#### Discover more from Paper Alfa - Macro & More

Thank you for all your feedback in regard to my previous post on **options**. Hearing from my followers how useful some of the theory and basics are I dug out an old presentation which gives a bit more detail and focuses mainly on fixed-income options.

Are you ready? Letâ€™s dig. I have also created a link **here** to download the whole thing for your own use.

I find the intro to the presentation excellent as it tries to simplify and give some intuition as to the pricing of options.

The example of an option on varying heights is quite a good one. This should all be clear here and the payoff also intuitive.

We then enter the option valuation game, understanding expected value, time decay and volatility.

So here are the inputs for your valuation:

Knowing the vol, you know the expected variability from todayâ€™s price, i.e. 6600 and 6800.

What are the values at expiration in each of the two scenarios? Simple and still quite intuitive, hopefully. Working through these examples will help you.

Again, shouldnâ€™t be overcomplicated to calculate the expected value of the option = 50 and the resulting discounted back value of 49.50. Just create a spreadsheet and work it out.

Now, this is quite useful in determining how an increase in volatility changes the price of the option. Higher vol, higher option price.

Intuition still in play? Well done.

Now some simple definitions regarding volatility.

For those of you not familiar, make sure you note that down.

We covered the measuring of volatility in the previous post, but good to hammer down the facts once again just for good measure.

If you donâ€™t have a spreadsheet open by now, Iâ€™d urge you to do this now and work through the below example in order to calculate price volatility yourself. It will help you understand things better.

**Step 1: Collect closing prices** **(I typed them up so you can copy paste them into your spreadsheet, good huh? )**

P1 = 99.48

P2 = 99.73

P3 = 99.00

P4 = 98.00

**Step 2: Calculate daily returns in %**

**Step 3: Calculate the daily standard deviation with those returns.**

Be no slacker here, do the calculations, do it.

So there you go, daily volatility is 0.663%.

Converting them into annual volatility is simple. An remember that volatility scales by square root of time.

**Step 4: Convert daily vol to annualised volatility. **

Here a handy guide as to how to scale any volatility measure up or down.

## Fixed income specifics in regards to volatility

This is quite specific. I personally use basis point (bps) volatility to measure fixed income, itâ€™s also more intuitive, personally.

So most pricing in fixed income will be price related. In order to convert price volatility to bps volatility we divide price vol by duration. See for other conversions below.

Here is a good example of calculating bps vol. Duration in the example is calculated by diving Dollar duration by the futures price. (456 / 108.53).

Right, I think thatâ€™s quite a lot to take in for now. I will come back and address the remaining parts in a follow-up post. We will then go through the greeks again and go through a bit more detail in regards to various strategies, etc.

Hope you enjoyed it!

Your,

Paper Alfa

## A Bit More on Options ...

Given this formula:

Annual volatility = sqroot(Time periods) x daily volatility

When calculating annual variance we multiply by 251 (time periods). So when translating this into the annual std dev, we also have to sqrRoot time periods right? i.e this is the reason for squaring root the time periods no?

If we want to compare implied vol and historical vol, for historical vol: do we take sample std or population std (i.e divide by n or n-1)?