Capital markets, derivatives & corporate finance

## Basics

* Opportunity set: a representation of the choices available to the investor
* Indifference or utility curve: a representation of the investor’s tastes or preferences
* Market clearing condition: The equilibrium interest rate is that rate at which the amount investors wish to borrow is equal to the amount investors wish to lend
* Security: think of a security as a legal contract representing the right to receive future benefits under a stated set of conditions
* Face value: the cash payment the investor will receive at maturity
* Return: Return will be measured by the sum of the change in the market price of a security plus any income received over a holding period divided by the price of a security at the beginning of the holding period; the difference between the purchase price and the face value constitutes the return the investor receives; For any security, the rate of return is defined as the payments to the owner plus the change in its value, expressed as a fraction of its purchase price
* Market portfolio: portfolio comprising all risky assets. Each asset is held in the proportion that the market value of that asset represents of the total market value of all risky assets
* Two mutual fund theorem: All investors will hold combinations of only two portfolios: the market portfolio (M) and a riskless security
* Callability means that the government can force the holder of the bond to sell the bond back to the government according to a fixed schedule of prices before maturity.
* Spread: The difference between the bid and ask prices is the spread
* Margin: If the investor utilizes borrowing as well as cash, the investor is said to purchase the securities on margin. An investor utilizing margin borrows money from the brokerage firm, which in turn borrows the money from a bank. The securities purchased serve as collateral for both the brokerage firm and the bank. The customer is charged an interest rate on the loan. This rate is determined by adding a premium (usually 1%) to the rate the brokerage firm is charged on its loan (designated as the call rate). The amount the customer can borrow to finance a purchase or short sale is carefully regulated; these regulations are referred to as initial margin requirements. There are separate regulations that monitor the amount of the loan relative to the value of the assets at each point in time; these are called maintenance margin requirements
	+ M
	a
	r
	g
	i
	n
	=

	market value of assets-amount borrowed
	market value of assets

	{\displaystyle \textstyle Margin=\tfrac{\text{market value of assets-amount borrowed}}{\text{market value of assets}}}
	+ Initial margin requirement (Sicherheitsleistung): The minimum amount of margin that must be in the account immediately after a security is purchased is called initial margin. The initial margin requirement is set by the board of governors of the Federal Reserve System. For example, if the initial margin requirement is 60% and you want shares for €5,000, you will need to fund o,6\*€5,000 yourself and can loan the rest (i.e. €2,000).
	+ Thus initial margin regulates the amount that can be borrowed at the time securities are purchased. The amount that can be borrowed can vary from zero to more than 100%. It could be more than 100% if the securities in the account had declined significantly in value, because at the time of any new purchase, initial margin requirements must hold for the whole account.
* Circuit breaks: not letting stock prices deviate too much from yesterday's price
* Maintenance margin: The minimum amount to which the margin can decline without an investor having to take action is called the maintenance margin. If the stockholder’s margin drops below the maintenance margin, then the brokerage firm issues a margin call. The shareholder must bring the margin above the maintenance margin by either adding additional cash or securities to the account or selling securities
* Securitization: Securitisations typically pool cash flows (assets) from many sources (e.g. individual mortgages, credit cards, corporate debts) and issue securities to fund their purchase. These securities came to have an explicit credit hierarchy such that losses from the pool of assets were transmitted first to the lowest-ranking security while continuing to make payments to higher-ranked securities. The different-ranked securities are known as tranches
* Call the return on a given security

r
s

{\displaystyle r\_s }

 and the return on your (own) cash investment

r
c

{\displaystyle r\_c}

.
* Example: If you purchased a share at €50 and the price increased by €5 over 6 months your

r

s

=

5
50

=
10
%

{\displaystyle r\_{s}={\tfrac {5}{50}}=10\%}

. Now assume the share was purchased with 50% margin and that the annual interest rate on the borrowing was 6% or 3% semiannually. With 50% margin the investor would put up $25 in cash, and the interest paid over the six months would be 0.03 (25)= $0.75. A $5 increase in share price results in a return on the cash investment (

r
c

{\displaystyle r\_c}

) of

r

c

=

5
−
0.75

25

=
17
%

{\displaystyle r\_{c}={\tfrac {5-0.75}{25}}=17\%}
* In general:

r
c

=

change in price - interest
margin x price

{\displaystyle \textstyle r\_c=\tfrac{\text{change in price - interest}}{\text{margin x price}}}
* Liquidity (of markets): Liquidity refers to the ability to transact a large number of shares at prices that don’t vary substantially from past prices unless new information enters the market. Liquidity is often subdivided into continuity and depth
	+ Price continuity: investor can expect to transact some shares at prices close to those at which the security recently traded absent any new information in the marketplace.
	+ Deep market: is one that has a large number of buyers and sellers willing to trade at close to the current transaction price, so that a large number of shares can be transacted without a substantial change in price
* Autocorrelation: Treasury bill returns, for example, tend to be highly autocorrelated, meaning that the return to investing in T-bills in one year does a good job at predicting the return to investing in T-bills the next year.
* Large cap/large capitalization company (sometimes "big cap") refers to a company with a market capitalization value of more than $10 billion. Large cap is a shortened version of the term "large market capitalization." Market capitalization is calculated by multiplying the number of a company's shares outstanding by its stock price per share
* Equity premium: The difference between the returns on large-company stocks and the U.S. 30-day T-bill return is called the equity premium. It is the amount of return that investors demand for holding a risky security such as stocks, as opposed to a riskless security such as T-bills. The annual equity premium is about 8.29% over the 1926–2011 period
* Eurobonds, which are denominated in a currency other than that of the country in which they are sold, are now the dominant security in the international bond market and have surpassed U.S. corporate bonds as a source of new funds
* Brokers: agents of investors who match buyers with sellers of securities
* Dealers: link buyers and sellers by buying and selling securities at stated prices
* Adverse selection: Adverse selection is the problem created by asymmetric information before the transaction occurs. Adverse selection in financial markets occurs when the potential borrowers who are the most likely to produce an undesirable (adverse) outcome—the bad credit risks—are the ones who most actively seek out a loan and are thus most likely to be selected
* Moral hazard is the problem created by asymmetric information after the transaction occurs. Moral hazard in financial markets is the risk hazard) that the borrower might engage in activities that are undesirable (immoral) from the lender’s point of view, because they make it less likely that the loan will be paid back
* Value at risk: VAR is an estimate of the highest possible loss in the value of a portfolio of financial assets and liabilities over a fixed time interval with a specific statistical confidence level (VAR assumes a predetermined & knowable probability distribution of future security prices)
	+ VAR calculation uses historical data to forecast future financial market performance
* VaR is that it completely relies on the probability of events and doesn’t consider their severity (the height of the incurred loss)
* Discounting: Essentially, the party that owes money in the present purchases the right to delay the payment until some future date.
	+ The discount, or charge, is the difference between the original amount owed in the present and the amount that has to be paid in the future to settle the debt
	+ The discount is usually associated with a discount rate, which is also called the discount yield. The discount yield is the proportional share of the initial amount owed (initial liability) that must be paid to delay payment for 1 year:

# discount yield

charge to delay payment for 1 year
debt liability

{\displaystyle \text{discount yield} = \textstyle \tfrac{\text{charge to delay payment for 1 year}}{\text{debt liability}}}

* + E.g. the present value of $250 to be paid in two years with i = 15%:

# present value

CF

(
1
i

)
n

=

$

250

(
1
0.15

)
2

=

$

189.04

{\displaystyle \text{present value} = \frac{\text{CF}}{(1+i)^n} = \frac{$250}{(1+0.15)^2} = $189.04 }

## Introduction

### Markets

 IndividualsCompanies

* Money markets (debt instruments with maturities of less than one year): money market instruments will offer different returns because of both differences in maturity and differences in the risk of the issuing institutions
	+ Treasury-bills (considered risk-free)
	+ Repos: A repurchase agreement is an agreement between a borrower and a lender to sell and repurchase a U.S. government security; The difference between the two prices represents the return to the lender; The maturity of a repo is usually very short (less than 14 days), with overnight repos being fairly common; The institution on the opposite side of the repo is said to have a reverse repo
	+ Negotiable certificate of deposit (CDs)
	+ Interbank lending/LIBOR
	+ Bankers’ acceptances
	+ Commercial paper
	+ Eurodollars

* Capital markets: Capital market securities include instruments with maturities greater than one year and those with no designated maturity at all; The market is generally divided according to whether the instruments contain a promised set of cash flows over time or offer participation in the future profitability of a company
	+ Fixed-income market/securities aka bond market (daily ~800 billion trading and 100 trillion outstanding; size of the US bond market is 35 trillion)
		- Government bonds: the quoted price of a fixed income security is not what the investor pays to purchase the security; rather, the investor pays the quoted price plus interest accrued since the last coupon payment. For example, US T-notes (1-10 years) and US T-bonds (+10 years). The trading volume in government bonds is very large, and they are highly liquid
		- Local government bonds: e.g. Federal Agency Securities. Also municipal bonds: debt instruments sold by political entities, such as states, counties, cities, airport authorities, school districts, and so forth, other than the federal government or its agencies; municipal bonds are backed by the (taxing power of) the issuer while revenue bonds are backed by the revenues of a particular project (e.g., a toll road) or the municipality
		- Corporate bond (backed by security or without): Preferred stock at first blush resembles an infinite life bond. It promises to pay to the holder periodic payments like coupons, but these are called dividends rather than interest; there is no return of principal. Given the illiquidity of the corporate bond market, the bid–ask spread will be much higher than in the government bond market
	+ Stock/equity market (Daily trading volume is 200 billion and market capitalization 53 trillion)
		- Preferred stock (takes precedence; dividend-like hybrid between bonds and shares; regular payment but if failure of payment there is no bankruptcy)
		- Common stock: i.e. ownership rights to company assets and profits but dividends are not guaranteed; The unique feature of common stock (unlike simply owning the business) is that the holder of common stock has limited liability → you lose at most your investment

* Primary market: not well known to the public because the selling of securities to initial buyers often takes place behind closed doors. An important financial institution that assists in the initial sale of securities in the primary market is the investment bank. It does this by underwriting securities: It guarantees a price for a corporation’s securities and then sells them to the public

### Trading

* Short-sale: sell a security you don't own - aim is profit from price fall
* Long-sale: manage portfolio risk
* Margin: (market value less the cost of borrowing) divided by market value
	+ Initial margin: tools necessary for the regulation of stock markets; the higher they are the more investors need to go into the game
	+ Maintenance margin:
* Margins magnify!

### Risk reduction

* Present value accounting
* Factors affecting future price and dividend: state of the economy, state of the sector, state of the firm
* Factors affecting risk:
	+ The maturity of an instrument (longer = riskier)
	+ The risk characteristic and creditworthiness of the issuer or guarantor of the investment
	+ The nature and priority of the claims the investment has on income and assets
	+ The liquidity of the instrument and the type of market in which it is traded
* Return risk (will I get my return), interest rate risk (market interest rate), risk of inflation/prices
* Allowing for the inflation risk (if inflation is 2%, your return has to incorporate these 2%): Fisher effect
* Goal is to maximize return for a given risk or minimize risk for a given return
* By mixing the assets in your portfolio, you play with the variance of your portfolio (while the mean is relatively uninteresting/stays)
* Diversification serves 3 purposes

1. Diversified portfolios have lower risk than more concentrated portfolios selected from the set of diverse forecasts
2. It is generally believed that analysts’ estimates have some information content but with lots of random noise. If the errors are uncorrelated, then a larger portfolio reduces the amount of random noise and increases the chance that the extra return is observed
3. Increasing the number of securities and reducing the amount invested in any single one reduces the amount invested in a security due to the extreme estimate of one analyst.

* The easiest way to ensure diversification is to put upper limits on the amount invested in each security. A 2% upper limit will guarantee at least 50 securities in any given portfolio.
* The simplest way to make forecasts less extreme and avoid the difficulties caused by this is to move all the forecasts part way to the mean and adjust the whole distribution of analysts’ estimates so that it has a mean consistent with what we believe is appropriate for the type of securities being examined.

## Interest rates

* Real interest rate: The real interest rate is defined as the nominal interest rate minus the expected rate of inflation
* A bond’s interest rate does not necessarily indicate how good an investment the bond is because what its rate of return does not necessarily equal its interest rate.
* Ex ante real interest rate: adjusted for expected changes in the price level
* Ex post real interest rate: adjusted for actual changes in the price level
* The nominal interest rate i equals the real interest rate r plus the expected rate of inflation or the real interest rate equals the nominal interest rate minus the expected inflation rate
* Fisher effect: When expected inflation rises, interest rates will rise
* Instead of using bond prices to determine interest rates, the liquidity preference framework determines the equilibrium interest rate in terms of the supply of and demand for money
	+ The starting point of Keynes’s analysis is his assumption that people use two main categories of assets to store their wealth: money and bonds
	+ In equilibrium: Bs - Bd = Md - Ms
	+ As the interest rate on bonds rises, the opportunity cost of holding money rises
	+ In Keynes’s liquidity preference analysis, two factors cause the demand curve for money to shift: income and the price level
	+ Keynesian economists typically emphasize the role of aggregate demand in the economy whereas monetarists emphasize the money supply in determining inflation
* There are four possible effects of an increase in the money supply on interest rates: liquidity effect, income effect, price-level effect and expected-inflation effect
	+ The liquidity effect: indicates that a rise in money supply growth will lead to a decline in interest rates
	+ The other effects work in the opposite direction. The evidence seems to indicate that the income, price-level, and expected-inflation effects dominate the liquidity effect such that an increase in money supply growth leads to higher—rather than lower—interest rates.
* The relationship among interest rates is called the risk structure of interest rates
* The "risk structure of interest rates" (the relationship among interest rates on bonds with the same maturity) is explained by three factors: default risk, liquidity, and the income tax treatment of a bond’s interest payments. As a bond’s default risk increases, the risk premium on that bond (the spread between its interest rate and the interest rate on other bonds such as a default-free Treasury bond) rises. The greater liquidity of Treasury bonds also explains why their interest rates are lower than those on less liquid bonds. If a bond has a favorable tax treatment, as do municipal bonds, whose interest payments are exempt from federal income taxes, its interest rate will be lower.

### Bond market

* When a bond issue is underwritten, one or more securities firms or banks, forming a syndicate, buy the entire issue of bonds from the issuer and re-sell them to investors. The security firm takes the risk of being unable to sell on the issue to end investors. Primary issuance is arranged by bookrunners who arrange the bond issue, have direct contact with investors and act as advisers to the bond issuer in terms of timing and price of the bond issue
* Types of risk related to fixed income assets (i.e. bonds)[1].
	+ Interest rate risk: rising interest rates mean falling bond prices - Interest rate movements are the major cause of price volatility in bond markets
	+ Inflation risk: If the rate of inflation outpaces the fixed amount of income a bond provides, the investor loses purchasing power.
	+ Credit/business/financial risk: possibility of an issuer defaulting on its debt obligation
	+ Liquidity risk: risk that an investor wishing to sell a fixed income asset is unable to find a buyer.
* Relation between price, yield and interest rate:

y
i
e
l
# d

coupon amount

p
r
i
c
e

{\displaystyle yield = \tfrac{\text{coupon amount}}{price}}

 so that if the price of the bond falls, the interest rate you receive on the bond (coupon) becomes relatively bigger, i.e. higher yield
* The yield-to-maturity is the expected rate of return on a bond if it is held until maturity. It takes into account the bond’s market value, the par value, the coupon interest rate and the time to maturity
* The supply and demand analysis for bonds provides one theory of how interest rates are determined
* A situation in which the quantity of bonds supplied exceeds the quantity of bonds demanded, is called a condition of excess supply (excess demand for vice versa)
* The asset market approach:emphasizes stocks of assets, rather than flows, in determining asset prices
* Determinants of asset demand: wealth increase (increase in wealth raises the quantity demanded of an asset); expected return (if expected return of an asset increases, demand rises); asset risk (if the asset's risk drops, demand rises); liquidity (the more liquid, the higher demand)
* Interest rates: (Ceteris paribus (CP)) Higher expected interest rates in the future lower the expected return for long-term bonds, decrease the demand, and shift the demand curve to the left. Likewise, lower expected interest rates in the future increase the demand for long-term bonds and shift the demand curve to the right
* Rate of inflation: An increase in the expected rate of inflation lowers the expected return for bonds, causing their demand to decline and the demand curve to shift to the left
* CP: In a business cycle expansion (firms invest, hence issue bonds), the supply of bonds increases
* An increase in expected inflation causes the supply of bonds to increase and the supply curve to shift to the right (because cost of borrowing drops). Put differently, if expected inflation rises, the expected return on bonds relative to real assets falls for any given bond price and interest rate, which decreases demand
* Higher government deficits increase the supply of bonds and shift the supply curve to the right. On the other hand, government surpluses, as occurred in the late 1990s, decrease the supply of bonds and shift the supply curve to the left
* A plot of the yields on bonds with differing terms to maturity but the same risk, liquidity, and tax considerations is called a yield curve [2], and it describes the term structure of interest rates for particular types of bonds, such as government bonds.
	+ Market segmentation theory states that the yield curve is determined by supply and demand for debt instruments of different maturities. The level of demand and supply is influenced by the current interest rates and expected future interest rates.
		- Preferred habitat theory: expected long-term yields are an estimate of the current short-term yields.; ...is a term structure theory suggesting that different bond investors prefer one maturity length over another and are only willing to buy bonds outside of their maturity preference if a risk premium for the maturity range is available
* Convertible bond: bond can be converted into stock
* Collateralized debt/bonds: Bonds that give investors the right to lay claim to the company’s underlying assets in case of non-payment
* Callable bond ('an option on the bond'): issuer can choose to pay bond off before the official maturity date (usuallay, to take advantage of a possible drop in interest rates at some point in the future because then the company can redeem the outstanding bonds and reissue the debt at a lower rate)
* (Bond) term spread: the difference between the 10-year yield and the short-term interest rate
* (Bond) term premium: calculation requires the computation of expected future short-term rates (i.e. based on market readings of expectations about future variables), and measures the premium an investor would earn holding a 10-year bond over the returns from investing in a succession of short-term paper
	+ If you could know exactly where short-term interest rates will be in the future, it would be easy to determine whether longer-term bonds offer attractive yields today. But that’s not the world we live in. Investors have to live with the risk that their best guesses about future borrowing costs won’t be good enough and hope the “term premium” embedded in bond yields is high enough to compensate for the uncertainty.
	+ Usually, the term premium is between 100 and 200 basis points (reflecting the higher premium for longer-term bonds). Since 2005, it has been below 50 basis points

### Present value

* Is the value of a cash flow estimated on this day
* Present (discounted) value: based on the commonsense notion that a dollar paid to you one year from now is less valuable than a dollar paid to you today (because if paid now, you could earn interest)
* Basically, a dollar received n years from now is worth only

$

1

(
1
i

)
n

)

{\displaystyle \tfrac{$1}{(1+i)^n)}}

 today
	+ Allows us to figure out today’s value (price) of a credit (debt) market instrument at a given simple interest rate i by just adding up the individual present values of all the future payments received.
	+ The present value is always less than or equal to the future value because money has interest-earning potential, a characteristic referred to as the time value of money, except during times of negative interest rates, when the present value will be more than the future value
	+ By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest
* Simple interest rate = interest divided by the amount of the loan
* Discounting the future: calculating today’s value of dollars received in the future: Present value =

Cash flow

(
1
i

)
n

)

{\displaystyle \tfrac{\text{Cash flow}}{(1+i)^n)}}
* 4 types of credit market instruments:
	+ Simple loan: amount of funds + interest to be repaid upon maturity date (e.g. many money market instruments)
	+ Fixed-payment loan/fully amortized loan: lender provides the borrower with an amount of funds, which must be repaid by making the same payment every period (such as a month), consisting of part of the principal and interest for a set number of years
	+ A coupon bond (e.g. T-bonds): pays the owner of the bond a fixed interest payment (coupon payment) every year until the maturity date, when a specified final amount (face value/par value) is repaid. A coupon bond is identified by four pieces of information
		- The bond’s face value
		- Corporation or government agency that issues the bond
		- Maturity date
		- Coupon rate (the yearly coupon payment expressed as a percentage of the face value)
* A discount bond/zero-coupon bond (e.g. T-bills): bought at a price below its face value (at a discount), and the face value is repaid at the maturity date
* If a bond's coupon rate is less than its YTM, then the bond is selling at a discount
* If a bond's coupon rate is more than its YTM, then the bond is selling at a premium.
* If a bond's coupon rate is equal to its YTM, then the bond is selling at par
* The operation of evaluating a present value into the future value is called a capitalization (how much will $100 today be worth in 5 years?)
* The reverse operation—evaluating the present value of a future amount of money—is called a discounting (how much will $100 received in 5 years—at a lottery for example—be worth today?).
* Net present value: NPV is the sum of all the discounted future cash flows (where discount rates are found from the term structure of interest rates); the NPV measures the excess or shortfall of cash flows, in present value terms, above the cost of funds; it compares the present value of money today to the present value of money in the future, taking inflation and returns into account

### Yield to maturity

* The yield to maturity for an instrument is the interest rate that equates the present value of the future payments on that instrument to its value today
* Yield to maturity: as the most accurate measure of interest rates. It's the interest rate that equates the present value of cash flow pay-ments received from a debt instrument with its value today
	+ For simple loans, the simple interest rate equals the yield to maturity
	+ For fixed-payment loans: the present value of the fixed-payment loan is calculated as the sum of the present values of all cash flow payments:

	loan value
	=

	FP

	(
	1
	i
	)

	FP

	(
	1
	i

	)
	2

	FP

	(
	1
	i

	)
	3

	.
	.
	.

	FP

	(
	1
	i

	)
	n

	{\displaystyle \textstyle \text{loan value} = \frac{\text{FP}}{(1+i)}+\frac{\text{FP}}{(1+i)^2}+\frac{\text{FP}}{(1+i)^3}...\frac{\text{FP}}{(1+i)^n}}
	+ Coupon bond: Price of coupon bond =

yearly coupon payment C

(
1
i
)

C

(
1
i

)
2

C

(
1
i

)
3

.
.
.

C

(
1
i

)
n

Face value

(
1
i

)
n

{\displaystyle \frac{\text{yearly coupon payment C}}{(1+i)}+\frac{C}{(1+i)^2}+\frac{C}{(1+i)^3}...+\frac{C}{(1+i)^n}+ \frac{\text{Face value}}{(1+i)^n}}

* When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate
* The price of a coupon bond and the yield to maturity are negatively related because the rise in i lowers the present value of all future cash flow payments for this bond-The yield to maturity is greater than the coupon rate when the bond price is below its face value.
* For example, you buy bond XYZ, which matures in 1 year and has a 5% interest rate (coupon) and has a par value of $100. You pay $90 for the bond. If you hold the bond until maturity, ABC Company will pay you $5 as interest and $100 par value for the matured bond. Now for your $90 investment, you get $105, so your yield to maturity is 15/90 =16.67%. If you had paid $105 for the same bond, you would get $5 when the coupon is paid. However you had to compensate for the extra $5 of your initial investment. Your gain is 0 and so is your yield to maturity
* Current yield ic: the yearly coupon payment divided by the price of the security; frequently used as an approximation to describe interest rates on long-term bonds
* More generally, the return on a bond held from time t to time t + 1 can be written as:

# R

C

P

t
1

−

P
t

P
t

=

C

P
t

P

t
1

−

P
t

P
t

{\displaystyle R = \frac{C+P\_{t+1}-P\_t}{P\_t} = \frac{C}{P\_t}+\frac{P\_{t+1}-P\_t}{P\_t}}

	+ where the first term is the current yield iC (the coupon payment over the purchase price) and the second term is the rate of capital gain g, or the change in the bond’s price relative to the initial purchase price
	+ Rewritting this as R = iC + g, which shows that the return on a bond is the current yield ic plus the rate of capital gain g
* A rise in the interest rate means that the price of a bond has fallen. A rise in interest rates therefore means that a capital loss has occurred
* Prices and returns for long-term bonds are more volatile than those for shorter-term bonds.
* The key to understanding why there is no interest-rate risk for any bond whose time to maturity matches the holding period is to recognize that (in this case) the price at the end of the holding period is already fixed at the face value
* Bonds whose term to maturity is longer than the holding period are subject to interest-rate risk: Changes in interest rates lead to capital gains and losses that produce substantial differences between the return and the yield to maturity known at the time the bond is purchased.
* Term structure of interest rates: The term structure of interest rates is the relationship between interest rates or bond yields and different terms or maturities. The term structure of interest rates is also known as a yield curve. The term structure reflects expectations of market participants about future changes in interest rates and their assessment of monetary policy conditions
* A rise in short-term rates will raise people’s expectations of future short-term rates. Because long-term rates are the average of expected future short-term rates, a rise in short-term rates will also raise long-term rates, causing shortand long-term rates to move together
* When the yield curve is upward-sloping, the expectations theory suggests that short-term interest rates are expected to rise in the future, as we have seen in our numerical example.
* Most often, the Treasury yield curve is upward-sloping. One basic explanation for this phenomenon is that investors demand higher interest rates for longer-term investments as compensation for investing their money in longer-duration investments. Because in the typical situation the demand for long-term bonds is relatively lower than that for short-term bonds, long-term bonds will have lower prices and higher interest rates, and hence the yield curve will typically slope upward.
* Liquidity premium theory: states that the interest rate on a long-term bond will equal an average of short-term interest rates expected to occur over the life of the long-term bond plus a liquidity premium (also referred to as a term premium) that responds to supply and demand conditions for that bond. Bonds of different maturities are assumed to be substitutes but not perfect substitutes. Investors tend to prefer shorter-term bonds because these bonds bear less interest-rate risk. For these reasons, investors must be offered a positive liquidity premium to induce them to hold longer-term bonds.
* Rising interest rates are associated with economic booms and falling interest rates with recessions. When the yield curve is either flat or downwardsloping, it suggests that future short-term interest rates are expected to fall and, therefore, that the economy is more likely to enter a recession. Indeed, the yield curve is found to be an accurate predictor of the business cycle
* A steep yield curve predicts a future increase in inflation, while a flat or downward-sloping yield curve forecasts a future decline in inflation → a steep yield curve indicating loose policy and a flat or downwardsloping yield curve indicating tight policy.

## The banking firm

* Not producing firms, hence to be analyzed in terms of its balance sheets
* Fractional reserve banking: deposit inflows lead to loan-making

Bank balance sheet
* Securities (government or corporate)
* Loans (commercial, real estate, consumer loans, interbank loans)
* Reserves (have to be liquid instruments)
* Physical assets
* Borrowings (CDs, repos)
* Own capital (retained profit or equity)

Conventional/FRB-view

which will be split into

Conventional/FRB-view
* Loans +

Post-Keynesian view
* Loans +
* Borrowing ++

### Bank management principles

* Liability management: by looking at reserves and the liabilites side; provide a cushion against a loss in liquidity on the liability side
	+ If the bank suffers a shock (loses deposits, i.e. the deposit/reserve ratio is insufficient), it can (working on the liabilities side): borrow on the open (interbank) market; it can borrow from the central bank (liabilities side); it can sell securities (working on the asset side); it can recall loans (i.e. stop rolling over debt)
* Include page 2 here
* Asset management
	+ Check the quality of loans: Consider the range, liquidity and quality of loans (screening & monitoring of loans)
	+ Capital adequacy management (can be determined as an abstract number like 8% or depend on the quality of the loans)
	+ Banks operate on the balancing act between solvency and profitability: if the quality of a bank’s assets declined, its solvency would come into question; the holders of bank liabilities would probably demand that the bank should honour its promises to pay. The solvency of the bank would be protected by its capital which could absorb losses generated by its assets; by this token, capital would also protect liquidity, since others would remain prepared to hold the bank’s promises to pay. However, there is no theoretical basis on which to ascertain the level of capital that would provide adequate protection to banks at a given point in time. Both solvency and liquidity are empirical categories, resulting from historical experience and particular institutional structures.

### Bank profit-making

* Managing liquidity and solvency with an eye to ensuring profitability is the cause of inherent instability in banking
* Profit after tax
* Return on assets

profit after tax
assets

{\displaystyle \frac{\text{profit after tax}}{\text{assets}}}

* Return on equity (also net assets or assets minus liabilities)

profit after tax
equity capital

{\displaystyle \frac{\text{profit after tax}}{\text{equity capital}}}

* Equity multiplier

assets
equity capital

{\displaystyle \frac{\text{assets}}{\text{equity capital}}}

* Equity multiplier x Return on assets = Return on equity

* Which is why banks don't like to raise equity or rather decrease equity capital

### Risk management

* As intermediaries between deposit and loan markets, commercial banks must manage a variety of risks:
	+ Credit risk: borrowers might default
	+ Interest rate risk: To the extent that the average interest rate on earning assets is sticky, earnings may be dramatically reduced by a rise in short-term borrowing costs. Banks must therefore "manage the book," by weighing the gains from borrowing short and lending long against the risk that spreads will turn negative as short-term rates rise
	+ Liquidity risk: banks must be able to meet unexpected demands for cash withdrawal, new credit, and transfers at clearing without hesitation (traditionally bank liquidity was provided by holding secondary reserves of short-term marketable assets- T-bills -that could easily be sold for cash when needed. More recently banks have come to depend primarily on borrowing through the CD and other wholesale markets or the Fed's discount window to meet any net cash drain ("liability management"))
* Banks engage in maturity transformation (from short-term borrowing to long-term lending)
* Differential movement in short- vs. long-term interest rates

Interest rate risk
	+ Variable rate loan
	+ Short-run securities
* Insensitive
	+ Reserves
	+ Long-term loans
	+ Securities
	+ CDs variable rates
	+ Money market + deposits
* Insensitive
	+ Other deposits
	+ Own capital
	+ Long-term CDs

* If the bank holds more sensitive liabilities than assets than a rise in interest rates leads to a decrease in profits
* If a number of loans go bad such that the net worth of the bank becomes negative (i.e. its insolvent), this need not affect the day-to-day business of the bank immediately. Only as the news spread will the solvency problem become a liquidity problem as people start to withdraw their deposits
* Gap analysis: Total difference between A-L
* Duration analysis: take the average duration and check how net worth responds to changes in interest rates
* Manage risk by changing structure of your balance sheet:
	+ Shorten the duration of bank assets and increase their insensitivity to interest rates (unhappy customers)
	+ Lengthen the duration of liabilities
* Or hedge through derivatives (i.e. without changing your balance sheets)
* Banks operate in an environment of regulation: think of regulation (e.g. reserve requirement) as tax on profits
* Technological change affected the relation between front and back office
* Banks began to look like other financial institutions and vice versa in the age of deregulation (universal banks were re-introduced with the repeal of Glass-Steagal-Act, i.e. banks that can do commercial & investment banking)
* Rise of non-interest income: banks began to make profit out of transactions
* Since 2009, banks operate in a near 0-interest rate environment: liquidity problems vanish (because the CB provides it) and profit-making is more difficult; given a decrease in financial market activity, they can also make less non-interest income
* Very interventionist not in terms of price

## Financial institutions

### Regulation

* Purchase and assumption method for insolvent banks: the Federal Deposit Insurance Corporation (FDIC) gives the insolvent bank a large infusion of capital and then finds a willing merger partner to take over the bank and its deposits
* Payoff method: pay only for deposit up to a limit (e.g. $250,000)
* With the increasing financial consolidation, government safety nets have to accomodate also linked financial institutions ("if I know that Bank XYZ will be bailed out, our pension fund need not monitor its risky behaviors"). Thus, financial consolidation of banks with other financial services firms means that the government safety net may be extended to new activities, such as securities underwriting, insurance, or real estate activities, as has occurred during the global financial crisis.
* Stress tests: calculate losses under dire scenarios and the need for more capital
* Value-at-risk (VaR) calculations, which measure the size of the loss on a trading portfolio—say, over a two-week period—that might happen 1% of the time
* Mark-to-market accounting: The rationale behind mark-to-market accounting is that market prices provide the best basis for estimating the true value of assets, and hence capital, in the firm
* Before mark-to-market accounting, firms relied on the traditional historical-cost (book value) basis in which the value of an asset was set at its initial purchase price
* Microprudential supervision, which focuses on the safety and soundness of individual financial institutions.
* Leverage cycle: in which there was a feedback loop from a boom in issuing credit, which led to higher asset prices, which resulted in higher capital buffers at financial institutions, which supported further lending in the context of unchanging capital requirements, which then raised asset prices further, and so on; in the bust, the value of the capital dropped precipitously, leading to a cut in lending
* Net stable funding ratio (NSFR), which is the percentage of the institution’s short-term funding in relation to total funding
* Because of financial institutions' loophole mining, regulation applies to a moving target

## Liquidity

* An illiquid asset is one in which the proceeds available from physical liquidation or a sale on some date are less than the present value of its payoff on some future date. The lower the fraction of the present value of the future cash flow that can be obtained today, the less liquid is the asset
* An investor’s demand for liquidity is greater the higher his (relative) risk aversion is, because liquidating early implies low consumption and, thus, high marginal utility of consumption.

### Banks & liquidity problems

* Distinguish between funding and market liquidity:
	+ Funding liquidity:ability of an institution to make payments promptly
	+ Market liquidity: ease of selling an asset in the market
* A bank’s liabilities are more liquid than its assets
* In deciding her capital structure, the banker now has to trade off liquidity creation against the cost of bank runs.
* The investor who doesn't know whether she will have to consume at T1 or T2 can get at least some insurance: Liquid assets that offer a smaller loss when liquidated early can provide indirect insurance.
* Investors demand liquidity because they are uncertain about when they need to consume and, thus, how long they wish to hold assets. As a result, they care about the value of liquidating their assets on several possible dates, rather than on a single date.
* Consumers, on the other hand: Creating deposits that are more liquid than the assets held by banks can be viewed as an insurance arrangement in which depositors share the risk of liquidating an asset early at a loss.
* Providing liquidity subjects the bank to runs. If a run is feared, it becomes a self-fulfilling prophecy
* Bank runs disrupt production because they force banks to call in loans early. This forces the borrowers to disrupt their production.
* Because moving away from a good equilibrium requires a large change in beliefs, the initiation of a run when none was expected requires something thatall (or nearly all) depositors see (and believe that others see).
* Suspension of convertibility is usually a discretionary policy. Another discretionary policy to prevent banks from liquidating illiquid assets and avoiding self-fulfilling runs is central-bank lending, financed by implicit taxation or money creation authority.
* The banker can issue demand deposits to the other lenders. The fragile deposit structure allows persistent relationship building to be delegated to the banker where the intermediary monitors on behalf of investors) because the banker can commit to paying depositors what she can extract from the entrepreneur on the basis of her specific loan collection skills.
* The bank shields entrepreneurs from the liquidity needs of depositors
* Argument against government deposit insurance: For the threat of disintermediation to provide full commitment, all depositors must have the option to withdraw (or at least enough depositors to fully disintermediate the bank), and not just those who need to withdraw for liquidity purposes.

### Narrow banking

* Separating the money-creating and investment part of banking. One balance sheet is about reserves vs. deposit and the other balance sheet is a non-deposit-taking loans & securities vs. own capital & borrowing
* With narrow banking, in the case of the private bank they have deposits as their liabilities and reserves as their assets. These latter are thehn held/converted into fiat money by the central bank (the CB's liabilities are the legal tender). The CB buys the securities that the state issues with its liabilities (fiat money)

## Optimal contract design

* Include specifications: time period, interest rate, collateral, actions left to the borrower, conditions for bankruptcy
* If monitored is prohibitively expensive, the optimal contract is a debt contract that specifies a face value f (amount to be paid in order to avoid liquidation) in such a way as to give the lender, at the least, her demanded profit.

### Monitoring

* Monitoring: to make sense, monitoring has to have economies of scale, size of projects is small, low costs of monitoring the bank
	+ Ex ante monitoring: project screening (avert adverse selection)
	+ Monitoring during completion (avoid moral hazard)
	+ Monitoring ex post (audit to deal with borrowers who fail to repay)

## Information and resource allocation

* 3 different ways of thinking about the role of financial market prices in allocating resources:
	+ Prices as indicators of value and scarcity
	+ The statistical use of prices to analyze risk
	+ Prices as aggregators of information; reveal agents' private information
* Under certain conditions, private information can be completely revealed through a loan market as well as through an equity market
* Informational efficiency and welfare (Pareto) efficiency are quite different things
* Sometimes more information leads to better investment decisions, but at the cost of more price variability
* Private signals obtained by investors could become incorporated in prices so that apparently private information became public. If an investor has favorable information, he will buy the security and bid up its price; if he has unfavorable information, he will sell it and bid down the price

### Risk

* Given normality, von Neumann-Morgenstern expected utility can be expressed as a function of the mean and variance of an investor's portfolio: Er = rF + β(ErM - rF
	+ Where Er = expected return on a stock, rF = risk-free rate, β =

	C
	o
	v
	(

	r

	s
	t
	o
	c
	k

	,

	r

	m
	a
	r
	k
	e
	t

	)

	V
	a

	r

	m
	a
	r
	k
	e
	t

	{\displaystyle \tfrac{Cov(r\_{stock}, r\_{market})}{Var\_{market}}}

	, and ErM is the expected return on the market
	+ Return on the market = portfolio consisting of all the stocks available weighted by value
* in order to make efficient decisions, that is, decisions that are both optimal from the point of view of shareholders and socially efficient, a manager needs to pay attention to only three pieces of information: expected return, the covariance of returns with those of the market portfolio, and the risk-return trade-off

## Capital structure

* The cost of capital is teh minimum risk-adjusted rate of return that a project must earn in order to be acceptable to shareholders
* What is the best source of funds? [3]
	+ Internal funds (e.g., cash reserves vs. cutting dividends)
	+ Debt (i.e., borrowing from banks or issuing bonds)
	+ Equity (i.e., issuing stock via VC or IPO)?
	+ Characteristics of financial claims
		- Payoff structure (e.g. fixed promised payment)
		- Priority (debt paid before equity)
		- Restrictive Covenants
		- Voting rights
		- Options (convertible securities, call provisions, etc)

### Modigliani-Miller Propositions

* The financing of debt vs. equity is randomly distributed among firms. If it had an influence, though, it wouldn't be randomly distributed
* Although the assumptions of MM are obviously false, it is still useful. That is, capital structure matters precisely because one or more of these assumptions is violated. It tells where to look for determinants of optimal capital structure and how those factors might affect optimal capital structure.
* Since the value of the firm depends neither on its dividend policy nor its decision to raise capital by issuing stock or selling debt, the Modigliani–Miller theorem is often called the capital structure irrelevance principle.
* The firm's value is that of the cash flows generated by its operating assets (e.g., plant and inventories) and financial policy simply divides up this cashflow pie among different claimants (e.g., debtholders and equityholders) while the absolute size of the value remains

### P1: Value of the firm

* Assumption: no personal or corporate taxes & no bankruptcy costs
* P1: The value of the firm is not affected by changes of the capital structure because the cash flows don't change and hence the value doesn't
* The value of the levered firm = the value of the unlevered firm + present value of the tax shield provided by debt (i.e. gain from leverage)
* Modiglioani-Miller P1: "The market value of any firm is independent of its capital structure and is given by capitalizing its expected return at the rate ρ appropriate to its risk class" - ceteris paribus, the value of a firm is independent of how its liabilities/debt+equity are partitioned
* Suppose an investor is considering buying one of the two firms, U or L. Instead of purchasing the shares of the levered firm L, he could purchase the shares of firm U and borrow the same amount of money B that firm L does (costs of borrowing for the investor and the firm are assumed to be equal here). The eventual returns to either of these investments would be the same. Therefore the price of L must be the same as the price of U minus the money borrowed B, which is the value of L's debt.
* r
E

(
Levered
# )

r
E

(
Unlevered
)

D
E

(

r
E

(
Unlevered
)
−

r
D

)

{\displaystyle r\_E(\text{Levered}) = r\_E(\text{Unlevered}) + \frac{D}{E}(r\_E(\text{Unlevered}) - r\_D)}

* If taxes are included now, note that firms can deduct interest payments from taxes and thus leverage lowers tax payments while dividend payments are non-deductible

### P2: Weighted Average Cost of Capital

* The WACC is not affected by the capital structure
* Although debt is cheaper for firms (usually around 6% to be paid vs. 13% for equity. However, this reasoning ignores the hidden cost of debt, namely that raising more debt makes existing equity more risky!

### Problems with MM: why debt matters

* Assumptions made implicitly or explicitly include:

1. Capital markets are fictionless
2. Individuals can borrow and lend at the risk-free rate - non-problematic if relaxed
3. There are no costs to bankruptcy or to business disruption - problematic if relaxed
4. Firms issue only two types of claims: risk-free debt and risky equity
5. All firms are assumed to be in the same risk class (operating risk), i.e. perfectly correlated expected future cash flows
6. Corporate taxes are the only form of government levy (i.e. no wealth taxes on firms and personal taxes) - problematic if relaxed
7. All cash flow streams are perpetuities (i.e. no growth)
8. corporate insiders and outsiders have the same information (i.e. no signaling opportunities)
9. Managers always maximize shareholder's wealth (i.e. no agency costs)
10. Operating cash flows are completely unaffected by changes in the capital structure

* Relaxing the assumption of risk-free debt will not change the results but dropping the no bankruptcy costs and no personal taxes assumptions will.
* Briefly put, debt imposes a fixed, external obligation on the borrower: interest and principal have to be repaid in pre-determined amounts at fixed times, or bankruptcy would result. Leverage, therefore, increases risk for functioning capitalists because, first, it makes the rate of profit of enterprise more variable (but also boosts it) and, second, it raises the danger of bankruptcy
	+ There is obviously a question of whether the equity is owned by the manager/entrepreneur himself or only by shareholders.
* Taxes (corporate & personal) make the firm's financial policy relevant because they determine the firm's tax bill (factor in the present value of the tax shield).
* Functioning capitalists must base their leverage decisions on the levels of leverage normally prevailing in their sectors, subsequently deciding on what is appropriate for their own enterprises. The empirical regularities of leverage reflect the characteristics of each sector preponderance of fixed over circulating capital, rapidity of turnover of capital, institutional practices of sale and purchase, access to telecommunications, but also plain custom and tradition. Needless to say leverage would also vary according to the phase of the economic cycle.
* Costs of Financial Distress (i.e. cash flow insufficient to pay obligations/debt payments): if debt burden increases, so does financial distress. However, to ascertain the costs of debt levering, one has to factor out the financial distress that is due to other reasons
	+ Since value is determined by cash flows, financial distress per se does not affect value. The fact that firms in financial distress often have falling sales, bad operating and poor financial performance is usually the cause, not an effect of financial distress
	+ Bankruptcy costs: include direct costs, for example legal costs such as courts and opportunity costs such as time spent by dealing with creditor), and indirect costs, such as debt overhang (inability to raise funds to undertake good investments, scaring customers). Shareholders of firms in financial distress are reluctant to fund valuable projects because most of the benefits go to the firm’s existing creditors
* No transaction costs for issuing debt or equity while in reality different transactions are taxed differently (interest payments are considered a business expense and tax exempt whie dividends/retained earnings are taxed) - call this the tax shield where the tax rate is multiplied by the (to be deducted) interest. Also restructuring the firm for higher gearing bears transaction costs
	+ Here you have an explanation for stock buyback (boosting financial ratios) in which the company issues debt in order to repurchas shares
	+ However, this also means that to the extent that taxes are not a big factor in the firm's financial policy (say because of tax haven possibility), the MM is again relevant
	+ Note that given that MM's assumptions, it is talking about **before-tax** cashflows
* No asymmetric information about the firm’s investments
* Capital structure does not influence managers’ investment decisions [4]
* Agency problems: managers will divert part of the company’s profits for their personal consumption (corporate jets etc.)
	+ Add more debt to add more weight/incentive to be profitable to pay interest, otherwise the firm goes bankrupt
	+ Managers are also more inclined to keep the firm operating even though liquidation is preferred by investors - debt mitigates these problems by

giving the debt holders the option to force liquidation if the firm’s cash flows are poor

### MM and DCs

* Most research on capital structure is carried out in developed countries
* Higher wealth concentration in D-ing countries leads to more concentrated ownership structure (complicated ownership structures often referred to as pyramids) of major firms than is usual in the developed countries, with an effect on capital structure decisions
* There are also conflicting views on whether capital structure decisions are more country or more firm specific: The variables that are relevant for explaining capital structures in the United States and European countries are also relevant in developing countries, despite the profound differences in institutional factors across these developing countries. The more profitable the firm, the lower the debt ratio, regardless of how the debt ratio is defined. This finding is consistent with the Pecking-Order Hypothesis (Booth et al., 2001) and supports the existence of significant information asymmetries. This result suggests that external financing is costly and therefore avoided by firms. The authors find evidence that firms' decisions are affected by the same variables as in developed countries (though they used data from 1980s)
* Weaker investor protection rights make it more difficult for small, dispersed shareholders to effectively control management whereas a single shareholder with a controlling stake can mitigate this problem. La Porta et al (1998) finds that the more effective the law is at protecting the rights of individual and minority investors, the less concentrated corporate ownership tends to be

## Efficient Market Hypothesis

* Markets price assets precisely at their intrinsic worth given all publicly available information
* The efficient market paradigm where changes in relative prices restore an efficient and desired equilibrium in production and exchange relations; asset prices aggregate and fully reflect all relevant fundamental information, and thus provide the proper signals for resource allocation.
* Models of the economy that incorporate the EMH are not models of decentralised market economies, but models of a centrally planned economy.
* Capital markets for loanable capital
* Operating rules: informational requirements (auditing) before listing
	+ Very regulated markets
	+ Stock brokers: agents for buying/selling - vs stock jobbers (trading on their own account and executing tasks by brokers) - this distinction
	+ Fixed commissions abolished → competition among traders
* Markets for primary securities - however, in practice secondary securities dominate (i.e. not so much direct finance)
	+ In theory, issued to finance a position of fixed capital
* Gild/bond market usually more sophisticated
* Indirect finance is big (i.e. not predominantly markets for channeling funds to borrowers)
* Price of newly issued securities reflects the secondary market - function of markets to ascertain the price of capital, which is important for the economy more generally; i.e. if the stock markets works out the prices properly, other financial markets work, too
* Perfect market
	+ Information costless and instantly received by all
	+ No transactions costs, taxes, and frictions
	+ Perfectly divisible/marketable assets
	+ Entry possible
	+ Rational utility maximizers
	+ Goods markets also competitive/efficient
	+ Only price takers
* Arbitrage until profit reaches opportunity costs of external funds
* Marginal rate of return is equalized for similar projects; i.e. marginal product = marginal costs
* Efficient capital market: where information is instantaneously reflected in prices and prices are accurate signals for capital allocation
* So long as transactions costs are fair returns for intermediaries, allocation is still efficient
* Efficiency is guaranteed if for a given risk, share prices correctly reflect the expected profitability of firms
* Share prices must reflect the information on future profits and economic fundamentals; share prices = fundamental values
* When someone refers to efficient capital markets, she means that security prices fully reflect all available information. A more realistic definition is that prices reflect information until the marginal costs of obtaining information and trading no longer exceed the marginal benefit
* Much of the efficient market literature is actually concerned with the speed with which information is impounded into security prices.
		- Current prices fully reflect all information about past prices and returns
		- No investor can make extra gains by
		- Widely accepted
		- Random walk (each change is uncorrelated with the previous change): best guess is today's guess; tomorrow’s price cannot be predicted & randomness reflects a disconnection from the fundamental rather than that ‘information
		- Tests of the predictability of returns (formerly tests of the weak form of the EMH) are in part tests of whether this type of trading behavior can lead to excess profits. If returns are not predictable from past returns, then new information is incorporated in the security price sufficiently fast that, by the time an investor could tell from the price movements themselves that there had been a fundamental change in company prospects, the fundamental change is already fully reflected in price
	+ Semi-strong: current prices reflect reflect all publicicly information (forecasts etc. included, i.e. not only past); no investor can make extra return by using such information
	+ If empirical tests find that future return cannot be predicted from past return, then trading rules based on an examination of the sequence of past prices are worthless. If the semistrong form of the hypothesis is supported by empirical evidence, then trading rules based on publicly available information are suspect.
	+ Strong: current prices fully reflect all information, both public and private/insider
* In perfect markets, price changes occurs only through new information and hence is independent from old information; i.e. large number of traders necessary.
* The price of a stock is the best estimator of its value: if an agent has private information and accordingly buys/sells, this information is signalled to others who follow suit
* The Random Walk Hypothesis states that increments in (the logarithm) of prices should be independently distributed with fixed and finite variance, while the EMH merely states that the current stock prices reflect all available information. Stock prices, by this theory, approximately describe “random walks” through time: the price changes are unpredictable since they occur only in response to genuinely new information, which by the very fact that is new is unpredictable.
* EMH: Balance the portfolio rather than trying to beat the market/find the winners; return is a fair award for bearing non-diversifiable risk. In an equilibrium where no one has any reason to trade, the market price of each security reflects the common or market information shared by all investors.
	+ If a market is efficient, investing is a fair game (i.e. the expected excess = 0 over the long-term)
* For random walk, all covariances must be 0, not so for the fair game
* The EMH can explain crisis only through a shift/shock of fundamentals
* One frequently reads that if the EMH holds, then the best estimate of tomorrow’s price is today’s price, or an expected return of zero. This is not a correct implication of the efficient market model. Rather, the implication is that the past information contains nothing about the magnitude of the deviation of today’s return from expected return

### Implications of EMH

* Technical analysis/chartism (trying to beat the market with historical data) is nonsense because the weak form of efficiency (all information that is publicly available has already been incorporated)
* More difficult to deal with the semi-strong form: what information is incorporated in the current prices
	+ Paradox of fundamental analysis: even with the semi-strong form you can't really make excess (but why then are there analysts and investment banks - the information is already incorporated in the price, so why pay for it?)
* Challenges to EMH comes from volatility & bubbles: stock prices deviate from fundamentals (because other factors than utility maximization factor into agent's choice principles)
* Allocation of capital through stock markets works only if their efficient
* What institutions might be absent/weak in countries where the stock market is inefficient?
	+ Auditing: There must be a legal framework that guarantees audit accounts
	+ Bankruptcy laws (including property rights)
	+ The exchange market itself: you need a board, regulatory institutions for inside and outside of the market
	+ Ranking organizations independent of the exchange and the firms
	+ Free flow of information about the macroeconomy/macro policy, and hence government policy - and hence a free press
	+ Absence of political manipulation of the stock market

## Derivatives

### General

* Based on credit but are not credit
* A derivative is a contract but the contract price is determined by the market price of the core asset
* Derivatives are a trading instrument par excellence in markets where participants are focussed on buying and selling repeatedly to capture price moves
* Virtually all of the growth has been in transactions between different forms of money and finance (financial futures and options, but especially interest rate and currency swaps - swaps are by far the most important of the three types (forward, options, swaps))
	+ Why the rise of swaps? → The demise of fixed exchange rates & need to hedge against x-rate or interest rate volatility
* OTC derivative gross notional outstanding is $631 trillion and gross market value 21tn
* Most is done OTC and by investment banks as dealers (with a high concentration of the market share in the hands of a few). The counterparts to OTC derivatives are other financial firms, both dealers themselves and non-bank financial firms. Non-financial firms are counterpart to less than 10% of the notional outstanding
* Uses: hedging, speculating, arbitraging
* Underlying can be commodity or other financial instruments. Participants act as if they are exchanging objects without having to go to the bother of actually doing so
* The problem of pricing multiple derivatives on the same underlying: derivatives allow differently structured claims on the same underlying which have different money prices but must be priced consistently for trading to occur, and require a translation mechanism between prices
* Two functions of derivatives:
	+ Binding: Some forms of derivatives bind the present to the future by reconciling prices today with prices tomorrow.
	+ Blending: blend apparently discrete types of assets by creating novel assets with characteristics that are a composite of these (once) discrete types (e.g. a blend of debt and equity, or a blend of stock prices and exchange rates).
* Instead of derivative markets reflecting spot/cash markets (where delivery is immediate), prices were being formed in derivative markets and were then running to cash markets. Spot markets have become 'derivative'

### Forwards

* Usually traded OTC
* Agreement to buy one unit of an asset worth ST (the spot price in, say, 6 months) for the price K (delivery price fixed in the contract) → Profit is made if there is a difference between the forward and spot price. If in 6 months, Forward (delivery price) < Spot price, the long position gains: the long positions gain is ST - K and the short position's gain is K - ST
* In practice, between two financial institutions, selling party is short, will deliver, and receives the specified price
* Forward price: price for some time in the future but determined now
* Spot price: going market prices; immediate deliver
* Difference forward & futures
	+ Forward contracts are private arrangements between two parties, whereas futures contracts are highly standardized and traded on exchanges. There is generally a single delivery date in a forward contract, whereas futures contracts frequently involve a range of such dates
	+ Under the forward contract, the whole gain or loss is realized at the end of the life of the contract. Under the futures contract, the gain or loss is realized day by day because of the daily settlement procedures
	+ A forward contract is not usually settled until the end of its life, and most contracts do in fact lead to delivery of the underlying asset or a cash settlement at this time.
* The forward price of a security providing no income is always higher than the spot price
* The value of a forward contract at the time it is first entered into is close to zero. At a later stage, it may prove to have a positive or negative value. It is important for banks and other financial institutions to value the contract each day. This is referred to as marking to market the contract.

### Futures

* Short: entering into a futures contract to sell asset in x months for y $
* Long: entering into a futures contract to buy asset in x months for y $
* The relationship between forward prices and spot prices: F0 = S0erT
* Although delivery rarely happens (most contracts are closed out early), it is the possibility of eventual delivery that determines the futures price
* A very high proportion of the futures contracts that are traded do not lead to the delivery of the underlying asset. Traders usually enter into offsetting contracts to close out their positions before the delivery period is reached.
	+ The decision on when to deliver is made by the party with the short position
	+ The usual rule chosen by the exchange is to pass the notice of intention to deliver on to the party with the oldest outstanding long position. Parties with long positions must accept delivery notices.
	+ The first notice day is the first day on which a notice of intention to make delivery can be submitted to the exchange. The last notice day is the last such day. The last trading day is generally a few days before the last notice day.
* Is standardized and market-traded, settled in cash, not attached to two specific individuals. Largest is CME Group Inc. (Chicago Mercantile Exchange & Chicago Board of Trade)
* Whereas a forward contract is settled at the end of its life, a futures contract is settled daily.
* Contract details quantity and quality of the commodity carefully, the place and delivery are also detailed
* You can corner the market: by going long (getting the commodity in the future) and buying up the commodity now (i.e. cornering the market) to raise the prices - then you make a profit
* These markets are bets - no one wants the underlying actually
* Roles of the market itself: act as clearing house (traders hold an account with the clearing house; cleared daily; minimum safeguards such as margins and price limits) & take upon itself the credit default/counterpart risk
* Initial margin is not more than 10%; maintenance margin is 3/4 of initial margin; if failure, 'margin call' to stock up funds (called variation margin)
* Note that margin requirements are the same on short futures positions as they are on long futures positions. It is just as easy to take a short futures position as it is to take a long one. The spot market does not have this symmetry.
* The settlement price is the price used for calculating daily gains and losses and margin requirements.
* Securities are usually accepted as margin, although at a reduced value for margin purposes (called 'haircut')
* If prices fluctuates too much, trading is suspended
* The difference between spot and future price is the cost of carry: storage + interest paid to finance the asset - income earned from asset
* Basis risk:
	+ Basis = Spot price of asset to be hedged - Futures price of contract used
	+ Basis risk arises from uncertainty as to what the basis will be at maturity of the hedge
	+ Assume a company using a short hedge because it plans to sell the underlying asset. If the basis strengthens (i.e., increases) unexpectedly, the company’s position improves because it will get a higher price for the asset after futures gains or losses are considered; if the basis weakens (i.e., decreases) unexpectedly, the company’s position worsens
	+ Cross hedging occurs when the two assets are different (e.g. an airline that wants to hedge jet fuel but there are no such futures and so it might use heating oil futures contracts to hedge its exposure)
* The hedge ratio is the ratio of the size of the position taken in futures contracts to the size of the exposure. When the asset underlying the futures contract is the same as the asset being hedged, it is natural to use a hedge ratio of 1.0 (i.e. contracts = exposure)
	+ Higher for commodities
* Hedge ratio: choose the # of contracts that minimizes risk, i.e. 1st derivative

### Options

* Forward contracts are designed to neutralize risk by fixing the price that the hedger will pay or receive for the underlying asset. Option contracts, by contrast, provide insurance.
* On one side is the investor who has taken the long position (i.e., has bought the option). On the other side is the investor who has taken a short position (i.e., has sold or written the option).
* Buyers are referred to as having long positions; sellers are referred to as having short positions. Selling an option is also known as writing the option
* Call option: right to buy the underlying asset by a certain date for a certain price
* Put option: right to sell the underlying asset by a certain date for a certain price
* Exercise price or strike price: price in the contract
* Expiration date or maturity: the date in the contract
* Not traded on margin
* American options (=majority of options): can be exercised at any time up to the expiration date
* European options: can be exercised only on the expiration date itself
* In the exchange-traded equity option market, one contract is usually an agreement to buy or sell 100 shares
* Whereas it costs nothing to enter into a forward or futures contract, except for margin requirements, there is a cost to acquiring an option.
* The largest exchange in the world for trading stock options is the Chicago Board Options Exchange
* The price of a call option decreases as the strike price increases, while the price of a put option increases as the strike price increases.
* When a call option is exercised, the holder’s gain equals the excess of the stock price over the strike price. When a put option is exercised, the holder’s gain equals the excess of the strike price over the stock price.
* Options are referred to as in the money, at the money, or out of the money. If the strike price is better than the price in the market at maturity, the option is deemed in the money. If S = stock/spot price and K = strike price
	+ A call option is in the money when S > K, at the money when S = K, and out of the money when S < K
	+ A put option is in the money when S < K, at the money when S = K, and out of the money when S > K.
* Both types of option tend to become more valuable as their time to maturity increases
* With a call option, you buy more cheaply
* With a pu option, you sell more dearly
* Conversely, if you're afraid that the stock price of the shares you hold might decline, you can buy put options that guarantee you a given strike price ('lock in' your sale price) - The put option gives you a floor below which you won't fall
* A covered call would be to sell a call while having bought the stock
* Synthetic stock options are option strategies that copy the behavior and potential of either buying or selling a stock, but using other tools such as call and put options
* C + cash = P + underlying
* If volatility increases, it pays to buy and not to sell options
* Intrinsic value: (X - S)\*er(T-t). For example, you'd want to sell an American put as soon as (X - S)\*er(T-t) > X

* **Buy European call/long call position**: purchasing the right to buy; if the price of the underlying is below the strike price, you lose the price of the call option (cause you don't exercise it). Payoff: S-X. Graph: purchasing a contract consisting of 100 Google December call options with a strike price of $700
* **Sell European call/short call position**: selling right to buy; if the price of the underlying is above the strike price, you lose the price of the call option (cause you don't exercise it)
* **Buy a put/long put position**: purchasing the right to sell; if you believe that the price of the underlying security will go significantly below the striking price before the expiration date; payoff is X-S where X=strike price and S=underlying price
* **Sell/write a put/short put position/selling short**: selling the right to sell/letting someone insure themselves; do this if you expect that the stock rises above the strike price of the short put by expiration because then the put options expire worthless and entire premium from its sale is earned

#### Pricing

* Intrinsic value (i.e. what is the value of the option right now if it were at expiration, is S-K ≠ 0)
	+ By definition, the only options that have intrinsic value are those that are in-the-money
* A call option’s intrinsic value is defined as max(S - X,0), put option’s intrinsic value is max(X - S,
* Extrinsic/time value (has to do with how option prices fluctuate in addition to fluctuations of the underlying)
	+ Time to maturity: value of option increases the longer the time to maturity (TtM); after all the longer has all the benefits of the shorter TtM plus some more
	+ Instantaneous variance of the return of the asset (i.e. volatility). As option holder, more volatility means more exposure (higher profit and loss). The preference of investors is opposed: volatility is bad for underlying-holders and good for option-holders (of this underlying).
	+ Risk-free interest rate: as i increases, the value of the stock increases - and hence of the option decreases; we can construct an option portfolio that eliminates risk - hence risk-free interest rate is relevant

#### Put-call parity

* The value of a European call (put–call parity holds only for European options) with a certain exercise price and exercise date can be deduced from the value of a European put with the same exercise price and exercise date, and vice versa
* Allows a trader or investor to derive the price of a call from a put and vice versa
* Defines a relationship between the price of a EU-call option and EU-put option, both with the identical strike price and expiry: a portfolio of a long call option + a short put option is equivalent to/same value as a single forward contract at this strike price and expiry
* Simply, (the payoff of) a stock + a put option (i.e. to hedge against a price fall) can be mirrored by a bond with a certain payoff in the future + a call option (that I exercise if the stock price rises): S + P (purchase put + buying the fraction e-rt of one share of the stock) = B + C (purchasing one call that expires at time T and lending the present value of the strike price at the riskless rate of interest)
	+ Note that B stands for a bond. In the BS, this is simply the price at expiration/strike price discounted back to the present at the risk-free rate

#### Exercise an American call?

* It is never optimal to exercise an American call option on a non-dividend-paying stock before the expiration date
* Because you would make a higher profit if you sold the option on the open market at that point in time, rather than exercising it at that point in time due to the time value of money
* The value of the call option is always bigger than what you might gain from exercising it (hence reaping S-K)
* If you exercise now, you have to pay for the stock – money for which you cannot receive interest anymore (and the stock doesn’t pay dividend)!
* Also: assume the stock price falls even lower than your strike price – now imagine you had exercised and paid the expensive price rather than wait and pay the now-cheap stock price
* To summarize, there are two reasons an American call on a non-dividend-paying stock should not be exercised early. One relates to the insurance that it provides. A call option, when held instead of the stock itself, in effect insures the holder against the stock price falling below the strike price. The other reason concerns the time value of money. From the perspective of the option holder, the later the strike price is paid out the better.

### Trading

* Typically pension funds want to hedge the value of their portfolio
* If the holder of an asset wants to sell in the future or lock in the value of the portfolio by taking short future positions. If spot price falls, the hedge loses on the asset sale but gain in future sale; if the price rises, the hedge reduces the value (it's an insurance)
* Using futures to hedge is not very popular except for financial futures; if producers want to hedge they draft forward and not futures contracts
* The amount that must be deposited at the time the contract is entered into is known as the initial margin.
* At the end of each trading day, the margin account is adjusted to reflect the trader’s gain or loss. This practice is referred to as daily settlement or marking to market.
* In a day trade the trader announces to the broker an intent to close out the position in the same day. In a spread transaction the trader simultaneously buys (i.e., takes a long position in) a contract on an asset for one maturity month and sells (i.e., takes a short position in) a contract on the same asset for another maturity month.
* Trading volume is the number of contracts traded in a day
* Open interest: # of contracts outstanding, that is, the number of long positions or, equivalently, the number of short positions
* Cotango: futures price is an increasing function of maturity
* Backwardation: is sometimes used to describe the situation where the futures price is a decreasing function of the maturity of the contract
* Market order: request that a trade be carried out immediately at the best price available in the market.
* Limit order: specifies a particular price. The order can be executed only at this price or at one more favorable to the trader
* Stop order or stop-loss order: order is executed at the best available price once a bid or offer is made at that particular price or a less favorable price; becomes a market order as soon as the specified price has been hit; usually to close out a position if unfavorable price movements take place. It limits the loss that can be incurred.

### Speculating

* Scalpers: watching for very short-term trends and attempt to profit from small changes in the contract price. They usually hold their positions for only a few minutes
* Day traders: hold their positions for less than one trading day. They are unwilling to take the risk that adverse news will occur overnight. \* Position traders: hold their positions for much longer periods of time. They hope to make significant profits from major movements in the markets.
* Making choices about movements of prices, usually taking long positions (hoping that prices rise)
* Difference to speculating on the underlying: you don't need to make any initial cash payment
* Keynes argued that hedgers hold short positions (lock in value of what you got and avoid price falls) vs. speculators who go long
* Check 'normal backwardation' and 'cotango'

### Arbitrage

* Arbitragers enter two markets simultaneously and exploit disequilibrium
* There is a convenience yield to commodities (I own the commodity and hence can take advantage of local price changes)

### Hedging

* A short hedge is appropriate when the hedger already owns (or will own) an asset and expects to sell it at some time in the future. It locks in a specific sale price
* A long hedge is appropriate when a company knows it will have to purchase a certain asset in the future and wants to lock in a price now

## Options pricing

* The main features of these models are that they are arbitrage-free (meaning no profit can be made by simultaneous buying and selling at different prices), and they make commensurate an array of money prices by using a common pricing factor, e.g. implied volatility or implied correlation
* The models have three basic elements: they define the financial instrument being priced by specifying the payments to be made between counterparts for different states of an underlying reference price index, and these payments are then discounted and probability-weighted.
* The models need a common metric to compare tranches on the same underlying which will have incomparable money prices, and a mechanism to compare potential future prices with recently observed completed market prices
* Derivatives resist trading through incomparable money prices which must be compared. Models solve the problem and are therefore necessarily bound up with derivatives.
* The derivative claim lets participants act as if they are buying and selling the underlying index, but the claim itself is rarely bought and sold; instead, a second derivative transaction is added to the first etc
* As the pricing of new trades becomes increasingly reliant on the use of valuation models, model-generated prices come to appear to be the market price and the prices of completed transactions come to be seen as diverting from the true, or model, price

### Markov & Wiener process

* A Markov process is a particular type of stochastic process where only the current value of a variable is relevant for predicting the future. The past history of the variable and the way that the present has emerged from the past are irrelevant.
* When Markov processes are considered, the variances of the changes in successive time periods are additive.
* The mean change per unit time for a stochastic process is known as the drift rate and the variance per unit time is known as the variance rate.
* Wiener process as a drift + shocks
* Options as a random variable based on a random variable + time
* Eliminate all risk by holding a combination of option and underlying with the rate of return = to the risk-free rate
* Random walks on integers and the gambler's ruin problem are examples of Markov processes
* An examples of Markov processes is the Wiener process, also known as the Brownian motion process

### Black-Scholes model

* The key financial insight behind the equation is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk". This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula
* The Black–Scholes–Merton differential equation can be derived by setting up a riskless portfolio consisting of a position in an option on a stock and a position in the stock → the return from the riskless portfolio in any very short period of time must be the risk-free interest rate
* The reason a riskless portfolio consisting of a position in the derivative and a position in the stock (what follows from BS) can be set up is that the stock price and the derivative price are both affected by the same underlying source of uncertainty: stock price movements. In any short period of time, the price of the derivative is perfectly correlated with the price of the underlying stock. When an appropriate portfolio of the stock and the derivative is established, the gain or loss from the stock position always offsets the gain or loss from the derivative position so that the overall value of the portfolio at the end of the short period of time is known with certainty
	+ In Black–Scholes–Merton, the position in the stock and the derivative is riskless for only a very short period of time. (Theoretically, it remains riskless only for an instantaneously short period of time.) To remain riskless, it must be adjusted, or rebalanced, frequently
* Under the assumptions of rational pricing (which assumes that asset prices & asset pricing models reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away"), the BS assumes that derivatives can be replicated by portfolios of other securities, and thus their prices determined
* Inputs into the model: current stock price, options strike price, time until expiration (denoted as a percent of a year), risk-free interest rate, anticipated variance/implied volatility
* The final equation does not contain any variables on which attitudes towards risk might differ
* The key breakthrough of the Black and Scholes option model concerned the rate at which anticipated cash flows should be discounted
* Assumptions: European & can only be exercised at expiration; no dividends are paid out during the life of the option; markets are efficient (i.e., market movements cannot be predicted/random walk of asset prices); no transaction costs in buying option; the risk-free rate and volatility of the underlying are known and constant (volatility is derived from the historical prices of a given security); the returns on the underlying are normally distributed/assumes a lognormal (Gaussian) distribution of price changes for the underlying asset
* The European Call option is equal to a probability of an event occurring, times the stock price, minus a probability of an event occurring, times the present value of the exercise price
* The formula[5]:
* Call premium = Stock price x Cumulative Standard Normal (d1) - Option Strik Price x Cumulative Standard Normal (d2) x e-(risk-free IR) x time until expiration
* The first part is what you get and the second what you pay (discounted back to today at the risk-free rate, giving present value of K)

* C
0

=

S
0

N
(

d
1

)
−
K

e

−
r
t

N
(

d
2

)

{\displaystyle C\_0 = S\_0N(d\_1) - Ke^{-rt}N(d\_2)}

 and

P
0

=
K

e

−
r
t

N
(
−

d
2

)
−

S
0

N
(
−

d
1

)

{\displaystyle P\_0 = Ke^{-rt}N(-d\_2) - S\_0N(-d\_1) }

* + Where:
	+ N
	(

	d
	2

	)

	{\displaystyle N(d\_2)}

	 is the probability that a call option will be exercised in a risk-neutral world.

	d
	1

	,

	d
	2

	{\displaystyle d\_1, d\_2}

	 can be interpreted as measures of moneyness (in standard deviations) and the terms

	N
	(

	d
	1

	)
	,
	N
	(

	d
	2

	)

	{\displaystyle N(d\_1), N(d\_2)}

	 are the *probabilities of the option expiring in-the-money*.
	+ S
	0

	N
	(

	d
	1

	)

	e

	r
	t

	{\displaystyle S\_0N(d\_1)e^{rt}}

	 is the expected stock price at time T in a risk-neutral world when stock prices less than the strike price are counted as zero
	+ d
	1

	=

	l
	n
	(

	S
	K

	)
	(
	r
	0.5

	σ
	2

	)
	t

	σ
	∗

	t

	{\displaystyle d\_1 = \frac{ln( \frac{S}{K})+(r+ 0.5\sigma^2)t}{\sigma\* \sqrt{t}}}

	 Note that this first term describes the current value of the stock and multiplies it by a probability (d1); it multiplies the price by the change in the call premium in relation to a change in the underlying price. This part of the formula shows the expected benefit of purchasing the underlying outright
	+ d
	2

	=

	l
	n
	(

	S
	K

	)
	(
	r
	−
	0.5

	σ
	2

	)
	t

	σ
	∗

	t

	{\displaystyle d\_2 = \frac{ln( \frac{S}{K})+(r- 0.5\sigma^2)t}{\sigma\* \sqrt{t}}}

	 or put differently

	d
	2

	=

	d
	1

	−
	σ
	∗

	t

	{\displaystyle d\_2 = d\_1 - \sigma\*\sqrt{t}}

		- Note that this second term takes the present value of the strike price at expiry (discounted to today, and subtracts it from the first term); the second part provides the current value of paying the exercise price upon expiration. The value of the option is calculated by taking the difference between the two parts, as shown in the equation
	+ The reason for the

	1
	2

	σ
	2

	{\displaystyle \frac{1}{2}\sigma^2}

	 factor is due to the difference between the median and mean of the log-normal distribution; it is the same factor as in Itō's lemma applied to geometric Brownian motion

* As volatility increases,

d
1

{\displaystyle d\_1}

 increases and

d
2

{\displaystyle d\_2}

 decreases
* The higher the σ of our log returns, the bigger d1 will get (since in the numerator σ is squared, whereas in the denominator not). In d2 the same applies but we will subtract σ - a higher σ will reduce d2. This will all raise the value of the call

#### The Greeks

* Replicating portfolio: a position consisting shares & (risk-free) borrowed money, which will produce identical cash flows to one option on the underlying share - given an asset or liability, an offsetting replicating portfolio is called a static hedge or dynamic hedge; fundamental to rational pricing, which assumes that market prices are arbitrage-free – concretely, arbitrage opportunities are exploited by constructing a replicating portfolio
	+ Static replication: portfolio has the same cash flows as the reference asset (e.g. in put-call parity)
	+ Dynamic replication: portfolio does not have the same cash flows, but has the same "Greeks" (measures of sensitivity of the price of derivatives to a change in underlying parameters on which the value of a portfolio is dependent) as the reference asset (i.e. for small/infinitesimal changes to underlying market parameters, the price of the asset and the price of the portfolio change in the same way); fundamental to BS
* Delta:

∂
C

∂
S

{\displaystyle \frac{\partial C}{\partial S}}

	+ In the BS, Delta

	Δ

	{\displaystyle \Delta}

	, measures the rate of change of the theoretical option value with respect to changes in the underlying asset's price. Delta is the partial derivative/first derivative of the value of the option with respect to the underlying instrument's price *S*.
	+ Represents the sensitivity of option price to the intrinsic value (how stock & strike price stand to each other) - the amount an option price is expected to move based on a 1$ move up of the underlying.
	+ Calls have positive delta, puts a negative - between 0-1
	+ The theoretical change in premium for each basis point or $1 change in price of the underlying is the delta, and the relationship between the two movements is the hedge ratio
	+ For example, the price of a call option with a hedge ratio of 0.60 will rise 60% of the stock-price move if the price of the underlying stock increases by $1. Imagine an investor who has sold call options to buy 2,000 shares of a stock. The investor’s position could be hedged by buying 0.6 x 2000 = 1,200 shares so that the overall delta is zero/neutral → since the delta of an option does not remain constant, the trader’s position remains delta hedged (or delta neutral) for only a relatively short period of time. The hedge has to be adjusted periodically through rebalancing. If, for example, delta goes up by 0.05, 0.05 x 2000 = 100 shares would have to be bought to maintain the hedge → this procedure is called dynamic hedging
	+ An options position could be hedged with options with a delta that is opposite to that of the current options position to maintain a delta neutral position. A delta neutral position is one in which the overall delta is zero, which minimizes the options' price movements in relation to the underlying asset. Options can be valued by setting up a delta-neutral position and arguing that the return on the position should (instantaneously) be the risk-free interest rate
	+ Delta hedging aims to keep the value of the financial institution’s position as close to unchanged as possible.
	+ It is the slope of the curve that relates the option price to the underlying asset price.
* Gamma:

∂
2

C

∂
2

S

{\displaystyle \frac{\partial^2 C}{\partial^2 S}}

	+ Rate of change/acceleration amount of delta: Gamma tells the trader how much the hedging ratio changes if the stock price changes
	+ The greater is Gamma, the more "out-of-line" a hedge becomes when the stock price changes, and the more frequently the trader must adjust the hedge.
	+ If delta is the speed of option price change, gamma is the accelerations; options with high gamme are highly responsive to change in the price of underlying; the convexity of the derivative value with respect to the underlying value
	+ Gamma is almost zero for deep-in-the-money and deep-out-of-the-money options, but it reaches a peak for near-the-money options → traders holding near-the-money options will have to adjust their hedges frequently and sizably as the stock price vibrates.
* Vega:

∂
C

∂
σ

{\displaystyle \frac{\partial C}{\partial \sigma}}

 impact of 1% change in volatility on option price
	+ A call option’s Vega conforms closely to the pattern of its Gamma, peaking for near-the-money options and falling to zero for deep-out or deep-in options. Thus, near-the-money options appear to be most sensitive to changes in volatility
* Theta: time to maturity (how much value of option decays the close it gets to expiration); always negative; this decay accelerates more the close we get
* Rho:

∂
C

∂
ρ

{\displaystyle \frac{\partial C}{\partial \rho}}

 sensitivity of the option price to a change in the interest rate

#### Volatility in the BS-model

* The implied volatility is the volatility that makes the Black–Scholes-Merton price of an option equal to its market price. The implied volatility is calculated using an iterative procedure
* The volatility σ of a stock is a measure of our uncertainty about the returns provided by the stock
* Generally speaking, volatility matters for option pricing because options provide an upside (a potentially very large one) while limiting the downside (to a possibly very small upfront premium sum), option buyers would enjoy the sight of the underlying asset dancing vertiginously, as movement can only deliver benefits on a net basis: the more movement, the higher the potential for a large gain all the while keeping the loss perennially constant
* Volatility is to a large extent caused by trading itself
* Implied volatility is a poor forecast of actual volatility: it is a biased forecast because the forecast error doesn't have a mean of zero
* Volatility aids us in option pricing by incorporating a reality-informed view as to the underlying asset´s possibilities, beyond the irrepressibly isolated picture provided by the asset´s spot price at any particular point → we need to make presence for a variability-representing parameter when trying to ascertain the proper value of a variability-enjoying instrument
* Hence, the asymmetry of options payouts determines that an asset with the capacity to swing is a more attractive underlying than one without such capacity
* BUT: which number do we plug in for volatility? try to (subjectively) predict turbulence from here till expiration date? → The benefit of the volatility parameter would thus be directional, rather than precisely numerical: revise it up or down based on recent market events, but don´t presume to get the right future figure at three decimal points
* The implied volatilities of actively traded options on an asset are often used by traders to estimate appropriate implied volatilities for other options on the asset
* The question is this: Why should an underlying asset´s volatility matter when pricing and valuing an option? → Unless the underlying is itself directly referenced to volatility, the final payout will not be directly determined by volatility. So shouldn´t we only care about the actual spot and forward prices of the asset and not about how much those happen to move around?
* What has perhaps traditionally been the most discussed and controversial aspect of the model: what number should we insert under the formula´s volatility parameter and what should that number truly stand for. The answers to those questions can lead to vastly different option prices and to vastly different interpretations of those prices
* Berkshire Hathaway uses the famed Black-Scholes option pricing model to calculate its liabilities on the massive long-term equity index puts it sold between 2004 and 2008. The fair value of the options, as churned out by the model, equals Berkshire´s discounted theoretical expected cash obligations on the trade or, in other words, the liquidation cost of the portfolio (should Berkshire be able to find someone willing to take the risk off its hands for precisely that amount)
* A key characteristic of modern option markets is that traders would use a different volatility number depending on the moneyness of the contract at any given time, in contrast to pure Black-Scholes that assumes constant volatility independent of moneyness intensity
* Implied volatility [6]
	+ is forward looking (vs. historical volatility which is the annualized σ of past stock price movements)
	+ overstates the actual expected move of a stock in nearly all long-term cases - i.e. implied volatility is usually > historical volatility (except in GFC) → hence options are usually overpriced (assume to much volatility) which means that sellers are better-off than buyers
	+ in the normal distribution, low volatility means a high curve/thin tails because most movement is around the mean, whereas high volatility is represented by fat tails
	+ if (your assumed) IV goes up while the stock price didn't change at all, the price of the option still rises
	+ is (a measure of) the estimation (based on probability) of the future variability for a security's price/the asset underlying the option contract. In general, implied volatility increases while the market is bearish (because bearish markets are believed to be riskier), when investors believe the asset's price will decline over time, and decreases when the market is bullish, when investors believe that the price will rise over time.
	+ is based on market participant action/where and how much the market expects prices to move, i.e. a bet and not necessarily where the price is actually going to move
	+ expressed as % of stock price. If stock price=$50, implied volatility=20% (i.e. $10 in either direction)
	+ approximates the future value of an option, and the option's current value takes this into consideration.
	+ is directly correlated with market opinion (again, Keynes: traders trying to predict not the true value but the average opinion of the true value), which does in turn affect option pricing
	+ is not the same as historical volatility, also known as realized volatility or statistical volatility
	+ is the only factor in the model that isn't directly observable in the market; rather, the option pricing model uses the other factors to determine implied volatility and the option's premium
	+ adds a weighting effect against a stock's price today and the stock's discounted price at expiry
	+ is determined by D & S: When a security is in high demand, the price tends to rise, and so does implied volatility, which leads to a higher option premium due to the risky nature of the option. Conversely, when S>D the implied volatility falls and the option price becomes cheaper.
* In fact, when the implied volatilities for options with the same expiration date are mapped out on a graph, a smile or skew shape can be seen even though a graph of the implied volatilities against any economic variable should show a flat line. Thus, the Black-Scholes model is not efficient for calculating implied volatility → Stated differently, deep-out and far-in options trade "rich" (overpriced) relative to near-the-money options. This has been explained by the fat tails in the frequency distribution of changes in the log of stock prices
* Importantly, by choosing the lognormal distribution, the price of any option on a given underlying provided by the model corresponds to a single variance number (known as volatility and given as v² in the formula above)
* The first logical step is to input observed prices of completed transactions and to obtain a single, corresponding implied volatility. In the second step the model is run in the reverse direction; the implied volatility (of the observed transaction) is input in order to calculate a consistent (but different) money price for a potential future transaction, i.e. for a different derivative contract on the same underlying
* Crucially, the likelihood of a call option to be in-the-money, given high volatility, will be much greater than that of a similar security with less volatility through maturity because bigger price swings benefit but do not subtract (because you simply don't exercise), i.e. the curve is flatter

* For puts, the terms are reversed such that X-S[7]
* The pivot is σ because past values are not a good estimate
* Derivatives do not affect volatility

#### Problems with BS

* Assessments of a model’s validity can be done in two ways. First, the model’s predictions can be confronted with historical data to determine whether the predictions are accurate, at least within some statistical standard of confidence. Second, the assumptions made in developing the model can be assessed to determine if they are consistent with observed behavior or historical data.
* It is well-known that Black-Scholes' main assumptions (that every option for every strike on a given stock and given expiry has the same volatiliy) simply do not hold water. Financial prices do not rule out extreme moves, tend to jump and present a volatility that is not constant
* The key strategy behind the formula - so-called dynamic hedging, a mathematical recipe for constructing a portfolio, made up of the underlying asset and cash, that will theoretically mirror the option's value at all times - is, sadly, not really feasible in the real world.
* The Black–Scholes–Merton option pricing model assumes that the probability distribution of the stock price in 1 year (or at any other future time) is lognormal. It assumes that the continuously compounded rate of return on the stock during the year is normally distributed
* Haug & Taleb, 2008:
	+ BS argumet is that an option can be hedged using a certain methodology called “dynamic hedging” and then turned into a risk-free instrument, as the portfolio would no longer be stochastic.
	+ BS assumes knowledge of the probabilities of future events (in a neoclassical Arrow-Debreu style). This is the most problematic assumption: certain knowledge of future delivered variance for the random variable (or, equivalently, all the future probabilities). This is what makes it clash with practice –the rectification by the market fattening the tails is a negation of the Black-Scholes thought experiment.
	+ BS is in fact not used (even by those who think that they use it): option prices may be simply the result of supply-demand interaction, with no model involved at all - options often are priced via other options, through a simple conduit known as put-call parity
	+ Volatility fudging means abandoning the original BS formula (BS didn't have volatiliy fudging in mind): Certain inputs must be inserted in order to obtain some output (price of option) - the underlying asset's volatility, however, is not directly available and traders must estimate it before feeding it into the BS; pure version of BS assumes that volatility should be constant, so that if one were to select all the listed options on e.g. IBM for the same maturity, one should obtain the same volatility for each contract independent of its particular strike level → IBM should have just one volatility expectation per maturity date
		- The strike price-implied volatility graph should be a horizontal line, whereas in fact it is the famous 'volatility smile' - reflects the fact that the volatility input is being fudged, or manipulated, so as to obtain more realistic option prices, essentially to correct for the formula's unrealistic assumptions and to allow traders to freely express their opinions
	+ BS indeed "smoothes" out risks but exposes the operator to massive tail events – reminiscent of such blowups as LTCM
	+ It is this central removal of the “risk premium” that apparently was behind the decision to confer the prize: “Black, Merton and Scholes made a vital contribution by showing that it is in fact not necessary to use any risk premium when valuing an option. This does not mean that the risk premium disappears; instead it is already included in the stock price.”