Trading, Investment Companies, and Derivatives: A Comprehensive Analysis

Posted on Feb 2, 2025 in Finance

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Lecture 16: Trading Mechanics and Costs

Trading Commissions: Discount brokerages profit from Payment for Order Flow (PFOF) and asset management products (explicit). PFOF: Customer to broker, broker sends to wholesaler who gets a fee, wholesaler provides best execution back to broker to customer.

Spread: Difference between bid (buy) and ask (sell) price (implicit).

Investors sell at the bid and buy at the ask.
Transaction cost for investors, source of profit for market makers (e.g., dealers).

Some trades are executed inside the quoted spread. Traders break up larger orders into smaller blocks to limit price impact.

Market Depth: Total number of shares offered for trading at best prices, a measure of liquidity, larger for larger stocks.

Market Order: Executed immediately. Trader receives current market price.

Walk down the book at ask prices (for buy), buy the cheapest first. The book bid side looks the same.

Price-Contingent Order: Traders specify buying or selling price.

Limit order: Sell when the price rises to a certain level (ask price), buy when the price falls to a certain level (bid price).

Like a market order, it stops when you hit the price limit. If it’s an average, you do it out of the shares you immediately purchased, not the total.
After the order, add to the bid side the price and quantity left to fulfill the original order.

Stop order: Sell when the price falls, buy when the price rises. Can be used to insure a short position.

Margin = Equity in Account / Value of Stock

Set by the Fed (currently at 50%).
Maintenance Margin: Minimum equity that must be kept in a margin account.
Margin call if the value falls too much.

Practice Problem: Initial Value – Initial Loan Amount = Margin

Loss = 3000, then subtract that from the old margin. Margin percent equals the new margin divided by the number of shares times the new share price.

Short Sales:

Borrow stock through a broker or dealer.
Sell it and deposit proceeds and margin in an account.
Must post additional margin (margin call) if losses decrease margin below the maintenance level.
Closing out the position: Buy the stock and return it to the party from which it was borrowed.

Profit = Decline in share price x number of shares sold short.

Investors sell at the bid and buy at the ask.
Transaction cost for investors, source of profit for market makers (e.g., dealers).

Some trades are executed inside the quoted spread. Traders break up larger orders into smaller blocks to limit price impact.

Market Depth: Total number of shares offered for trading at best prices, a measure of liquidity, larger for larger stocks.

Market Order: Executed immediately. Trader receives current market price.

Walk down the book at ask prices (for buy), buy the cheapest first. The book bid side looks the same.

Price-Contingent Order: Traders specify buying or selling price.

Limit order: Sell when the price rises to a certain level (ask price), buy when the price falls to a certain level (bid price).

Like a market order, it stops when you hit the price limit. If it’s an average, you do it out of the shares you immediately purchased, not the total.
After the order, add to the bid side the price and quantity left to fulfill the original order.

Stop order: Sell when the price falls, buy when the price rises. Can be used to insure a short position.

Margin = Equity in Account / Value of Stock (set by the Fed, currently at 50%).

Maintenance Margin: Minimum equity that must be kept in a margin account.
Margin call if the value falls too much.

Practice Problem: Initial Value – Initial Loan Amount = Margin. THEN loss = 3000, and then subtract that from the old margin. Then, the margin percent equals the new margin divided by the number of shares times the new share price.

Short Sales:

Borrow stock through a broker or dealer.
Sell it and deposit proceeds and margin in an account.
Must post additional margin (margin call) if losses decrease margin below the maintenance level.
Closing out the position: Buy the stock and return it to the party from which it was borrowed.

Profit = Decline in share price x number of shares sold short.

Lecture 17: Investment Companies

Investment Companies: Pool funds of individual investors and invest in a wide range of securities or other assets. Small individual investors get the benefits of acting as one large investor and pay a fee for it.

Services provided:

Administration & record keeping
Diversification & divisibility
Professional management
Reduced transaction costs (economies of scale)

NAV=(Market value of assets minus liabilities)/(Shares Outstanding)

Gross return: Fund return before fees and expenses; i.e., the return on the fund’s investments. Note: Gross return is return after transaction costs.

Net return: Fund return after fees and expenses.

𝑅𝑛𝑒𝑡=𝑅𝑔𝑟𝑜𝑠𝑠−𝐸𝑥𝑝𝑒𝑛𝑠𝑒 𝑅𝑎𝑡𝑖𝑜 (𝐸𝑥𝑝𝑒𝑛𝑠𝑒 𝑅𝑎𝑡𝑖𝑜 includes management fees and 12b-1 fees)

𝑅_𝑛𝑒𝑡= [𝑁𝐴𝑉1−𝑁𝐴𝑉0+𝐼𝑛𝑐𝑜𝑚𝑒 𝑎𝑛𝑑 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛𝑠 ]/ (𝑁𝐴𝑉0)
NAV is net of expenses

Premium/Discount= (Price- NAV)/NAV

Your return: Return after paying commissions.

𝑅_𝑖𝑛𝑣𝑒𝑠𝑡𝑜𝑟=(1+Rnet )(1−FrontendLoad)(1−Backend Load)−1

𝑅_𝑖𝑛𝑣𝑒𝑠𝑡𝑜𝑟=($ 𝑂𝑢𝑡)/($ 𝐼𝑛)−1

How funds are sold:

Direct-marketed funds (e.g., Vanguard)
Distributed sales force (e.g., broker or financial advisor)
Financial Supermarkets

Mutual Fund Expenses:

1. Operating expenses (often expressed as an annual expense ratio)

Overhead and fees paid to fund manager (periodically deducted from assets)

0.2% of total AUM (even less for some index funds) to ~2% per year

2. 12 b-1 charge (added to operating expenses to get true annual expense ratio)

Used for marketing and distribution costs (e.g., broker commissions)

Capped at 1% per year (also periodically deducted)

3. Front-end load

Up-front commission / sales charge to purchase fund

Can be as high as 8.5% (typically lower)

Implication: Pay $100, get shares with NAV of $95 (example with 5% front-end load)

4. Back-end load

Redemption / exit fee to sell shares

Typically start at 5-6% and decrease over time funds are invested

Functions as an incentive to keep funds invested in a mutual fund

Other: transaction costs and taxes

Mutual Fund Taxation: Prefer to defer taxes as long as possible.

Mutual funds are pass-through entities.
Mutual funds are not taxed (i.e., no corporate income tax on the fund).
Instead, dividends and capital gains are passed through to investors.
Investors get distributions for dividends and realized capital gains.

Firms with Higher turnover will generate capital gains earlier: Selling by a mutual fund (e.g., in response to withdrawals by other investors) will trigger capital gains realizations even for investors who don’t withdraw.

Mutual Fund Performance: Mutual funds on average underperform the market by about 1%.

Past losers tend to keep losing.
Investors keep investing in them because of diversification benefits, expertise has more value, they believe markets are inefficient, and behavioral biases.

Arbitrage Assures NAV is close to Share Price: If demand for an ETF is high, the price of ETF shares exceeds NAV, and there is an arbitrage (buy the underlying portfolio, trade for ETF shares, sell ETF shares): Increases shares outstanding.

A similar arbitrage is possible if the ETF price is lower.

Securities Act of 1933: Must disclose:

Information about their management.
A description of the securities they own and offer.
Financial statements verified by independent accountants.

Exception: Only to accredited investors (those who make $200,000+ per year or have $1,000,000 in total net worth, not including their primary residence).

Hedge Funds:

Structured as a private partnership: Less SEC regulation than mutual funds. Only open to wealthy investors and institutions.

Lock-ups restrict when investors can withdraw funds: Allows funds to pursue less liquid investments. Makes hedge funds illiquid to investors.

Wide range of investment assets and strategies: Including short sales and derivatives. The goal is to generate excess returns (alpha) without exposure to systematic risk (beta). Secretive, do not have to disclose strategies to the public.

High fees: Standard is a flat fee (2% of assets) + performance fee (20% of returns above a benchmark). Recently a bit lower. Often a “high water mark” to ensure that the hedge fund makes up previous losses before getting a performance fee.

Hedge Fund Fees: Afterfee return =Beforefee return – percent fee

Growth of hedge funds:

Hedge funds generate alpha: Hedge fund managers are the smartest!
Hedge funds hedge! They provide non-systematic returns. Hedge funds are less constrained and can invest in a variety of assets.

Academic evidence: Some hedge funds probably have positive alpha.

Majority view: Hedge funds outperform the market, at least before fees (but this is not an entirely settled question). Fund of funds probably don’t have positive alpha (because of additional fees).

Private Equity: Ownership of shares that trade in private markets. Most generally, this includes venture capital (VC) (VC is a small portion of all PE funds).

Lecture 18: Evaluating Investment Returns

When evaluating returns, we ask:

Risk Adjustment
Performance attribution (comparison)
Luck vs. Skill

Universe Comparison: Compare returns to a benchmark and universe of similar funds.

Benefits:

Can control for risk, style, cost of investment strategy, and specific market conditions.
Easy to implement.

Costs: No clear way to choose a peer group, and results are sensitive to the group/benchmark.

Jensen’s Alpha: Intercept from the regression of excess returns on excess market returns (CAPM) or excess market returns and other factors (multifactor model) – for performance attribution.

CAPM: 𝑟 ̅𝑃−𝑟 ̅𝑓= 𝛼𝑃+𝛽(𝑟 ̅𝑀−𝑟 ̅𝑓 )

3-factor: 𝑟𝑃−𝑟 𝑓= 𝛼𝑃+𝛽 (𝑟 ̅𝑀−𝑟 ̅𝑓 )+𝛽_(𝑃,𝑆𝑀𝐵) (𝑆𝑀𝐵) ̅+𝛽_(𝑃,𝐻𝑀𝐿) (𝐻𝑀𝐿) ̅

HML is the return on a portfolio of high book-to-market firms (value firms) minus the return on a portfolio of low book-to-market firms (growth firms).
SMB is the return on a portfolio of small firms minus the return on a portfolio of big firms.

Alpha measures the return in excess of an investment’s required risk premium, under the assumption that the model correctly measures the risk premium. If the CAPM is the correct model of investment risk, CAPM alpha is the correct measure of risk-adjusted returns.

The risk-adjustment argument for multifactor models is weaker because it is less clear that they capture risk.

Fama and French interpret the 3-factor model as capturing risk premia, but it is really just an empirical model of returns.

Style Analysis: The three-factor model is one approach.

What you learn:

What is the fund’s actual investment style? Regression coefficients sometimes tell a different story than how the fund markets itself (e.g., “value” funds with 𝛽_(𝑃,𝐻𝑀𝐿)≈0).

How closely do the fund’s returns track its investment style? Tracking error can be measured by regression R2 or residual standard deviation. Low tracking error means the fund is basically the same as an index fund (closet indexing); high tracking error reflects active investing.

What is the fund’s style-adjusted performance? Caveat: In practice, fund-specific alpha measures are too noisy to be of much value.

Selection Bias: Need to think about why you decided to evaluate a particular fund.

Survivor Bias: Don’t report data. Relevance for performance evaluation: We sometimes only have data on surviving funds.

Particularly big problem if data is back-filled (the data set adds historical returns when a fund is added).

We have pretty good survivor-bias-free mutual fund data, but this remains a concern for hedge fund research.

Mutual Fund Performance: Jensen: Average alpha net of fees: -1.1%. Average alpha gross of fees: -0.4%.

Pastor and Vorsatz: They find funds tended to underperform the S&P 500. 74% of active funds underperform the benchmark. Average -5.6% return during a 10-week period (-29% annualized) compared to the S&P 500.

Some Private Equity: Private equity returns are not observable.

What’s the return?
What’s the variance?
What’s the covariance with factor X?

This makes it difficult to risk-adjust returns (i.e., compare returns relative to systematic market risk).

The industry commonly reports IRR. What’s the right benchmark? S&P 500, small-cap value, etc. Reported volatility is downward biased.

PE funds have about the same returns as public equity indices since at least 2006, and yet have collected $230B in performance fees. High fees persist due to “multiple layers of agency conflicts and the complexity of measuring risk and returns.”

Stafford: PE does not outperform. The IRR of a portfolio is 3% higher than PES once you leverage a portfolio of public stocks to match fund-level cash flows as typical LBOs.

Lecture 17: Investment Companies

Services provided:

Administration & record keeping
Diversification & divisibility
Professional management
Reduced transaction costs (economies of scale)

NAV=(Market value of assets minus liabilities)/(Shares Outstanding)

Gross return: Fund return before fees and expenses; i.e., the return on the fund’s investments. Note: Gross return is return after transaction costs.

Net return: Fund return after fees and expenses.

𝑅𝑛𝑒𝑡=𝑅𝑔𝑟𝑜𝑠𝑠−𝐸𝑥𝑝𝑒𝑛𝑠𝑒 𝑅𝑎𝑡𝑖𝑜 (𝐸𝑥𝑝𝑒𝑛𝑠𝑒 𝑅𝑎𝑡𝑖𝑜 includes management fees and 12b-1 fees)

Premium/Discount= (Price- NAV)/NAV

Your return: Return after paying commissions.

𝑅_𝑖𝑛𝑣𝑒𝑠𝑡𝑜𝑟=(1+Rnet )(1−FrontendLoad)(1−Backend Load)−1

𝑅_𝑖𝑛𝑣𝑒𝑠𝑡𝑜𝑟=($ 𝑂𝑢𝑡)/($ 𝐼𝑛)−1

How funds are sold:

Direct-marketed funds (e.g., Vanguard)
Distributed sales force (e.g., broker or financial advisor)
Financial Supermarkets

Mutual Fund Expenses:

1. Operating expenses (often expressed as an annual expense ratio)

Overhead and fees paid to fund manager (periodically deducted from assets)

0.2% of total AUM (even less for some index funds) to ~2% per year

2. 12 b-1 charge (added to operating expenses to get true annual expense ratio)

Used for marketing and distribution costs (e.g., broker commissions)

Capped at 1% per year (also periodically deducted)

3. Front-end load

Up-front commission / sales charge to purchase fund

Can be as high as 8.5% (typically lower)

Implication: Pay $100, get shares with NAV of $95 (example with 5% front-end load)

4. Back-end load

Redemption / exit fee to sell shares

Typically start at 5-6% and decrease over time funds are invested

Functions as an incentive to keep funds invested in a mutual fund

Other: transaction costs and taxes

Mutual Fund Taxation: Prefer to defer taxes as long as possible.

Mutual funds are pass-through entities.
Mutual funds are not taxed (i.e., no corporate income tax on the fund).
Instead, dividends and capital gains are passed through to investors.
Investors get distributions for dividends and realized capital gains.

Mutual Fund Performance: Mutual funds on average underperform the market by about 1%.

Past losers tend to keep losing.
Investors keep investing in them because of diversification benefits, expertise has more value, they believe markets are inefficient, and behavioral biases.

A similar arbitrage is possible if the ETF price is lower.

Securities Act of 1933: Must disclose:

Information about their management.
A description of the securities they own and offer.
Financial statements verified by independent accountants.

Exception: Only to accredited investors (those who make $200,000+ per year or have $1,000,000 in total net worth, not including their primary residence).

Hedge Funds:

Structured as a private partnership: Less SEC regulation than mutual funds. Only open to wealthy investors and institutions.

Lock-ups restrict when investors can withdraw funds: Allows funds to pursue less liquid investments. Makes hedge funds illiquid to investors.

Hedge Fund Fees: Afterfee return =Beforefee return – percent fee

Growth of hedge funds:

Hedge funds generate alpha: Hedge fund managers are the smartest!
Hedge funds hedge! They provide non-systematic returns. Hedge funds are less constrained and can invest in a variety of assets.

Academic evidence: Some hedge funds probably have positive alpha.

Majority view: Hedge funds outperform the market, at least before fees (but this is not an entirely settled question). Fund of funds probably don’t have positive alpha (because of additional fees).

Private Equity: Ownership of shares that trade in private markets. Most generally, this includes venture capital (VC) (VC is a small portion of all PE funds).

Lecture 18: Evaluating Investment Returns

When evaluating returns, we ask:

Risk Adjustment
Performance attribution (comparison)
Luck vs. Skill

Universe Comparison: Compare returns to a benchmark and universe of similar funds.

Benefits:

Can control for risk, style, cost of investment strategy, and specific market conditions.
Easy to implement.

Costs: No clear way to choose a peer group, and results are sensitive to the group/benchmark.

Jensen’s Alpha: Intercept from the regression of excess returns on excess market returns (CAPM) or excess market returns and other factors (multifactor model) – for performance attribution.

CAPM: 𝑟 ̅𝑃−𝑟 ̅𝑓= 𝛼𝑃+𝛽(𝑟 ̅𝑀−𝑟 ̅𝑓 )

3-factor: 𝑟𝑃−𝑟 𝑓= 𝛼𝑃+𝛽 (𝑟 ̅𝑀−𝑟 ̅𝑓 )+𝛽_(𝑃,𝑆𝑀𝐵) (𝑆𝑀𝐵) ̅+𝛽_(𝑃,𝐻𝑀𝐿) (𝐻𝑀𝐿) ̅

1. HML is return on portfolio of high book-to-market firms (value firms) minus return on portfolio of low book-to-market firms (growth firms) 2. SMB is return on portfolio of small firms minus return on portfolio of big firms

Alpha measures return in excess of an investment’s required risk premium, under the assumption that the model correctly measures the risk premium = If the CAPM is the correct model of investment risk, CAPM alpha is the correct measure of risk-adjusted returns

Risk-adjustment argument for multifactor models is weaker because it is less clear that they capture risk
– Fama and French interpret the 3-factor model as capturing risk premia, but it is really just an empirical model of returns

Style Analysis: three factor model is one approach

what you learn: What is the fund’s actual investment style?: Regression coefficients sometimes tell a different story than how the fund markets itself (e.g., “value” funds with 𝛽_(𝑃,𝐻𝑀𝐿)≈0)
How closely do the funds returns track its investment style?: Tracking error can be measured by regression R2 or residual standard deviation: Low tracking error means fund is basically the same as an index fund (closet indexing); high tracking error reflects active investing
What is the fund’s style-adjusted performance — Caveat: In practice, fund-specific alpha measures are too noisy to be of much value

Selection Bias: Need to think about why you decided to evaluate a particular fund

Survivor Bias: dont report data:Relevance for performance evaluation: We sometimes only have data on surviving funds
Particularly big problem if data is back-filled (data set adds historical returns when a fund is added)
We have pretty good survivor-bias-free mutual fund data, but this remains a concern for hedge fund research

Mutual Fund Performance: Jensen: Average alpha net of fees: -1.1% , Average alpha gross of fees: -0.4%

Pastor and Vorsatz: They find funds tended to underperform the S&P 500, 74% of active funds underperform the benchmark- Average -5.6% return during 10-week period (-29% annualized) compared to S&P 500

Some Private Equity:Private equity returns are not observable: – What’s the return? -What’s the variance?- What’s the covariance with factor X?- This makes it difficult to risk-adjust returns (i.e., compare returns relative to systematic market risk)

Industry commonly reports IRR- What’s the right benchmark? S&P500, small-cap value, etc. – Reported volatility is downward biased

PE funds have about the same returns as public equity indices since at least 2006, and yet have collected $230B in performance fees, High fees persist due to “multiple layers of agency conflicts and the complexity of measuring risk and returns”

Stafford: PE does not outperform: IRR of portfolio is 3% higher then PES once you leverage a portfolio of public stocks to match fund level cashflows as typical LBOS

Lecture 19:

Option:

contract conveys the right (but not the obligation) to purchase or sell an asset at a pre-specified time (or range of times) and price- In return for this right, option buyer pays option seller (writer) a premium

– options have LEVERAGE RETURNS

Examples: Call Option: Right to buy the underlying asset at a predetermined price = Positive exposure to stock price
Put Option: Right to sell the underlying asset at a predetermined price= Negative exposure to stock priceStrike (or exercise) Price= Price at which underlying may be bought or sold
Maturity (or expiration) Date = Date on which the option expires

Options: Leverage: For a small premium, investors can gain large exposure to asset price movements
Flexible payoffs: Options can be used to create arbitrary payoffs as a function of underlying asset prices (financial engineering)

Useful for: Speculation:-1.Take leveraged (long or short) positions in an asset 2. Bet on specific payoffs (e.g., speculate only on upside) 3. Options may also be more liquid than underlying assets
Risk management: 1.Cater payoff profile to specific risk preference 2. Hedge risk of existing (long or short) positions in the underlying asset

Call —- btw ST= price of underlying asset and x is strike price
Exercise when ST > X, CT = ST − X (“in-the-money”)
Do nothing when ST Hence, call payoff at maturity:
𝐶_𝑇=max⁡[0, 𝑆𝑇−𝑋]
Payoff structure implies for t 0 ≤𝐶𝑡≤ 𝑆𝑡
Put
Exercise when X > ST , PT =X − ST (“in-the-money”)
Do nothing when X Hence, put payoff at maturity: 𝑃_𝑇=max⁡[0, 𝑋−𝑆_𝑇]
Payoff structure implies for t 0≤𝑃_𝑡≤ [ 𝑋/ (1+rT−t)^(𝑇−𝑡)

Net Payoff = Payoff at maturity – cost of establishing position (option price/premium)

Bet Underlying will increase in price: 1)buy a call 2) sell a put

Bet Underlying will decrease in price: 1)buy a put 2) sell a call

Riskiness: Options in isolation are much riskier than stocks. That is, with the same amount of money invested in a portfolio of options instead of a stock, one can achieve a much higher risk exposure & Higher riskiness is achieved through leverage

Portfolio insurance / protective puts= A popular hedging strategy is – Buy one put for each share of the underlying

– eliminates the downside and preserves the upside

ex: Buy 100,000 1-year put options with strike price $90.
100,000 ST + 100,000 max [0, 90 − ST]
= 100,000 max [ST , 90]
= max [100,000 ST , 9M]

Long straddle: Buy call and put with same exercise price and maturity (buying volatility) – The straddle is a bet on volatility
To make a profit, the change in stock price must exceed the cost of both options- You need a strong change in stock price in either direction- The writer of a straddle is betting the stock price will not change much

– straddle graph looks like a V

Put- Call Parity: dYnN2SkAAAAASUVORK5CYII=

– if it is a put GL+l3fZECaAAAAAASUVORK5CYII=

a call and a put with the same strike price and maturity, EUROPEAN OPTIONS, if this doesn’t hold we can construct an arbitrage

for dividend paying stocks: c = p +s + I – ( x/ 1+rt)^T where I is where I is the PV (at the riskless rate) of the dividends to be received until maturity

Lecture 20: Option value can be decomposed into two parts:
1. Value of option if exercised today (intrinsic value)
Max[0, S-X] for call where S is the stock and X the exercise price
Max[X-S, 0] for put

2. Extra value of waiting to execute option (time value)
Delay strike price payment= Positive value for call options, Negative value for put options
Miss dividend payments= Negative value for call options = Positive value for put options
Asymmetry / insurance against adverse price movements because of limited liability of option (volatility value)= always positive

European option = May only be exercised on the expiration date

American option = May be exercised at any time prior to expiration

𝑃𝑟𝑖𝑐𝑒(“American Option” )≥𝑃𝑟𝑖𝑐𝑒(“European Option”) 1. Only exercise early if it is beneficial 2.Thus, at worst value is the same as the European Option 3. Difference is referred to as early exercise premium

Early Exercise of an American Call: NEVER optimal to exercise early an American call on a non-dividend paying stock

so PRICE OF AMERICAN CALL NONDIV = EUROPEAN CALL

but if American option is lower then arbitrage opportunity – buy low and sell high

no-arbitrage strategies 1. Replicate option payoff with another portfolio; option price must equal cost of replicating portfolio (Law of One Price) 2. Use option and stock to create risk-free payoff, then discount at the risk-free rate

Binomial Pricing: AR1MOWznQjEkAAAAAElFTkSuQmCC

of a call with x d -> 𝐶0=𝑎𝑆0+𝐹/(1+𝑟𝑓)

Basic idea: can replicate a call or put option by trading the right combination of risk-free asset and underlying stock.
For call option: 0≤𝑎≤1, 𝐹≤0. Call option is a position in the underlying partially financed by borrowing.
Call option is leveraged position in the underlying stock.
For put option: −1≤𝑎≤0, 𝐹≥0. Put option is bond position partially financed by shorting the underlying stock

𝐻𝑒𝑑𝑔𝑒 𝑟𝑎𝑡𝑖𝑜= (𝐶𝑢−𝐶𝑑)/(𝑢𝑆0−𝑑𝑆0 ) and in NO arbitrage pricing : JJaFEEK0FGtRJgsCFfSEECLPSCwLIYQQQggRgwL8hBBCCCGEiEFiWQghhBBCiEiC4P8H+0LBKI4nhdkAAAAASUVORK5CYII=

Black- Scholes Option Pricing: rOYAAAAASUVORK5CYII=

1bgTI9e9WuXAAAAAElFTkSuQmCC

C0 = Current call option value S0 = Current stock price N(d) = probability that a random draw from a standard normal distribution will be less than d

if DIVIDENDS are involved = replace So with So-PV(dividends)

Lecture 21: Forward contract- An obligation to buy/sell the underlying asset at a specified price (forward price) and specified time (maturity or expiration date) – settlement at maturity

Futures contract: 1. It is standardized and traded on an exchange 2.Counterparty is clearinghouse 3. Buyers and sellers must post margin (initial and maintenance) 4. Gains and losses are settled daily 5. Contracts are “marked to market” 6. Futures contracts are more liquid than forwards because they are standardized – settlement daily

Open Interest: Number of contracts outstanding : Since there is a short position for every long, the market has net zero exposure to any commodity

Only 1-3% of contracts result in actual delivery of underlying commodity= most positions “reversed” by entering into an offsetting position

Long Position(buy): St-Futures price & Short Position(sell): its F- St

Speculation: better options
Buy S&P futures, speculating that the S&P500 will go up

Hedging: better futures
An investor has a long position in the S&P500. To reduce the risk of the position, the investor can sell S&P futures.

No arbitrage pricing: F0 = S0(1+rT)^T

BINOMIAL FOR FUTURES is the same but its -F

Subtract present value of dividends from stock’s price
𝐹0=[𝑆0−𝑃𝑉(𝐷)] (1+𝑟_𝑇 )^𝑇 —–> from ex: use the spot rate for the year the maturity of contract

Equivalently, adjust the interest rate by the dividend yield (spot-futures parity theorem)
𝐹0=𝑆0 (1+𝑟𝑇−𝑑)^𝑇
𝑑 is the annual dividend yield (rate of dividend payments)

𝑟_𝑇−𝑑 is the cost of carry
To hold the stock you pay an opportunity cost of 𝑟𝑇 for tying up your capital and get a dividend yield of 𝑑

𝐹0=𝑆0 (1+𝑟_𝑇+𝑐)^𝑇

Let 𝑐 be the net carrying cost (carrying cost minus convenience yield)

BINOMIAL MODEL EX: Replicating portfolio using commodity and risk-free bond has price aS0 + f/(1 + r)

Lecture 22:

Hedged Portfolio: Long Position $ / S0 = shares we should sell

value of portfolio = (shares we should sell) x St+𝐷−𝑦(St−𝐹) and value is independent so SWSS-y=0

Index Value = S’ and Futures Price= F’

Value of our = Stock portfolio is 15,476 𝑆’+𝐷

Futures position is 15,476 x (sp500futuresprice− F’).

Hedged portfolio is 15,476 x (𝑆′+3,231.00−𝐹’)+𝐷

Why Should the Firm Not Hedge? —> 1Investors are more capable of diversifying risk than is management 2.Risk management does not add value to the firm 3.Management conducts hedging strategies for their own benefit at the expense of the shareholders 4. Managers cannot out guess market – value of hedging is zero 5. Accounting reasons motivate reduction in cash flow variability 6.The risk has been factored into the market value

Why should hedge? 1. Reduction of risks of future cash flows improves the planning capabilities of a firm 2. Hedging reduces the probability of financial distress 3. Managers have a comparative advantage over stockholders in knowing specific risks of firm 4. The markets are usually in disequilibrium because of structural or institutional imperfections

Portfolio value= $10 million & Modified duration=9 years
If rates rise by 1 basis point (.01%), then Change in value = ( 9 ) ( .01%) = .9% or $9,000
Price value of a basis point (PVBP) = $9,000 per basis point

To hedge, we need an offsetting position that will increase in value by $9,000 per basis point change in interest rates

Idea: take offsetting position in interest rate futures Specifically, sell futures contract on 20-year maturity, 6% coupon bond Bond has futures price of $90 (per $100 par value) and modified duration of 10 years
To offset original portfolio’s interest rate risk, need to sell futures on $10 million of bonds (par value)

UK Pension Fund Crisis (Oct 2022): UK Defined Benefit pension funds took risky bets in Liability-Driven-Investment (LDI) funds used interest rate derivatives with high leverage When interest rates rose in Sep 2022, funds forced to liquidate gilts to meet margin calls

Silicon Valley Bank lost ~1.8B when it sold long-term treasury bonds to meet deposit withdrawals from tech startups and crypto companies——SVB mismanaged risks (did not have CRO for 7 months!)