here's an example. Let's say XYZ is at 120. In our hypothetical example, there is a 15% probability XYZ will be at 110 at expiration, a 20% probability it will be at 115, a 30% probability it will be at 120, a 20% probability it will be at 125, and a 15% probability it will be at 130. These are the only 5 possibilities in this example. Note that they add up to 100%.

So how to determine the fair value of the XYZ 120 call? Again - list each possible outcome, and multiply its probability times its payoff:

Outcome Probability Payout
110.........15% ... x \$ 0.00 = \$0
115.........20% ... x \$ 0.00 = \$0
120.........30% ... x \$ 0.00 = \$0
125.........20% ... x \$ 5.00 = \$1.00
130.........15% ... x \$10.00 = \$1.50

Total.................................\$2.50

So in this example, the fair value of the xyz 120 call is \$2.50.

In option pricing, the tricky part is to determine the probabililty of each outcome. Determining payout at expiration is easy - it's just the intrinsic value of the option. That's what the model uses time to expiration and volatility for - to determine that probability curve.

10. ### dagnyt

Jack gave you an excellent reply.

Insurance would cost at least \$50 because no one would sell it to you for less when the outcome is known immediately. If the result were not known for a long time you would get a small discount based on the interest than can be earned from the premium.

At \$50.01, you would never buy it. Thus: No one sells insurance on a fair coin toss; no one buys insurance.

Mark

#10     Mar 8, 2009
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