Pricing Deterministic Payouts (Finance Course 1.1)
The beginning of a multi-month twitter spree covering finance and option pricing.
I originally wrote this as a twitter thread, but I thought it might be useful to send it out on substack as well. The style and flow might be a little different for that reason, but you will get the key points and I’m sure you can forgive me for the lack of literary flair you have become accustomed too.
Deterministic Payouts
Every house needs a solid foundation, the following thread is nothing revolutionary or even particularly interesting, but we need to define the foundation for the Option information we will cover in the future.
Fair warning, these threads will not be for everyone.
We will be covering a lot of interesting things over the course of the next few months. I had to dust off some old textbooks and watch a few Youtube videos, but I think it will be worth it.
That being said, let’s start with Pricing Deterministic Payoffs. Suppose you can earn money at an annual interest rate (r) with a risk-free asset. Let’s take borrowing and lending USDC to Maker for example.
Lets assume that like a bank account Maker offers a fixed rate (r).
(Bank accounts have deterministic interest rates. So when you put your money in the bank you can measures mathematically how they screw you.)
Because we know r, we can determine how much $1 will be worth in the future.
Lets dive into an example. (Interest rate is denoted by r if you didn't catch that.)
At present value (PV) $1 = $1
How do we find Future Value (FV) after 1 year?
Answer: $(1+r).
Now if you wanted to find out how much our USDC would be worth in a few years, we would simply add time (T) into the equation.
FV= 1 +T*r
This is simple interest, in other words it doesn’t take compounding into account.
Now simple interest isn’t realistic, as most of us compound our lending fees frequently. So what would the equation look like if we compounded only once a year.
Its simply Future Value + whatever we earned in interest to the power of however many years (T), so if we wanted to know 2 years into the future it would be T=2.
FV+(1+r)T.
Being a sophisticated degenerate we will most likely compound our USDC more than 1 time a year. How is this calculated? First we need to define two more factors, n and m.
To clarify, r is the nominal interest rate. You divide r by compounding frequency (n), and you put that to the number of compound periods (m), which corresponds to the future time.
Example: 8% (r) / 4 times a year (n), put to 2 years * 4 times a year = 8 (m).
We can also find our effective annual interest rate or r prime (r’). First we find what our USDC would become in 1 year (1+r/n)n. This will be equal to our annual effective rate.
(1+r/n)n=1+r’
Example:
Say we wanted to find our effective annual interest rate by quarterly compounding at a nominal annual rate r = 8%. (For example sake lets say Maker is paying us 8% on our USDC)
(1+.08/4)4=1.0824= 1+0.0824
Therefore our effective annual interest rate would then be 8.24%.
When we get to future models it will look more elegant to use continuous compounding interest rate. We aren't going to spend a lot of time on this presently, but the calculation is below.
(Compounding will render more yield than simple, but up until what point?)
If you want to read about Exponential functions here is a link. https://alamo.edu/contentassets/afe30946fa58450c89840c1173f3b9d0/exponential/math1314-exponential-equations-base-e.pdf…
Lastly, if you want to find what your USDC would be worth after a specific amount of time continuously compounded, you simply multiply the exponent by T years as seen in the equation below.
Ok, enough of that.
Recap: So far we have calculated what your actual interest rate would be with and without compounding, nothing a simple excel sheet or calculator can’t do for you.
Now we need to touch on just one more thing and we will be finished for today. We want to now reverse everything we just talked about. This will be a little more complicated than it needs to be because we will carry this concept over into random payouts.
What is the value of the deterministic payoff in the future worth today? Put another way, if I have $108 dollars in the future, how much is that worth in today’s dollars?
Law of One Price: If two sequences of payments pay the same amount at the same time in the future, they should be worth the same price (value) today.
So if I know that that I can have $X(T) at time of T, by investing $X(0) today, then today price of X(T) should be X(0). As you will see later, you can create X(T) by buying a Bond or Option.
Note: For deterministic payouts, X(T), X(0) is called present value, PV(X(T)).
Example: If I know that Maker is paying a fixed 8% on USDC, and I invest $100 (X(0)), I know that I will have $108(X(T)) at year end.
Therefore, $108 a year from now is worth $100 now. This is how we are going to define present value (PV). PV(X(T)).
You can add in the compounded rate as well. To find this its just X(T) divided by the rate over n to the m.
This is equivalent to,
Which simply means if you invest X today and multiply it it by the rate (r) compounded over time, you get what your money will be worth in the future, in a certain number of periods from now.
Nothing like beating a dead horse eh?
To put it another way the present value is simply dividing the future value by the compounding factors. Like I said, this is more complicated than it had to be, but again we will need this for future threads on Options.
Stay tuned for more useful information in the future, like how to actually price Options and other financial products.
Thank you for coming to my Ted Talk.