Probability Basics and Expected Value

Probability is defined as a chance that a random number will have a particular outcome. For example, there is a 50-50 chance (hence 50% probability) that a fair coin will have a heads land up. If you gather all the possible outcomes of a random variable, such as the outcomes of the dice when it is rolled up, it generates a distribution which is the key in Probability space as it gives an insight of all possible values of the Random Variable.

What is a Random Variable: 
Random variable is just a name that can be anything where an analysis needs to be performed. Common examples are, outcome of a toss coin, outcome of a Dice roll, Daily returns of the financial instrument and so on. Once you know the outcomes of the Random variable, you can assign the probabilities attached to it to generate Probability Distribution. It is this probability distribution that helps to forecast future outcomes with certain confidence.

There are 4 main topics that one must consider while working with Probability.
1. CDF - Cumulative Distribution Function - CDF, F(.), of a random variable, X, is defined as:
                F(x) := P(X <= x)
                (Probability of X is less than or equal to x)

2. PMF - Probability Mass Function. p(.) is a Probability Mass function for the random variable, X,
                P(X in A) = Summation (p(x)) for all x in A

3. Expected Value: Expected value of a random variable, X, is given by the sigma of probability weights of the random variable.
                E(X) := Summation (Random value * Probability of Random Value)

4. Variance: Variance of a Random variable, X, is defined as
                Var(X) := E[(X - E[X]^2)]
                             = E[X^2] - E[X]^2
Variance is also known as Mean of the Squares - Square of the Mean.

Real World Application: 
What is the expected value of a fair dice roll?
As we know a fair dice will have 6 faces, numbered from 1 through 6 with each face with equal probability of 0.1667 or 16% chance of each number showing up.

Expected value then becomes:
0.1667 *(1) + 0.1667 *(2) + 0.1667 *(3) + 0.1667 *(4) + 0.1667 *(5) + 0.1667 *(6) = 3.5

Therefore one can say that the average value of a dice outcome is 3.5 or there is a 50-50 chance of beating the average value in which case 50% of the times, a dice can have a value higher than Expected value (4,5,6) or 50% chance that it can have lower than expected values of 3.5 (1,2,3).

Simple stuff - isn't it?

Nitin

High Frequency Trading Overview

High Frequency Trading Overview: 

In this post, I would like to talk about a little bit about High Frequency Trading, some of the key players around it and so on.

So what is HFT and how is it different than Traditional Trading? 
High Frequency Trading refers to fast allocation/re-allocation or turnover of trading capital. In Traditional trading, a trader will look at some of the components of the trading strategy such as Charts with indicators (moving averages, RSI etc...) before committing capital to the trade. On the other hand, in HFT, computers make the decisions to fire the trade based on some econometric model that is fully tested out (back-tested) by the trader. In HFT, trade execution speed is the key as the trades might be executed in a fraction of seconds/milliseconds/nanoseconds and traditional traders can't operate at such a low latency.

* In HFT, Low latency refers to the speed of executing an order. A low latency can be a trading strategy in its own right when the high speed of execution can be used an arbitrage strategy where there are price difference found on the same security on multiple exchanges for the same security.

There are 4 key characteristics of HFT:
1. Tick by Tick Data Processing
2. High Capital Turnover (HCT)
3. Intraday Entry and Exit
4. Algorithmic Trading

Some of the key players in HFT are: DE Shaw, Renaissance Technologies, and Towa Research Capital

3 Major components of HFT system are:
1. Highly Quantitative, Econometric models that forecast short term price moves based on contemporary market conditions
2. Advanced Computer systems built to quickly execute the complex econometric model
3. Capital applied and monitored with in Risk Management and Cost Management framework

In practice, an HFT firm will use 2 years of tick level data to backtest a strategy before committing live capital on it. Since setting up an HFT strategy requires pretty sophisticated level of understanding in Finance and Economics, most of the staffing at such firms are Ph.D in Quantitative Research (Finance/Economics/Physics)

Finally, an HFT operation is more likely to survive & proper if it had leverage and high Sharpe Ratios. Leverage helps to cover the operational costs and Sharpe Ratio helps in reducingthe risk of catastrophic losses and fund ruin.

Please feel free to comment should you have further questions related HFT.

-Nitin

What is Basel III

What is Basel III 

Basel Accord is a set of banking regulations & standards designed to maintain the integrity of financial system. The original set of standards were called Basel I, followed by II, and III. Basel III is a crucial regulatory response to financial crisis and a major step forward towards a stronger and stable banking system. These standards are developed by Basel committee in Switzerland and then each country copies the framework and applies into domestic financial sector. Basel III is a successor of Basel II and hence more rigid with its requirement so as to keep healthy financial system globally.

Basel III focuses on Capital adequacy requirements to have a more resilient banking system such that during stress times banks have enough liquidity to keep a fully functional banking system without government intervention. Also one of the goals of the Accord is to substantially lower the cost of economic loss given shocks in the economy.

Before we go any further on it, one must understand the functioning model of banking system. Banks have usually deposits as cash which we all deposit as a part of our salary, trading account or whatever else. Banks usually need to reinvest the capital it collected back into capital markets to generate more money. Banks usually use this capital to give out loans to consumers, bankers checks, short term loans to corporations or they even themselves go ahead and buy some assets such as US Treasury or Euro denominated bonds, Gold  etc...

During crisis, banks usually go through difficult circumstances such as :
1. Asset values might start to go down faster than banks original predicted. These assets might lose 20-30% in their value within weeks or so.
2. Depositors might start drawing cash faster than bankers originally anticipated.

If these situations continue for a prolonged period, banks might themselves run into liquidity crisis where they don't have any more cash left to give back to depositors or any sort of money to lend out. So, their assets on  the balancesheet are losing value and their cash deposits are depleting faster than anticipated. Under these circumstances, banks will run out to Central Government for protection. Governments also have to go through difficult choices at this time. Government usually have following 2 choices only:

1. They can let the financial institution fail and hence cripple the financial markets further. OR
2. Rescue them and hence adversing the moral hazard. (This one is really tricky as commercial banks have the benefit of taking risky bets knowing the central governments can bail them out while the end consumer might not receive any benefit from it. During good times banks can make money off the consumers and during bad times banks can make money off governments so they are in win-win situation)

So, now what Basel III Accord truly wants is to restrict banks from being too dependent on government bailout and hence making regulations for better functioning financial system.

4 Keys areas of Basel III regulations are:

1. Risk Management & Corporate Governance: During Financial crisis of 2008 till today, it was observed while Risk Management and Corp Governance policies were implemented across banking sector, they were pretty weak. Lack of governance influenced risky bets and hence the crisis. Basel III looks into this issue and make recommendation for improvement.

2. Better and More Capital: Basel III looks for better Quality and Quantity of Capital Requirement. The accord requires 7% of Risk Weighted Capital must be common equity with highly liquid assets. In Basel II, the requirement was only a mere 2%.

3. Systemic Risk: Systemic Risk is defined as the event in which the failure of few banking institutions can lead the financial system collapse and hence a system failure. Basel III looks into this issue and tries to provide solutions in at least 2 dimensions (Time and Cross Sectional)

4. Liquidity Management: Before financial crisis of 2008, all banks were taking a free ride on liquidity. It was treated as a free good with the impression that everyone will have access to it. However, after the crisis, it was no longer a free good. Central Governments across the globe (FED, BOE, ECB, RBA etc...) have to pump in liquidity to have money markets fully functioning. This is the first time when global liquidity rules are defined to ensure that there will be enough liquidity under stress times available in the banking sector to keep them functioning.

This is all for Basel III

-Nitin

Symptoms to Worry about Personal Finances!!

When to worry about your personal finances? 

While this observation started with noticing American behavior, the symptoms described below can be hold universally true regardless of the location. It's people's habits that puts them into financial trouble and not the location but American are used to exploit it to the extreme end of the spectrum.

Americans are known to live beyond their means. American Consumer Debt, US Govt Debt, and everything around Debt have sky-rocket records around Americans. Most of the Americans would look for path of least resistance to earn the extra dollar that they need. They rather buy Lottery, which is poor man's tax, than doing actual work which will make them millionaire. Statistics have shown that you have a better chance of getting hit by a lightning storm than winning a lottery yet people are willing to do the same thing over and over and expecting a different result each time. It's sad but the truth is most of the American Wealth is highly skewed to the right where few people own most of it and others barely put food to the table.

Following framework might show symptoms when to worry about your personal finances:
1. Your income is not increasing (or perhaps decreasing) but you are still spending too much money on credit cards. A simple outstanding balance of "Month Over Month", MoM, will paint the picture clearly.
2. You are doing credit card balance transfers too many times. With each Balance Transfer, credit card companies charge a one time processing fee. So, you already paid more money in Bank's pocket than keeping in yours.
3. You are applying for new credit cards all the times since you maxed out on your existing ones
4. You are missing payments on your credit cards or paying the minimum payment most of the times
5. You are using credit cards for every little purchase such as Gasoline or even to pay your rents
6. You are using credit cards with merchants even when they charge a convenience fee of 2.5 - 5%
7. You have converted your Amortized Mortgage to an Interest Only Mortgage to lower down your payment schedule
8. You do not have an accurate picture of how much debts you truly have (Exclude Primary Mortgage)
9. You might have borrowed money from your 401K or IRA or any other Retirement Fund
10. You have taken money from your family or friends that you are not ready to return yet

This framework gives some guidelines when people are already into the trouble or very soon will be entering into it if appropriate actions are not taken.

How to avoid these pitfalls or manage finance better? 
Money management is an art than science. Most of it comes with a discipline and strong adherence to the rules that money management brings. If you don't follow it or try to find back-door to escape from it, you will get into financial catastrophe.

Following framework might help you to manage your finance better:
1. If you can't pay your credit card balance completely with your monthly statement, don't use it. You should start getting into the habit of using ATM cards aka Debit Cards than credit cards. Since Debit cards debit your account right away, you will have a much clear picture of your savings. Don't get caught up by those Air miles on credit card if you can't make full payment on credit card by End of Month.
2. Check your Bank Account on daily basis or at least every other day to ensure that you don't have any surprised bank fees. If you see so, call bank immediately and ask for a one time waiver. Make sure you understand why such costs were there so you don't get into the same situation
3. Stop Spending money on things that are not needed until your credit cards are paid off. I know you would have to convince your wife to stop spending but it's worth it. In the long run, a wife with a debt free husband will be much happier than indebted husband. So, be ready for the fights but convince her that this is the right thing to do.
4. Pay the credit cards that have the least balance first. And, just follow the same protocol until you have paid off everything
5. Be proactive to make your payments. Set up alerts or automatic withdraw from your checking account. Don't make payments on the last day and pay convenience fee to the merchant. These are just scams.
6. Stop carrying too many credit cards in your pocket. In fact, don't ever bother to carry more than one. It's OK to not use them. It won't bring down your status in society
7. You must Save. You can open up a Systematic investment Plan with a broker or your own bank. Look more into it based on your preference.
8. Try to eliminate anything and everything where you are paying interests. If it means no discretionary spending for next 2 years, so be it.

That's all I can say about it. Please feel free to comment or any feedback you would like to share.

-Nitin

Portfolio With Options - Greeks Sensitivity

Portfolio With Options - Greeks Sensitivity

Put-Call Parity can be chosen as a primary tool for measuring portfolio risk. Not only it can help one to determine portfolio risk, but also to exploit Arbitrage in case it exists when Put-Call Parity equation is violated.

Put-Call Parity in a nutshell is defined as:

Put(K) + Stock - Call(K) = Riskfree Bond ---------------------------(1) 

Equation (1) is the heart of the option portfolio so it must be respected at all times. You can think of this equation as a Portfolio Replication such that their market price must match. In case the market prices do not match on both sides of the equation, the arbitrageur can sell the overvalued portfolio and can buy the undervalued portfolio and keep them on books until maturity and collect the risk free cash.

Also, the above scenario assumes that the options are European Style and their maturity is the same as the maturity of the risk free bond, and no dividends will be offered. The result of the above portfolio is to generate the Riskless Cash flow equal to the Strike of the option at expiration.

Let me explain why would it work:
Scenario 1 - Stock Price > Strike Price
Here the short call will be exercised, put will expire worthless, and you WILL BE FORCED to sell the stock at Strike Price

Scenario 2 - Stock Price < Strike Price
Here the short call will expire worthless, put will be exercised, and you WILL sell the stock at Strike Price

Scenario 3 - Stock Price = Strike Price
Here both options will expire worthless and stock will be sold for Strike price which should be same as market price at that moment

So, it must be clear the total payoff in all scenarios will the same amount, the strike price K. Only a  Riskfree bond with a maturity value of K can give the same payoff as every other bond will carry risk and your final payoff can be substantially different than K based on other risk factors.

Such portfolios are called Riskless Portfolios, Market Neutral Portfolios, Delta Neutral Portfolio, Riskless Hedged Portfolios and so on. They all mean same thing though that there is no risk if either of these  portfolios are held until maturity.

Below table shows a little bit of Options Greeks math to convey the same result that total exposure of the portfolio is 0.

Option Portfolio Sensitivity

Sorry, don't know why this image is constantly getting rotated but if you don't mind bending your neck, you will truly appreciate the Greeks math behind the portfolio. :). Also, BTW my handwriting is not bad, it's actually the font I am using which might not be the best as per the taste of the reader. :)

I hope you enjoyed reading it.

-Nitin

CDO - Collateralized Debt Obligations

What is a CDO?

CDOs are structured products created using financial engineering. These are products which are backed by cash flow from some sort of debt securities.

Securitization: It is a process of structuring the future cash flow in a more systematic manner such as getting coupons from bonds of various maturities and creating a single product with only one date for cash flow distribution to investors.

Under the CDO hood, anything can reside that has cash flow stream attached to it. You can think any bonds, loans such as credit card loans, auto-mobile loans, Mortgage loans etc.. can be used to create the CDO.

So, how is CDO created and structuring of cash flow works?

Usually a Bank has many loans with different customers. Banks want to get rid of these loans from their portfolio books. Some of these loans can be very risky but banks don't care about that part. They just want to get rid of these loans from their books. So, here is the process of creating the product.

1. Bank will bundle all the loans into one big chunk. Let's say the total value is 100 Million.

2. Bank will divide the bundle into multiple parts and these parts are called Tranches. The top most part is called Senior Tranche which usually has AAA rating. Other tranches are called Subordinate/Mezzanine tranches and will carry their own credit ratings.

3. Last tranche is called Equity tranche which almost never has any rating. Also, this tranche is usually still held in bank's books

Let's take a scenario where all the original 100 Million dollars loans were very risky loans and they all carried  BBB rating which means the loans might default any time. Now, banks use financial engineering process to create a AAA security from BBB security and that's innovation.

All the tranches carry 2 sources of income. Interest Income and/or Principal Income. They all carry one main Risk which is Default Risk and that is the most important risk.

Cash flows are distributed from the most senior tranche (AAA) to the most junior tranche (Equity). But losses are distributed from most junior (Equity) to the most senior (AAA) tranche.

Since BBB rating will provide higher yield than AAA due to its riskiness, banks usually try to create the AAA tranche with most amount of the CDO amount so that they have to pay the least interest to the tranche investors and can keep most of the interest income in the equity tranche (see below).

Let's imagine the original BBB bonds worth 100 Million were broken as follows under CDO:

60M - AAA (Senior Tranche) (Tranche A)                          Interest Offered (3%)

25M - BB (Second Tranche)  (Tranche B)                           Interest offered (6%)

15M - Equity Tranche (Usually Denoted as Tranche Z)          All residual goes here

Now the cash flow will be distributed as this:

Bank receives 8M in the interest income (8% of 100 M and considering no one has defaulted)

Bank will usually charge a service fee for processing payments (Can be anywhere 8-10% of yearly interests received)

Let's say Bank now has 7 Million to distribute

Bank will give 1.8M to Tranche A (3% of 60M)

Bank will give 1.5M to Tranche B   (6% of 25M)

Bank will give 3.7M to Equity Tranche so Equity tranche keeps on increasing in value

As you can see, banks love Securitization process. Why?? Because first they write bad loans and charge customers the loan origination, service fee and other fees plus some more fees. Now they package all these loans and sell them as CDOs and get their cash back that they lent to original customers plus they charge CDO investors also fee such as CDO origination fee, trustee fee, administration fee, and more fees. Additionally, with every cash distribution, they pay themselves first for processing the cash flow before investors see any cash. In the end, Banks make money while everyone else might be losing.

This securitization process was what caused CDO market collapse in America where CDO tranches were rated as AAA while underlying securities were BBB and AAA started to default which according to AAA definition should not have happened to that massive scale.

Key note is - Banks will find a way to offload their risk to another investor and make money. Bank love this process. So, be sure to understand the key characteristics of the underlying security before buying structured securities. Understand the Risk before understanding the Promised Returns

-Nitin

Debt and Equity Valuation using Option Pricing

Debt and Equity Valuation using Option Pricing:

Here, I will talk about how to value Debt and Equity of the firm using Option Pricing formula and the heart of Option Pricing Put-Call Parity.

Asset of the firm can be defined as the total value of its Debts and Equity.
Hence, Assets = Debt + Equity ---------------(1)

Using Put Call Parity from Black Scholes and Merton,
Call Option + PV (Strike) = Underlying + Put Option (Strike)  -----------(2)

Now, PV(Strike) can be thought of a Risk Free Bond that has the same face value as Strike price so you are confident that you have the money K at expiration. K is thought of the total DEBT value that firms owes to its debt holders at maturity. Underlying in this scenario can be treated as the assets of the firm itself. So, equation (2) can be re-written as:

Call + Risk Free Bond = Asset + Put Option --------------------(3)
From Equation (1), Debt = Asset - Equity --------(4)
Equity holders are nothing but Call holder with the strike as K as once the Debts are paid off, Equity holders own everything in the firm.
So, Debt = Asset - Call Option -------------------(5)

Using Call valuation from Equation 3, we can re-write, equation 5 as:
Debt = Asset - (Asset + Put Option - Risk Free Bond)

Debt = Risk Free Bond - Put Option ----------------------(6)

Which makes sense also, as Debt holders have their payoff maxed at K if firm value >= K else they own the assets which should be K - Put Pay off

Below image might make this intuitive sense clearer.

Similarly, Equity Holders can be valued as:
Equity = Asset + Put Option - Risk Free Bond --------------------------(7)

Intuitively, you can think of Equity Holders own the Asset and Put option to sell the assets after paying off the Debts (Risk Free Bond).

Nitin