Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Topics Covered Since Midterm

Final Exam Review

FINTECH 522

Markets and Market Microstructure (from Trading Day)

· The Order Book

· Bid-ask Spread

· Limit Order – buy < $50 sell > some stock price (e.g $50)

· Market Order – current market price

· What happens if a lot of short sellers all want to buy to cover their short positions at the same time (“Short Squeeze”)? Rush to cover short interest -> ^ stock price

· What happens if, by mistake, the algorithms that buy and sell large blocks of stocks (dark pools) happen to all place a large amount of market orders to sell a certain stock at the same time (flash crash)?

· Execution Risk – not being able to fully complete an intended portfolio

· What is a Market Maker?

· What are Dark Pools?

Options Symbols and Terms:

· OPTIONS ARE SECURITIES (T/FALSE)

· Put: Contract that gives the Holder the right but not the obligation to SELL the underlying asset TO the Writer of the contract at a pre-agreed price X.

o Holder’s perspective: The right, but not the obligation to force Writer to BUY

§ Is a valuable ability, so Holder must PAY money to acquire it.

o Writer’s perspective: The promise (obligation) to BUY stock if asked by Holder.

§ Is a burden and a risk, so writer RECIEVES money in exchange.

· Call: Contract that gives the Holder the right but not the obligation to BUY the underlying asset FROM the Writer of the contract at a pre-agreed price.

o Holder’s perspective: The right, but not the obligation to force Writer to SELL

§ Is a valuable ability, so Holder must PAY money to acquire it.

o Writer’s perspective: The obligation to SELL stock if asked by Holder.

§ Is a burden and a risk, write RECIEVES money in exchange.

· Selling a put or call

o Holder may sell position on an options exchange.

o If so, Holder is now Seller

o The Buyer of Holder’s position becomes the new Holder

o Writer _______(may/may not) sell position

o Writer can be thought of as the “ORIGINAL seller”.

· P, C, price: The amount of money exchanged to buy or sell a position in an option

· X, exercise (or strike) price: Pre-agreed price at which the underlying will be bought/sold.

· T, expiry date: Date on which the option expires

· T-t, tenor: The amount of time left until expiry for a contract at any time t.

· Contract Size: What it means, and how to handle it (HW3 part 1 Question 2)

· Naked, or uncovered: JUST buying or writing the option

· Covered, or Clothed: you DO own the underlying

· American style: Option can be exercised at any time until expiry

· European style: Option can only be exercised on the date of expiry

Short Selling:

· Is a service offered by (choose: the SEC, the Stock Exchange, your broker)

· Your execution system keeps up-to-date with what stocks are shortable

· If you short sell a stock, your Broker will sell stock owned by other clients and deposit the money into your account

· You are obligated to buy this stock back to cover, subject to the rules on your margin account

· If you are short a stock when it pays a dividend, THE OWNER OF THE POSITION (YOU) pays the dividend to the stock’s owner

Payoff Diagrams:

Tip: Changing the chart type to LINE (not scatter) on your Payoff Diagram Calculator will cause the x-axis on the chart to auto-update without using a macro (Credit to John Lozano)

· BUYING a CALL will rotate the payoff line counterclockwise on the right-hand side the X price

· BUY a PUT will rotate the payoff line clockwise on the left-hand side the X price

· WRITING a CALL will rotate the payoff line clockwise on the right-hand side the X price

· WRITING a PUT will rotate the payoff line counterclockwise on the left-hand side of the X price

· Taking a LONG position anything makes the Cost of Position more positive (money OUT)

· Writing a SHORT position anything makes the Cost of Position more negative (money IN)

These next three are less useful but practically, but can certainly help if you’re looking to identify an equivalent short (or long) position for a particular spread, or what would be required to invert a spread (by flipping it around the x-axis), etc.

· You can shift a payoff diagram up or down by short & buy the stock at different prices.

o …but this can be dangerous in practice because it exposes you to directional risk and you might have to buy to cover.

· You can flip a payoff diagram around the x-axis by multiplying all positions by -1.

· In the special case of equal costs for puts & calls you can flip a payoff diagram’s Cost of Position by switching contracts: change puts to calls and calls to puts at the same X-price, and change long stock positions to Short and vice versa.

Example:

· Review and understand:

o Bull Spread [Benninga, p. 611],

o Bear Spread [Benninga, p. 612],

o Butterfly [Benninga, p. 612-13], or Condor [wings]

o Straddle [called an “opposite butterfly” in Benninga, p. 613] Strangle

o … and their short versions

Translating a private belief into a financial position

Example: Trader is compiling data on recent house sales to use in a forecasting model. A name in one of the records catches her eye – it’s the CEO of a major, publicly traded company! The property is a large, comfortable house on Martha’s vineyard, complete with swimming pool and guest house. It’s located between Martha’s Vineyard Hospital and Farm Neck Golf Club.

Trader recalls a few comments from recent earnings calls about the CEO’s manageable, but serious medical condition. The trader also knows (as does everyone else) that the CEO loves golf. After pulling up the CEO’s compensation information (publicly available on SEC’s website) the trader realizes in a flash: the CEO plans to retire early!

Trader doubts that many (if any) other people have come to this same realization.

The trader also knows that there is no way of predicting how the market will react to the news – the new CEO could be received much more enthusiastically than the current one… or the opposite.

Possible Questions:

· Although the direction of the change in the CEO’s company’s stock price is impossible to predict at this time, it’s a pretty sure bet that the VOLATILITY of the stock price will INCREASE significantly in reaction to the retirement / new CEO news.

· What are some possible options structures that could take advantage of that fact?

· What happens if T for the options Trader uses occurs before / after the surprise announcement?

o Before: stock price might not have been volatile enough because the new CEO hasn’t been announced by expiry

o After: more likely to catch the higher volatility

· Is it safer to bet with longer T, or shorter T? beneficial to err with longer T

· What happens if T is too long? Everyone will have gotten used to the new news and the volatility of the underlying will mean regress to its old value

Translating a financial position into a private belief

What does the trader who bought this believe about the future behavior of the price of the underlying?

What about this one?

No Arbitrage

(analogous to Efficient Market Hypothesis)

Put-Call Parity for European Options

Short form:

See PS3, Part 1, Question 3-3

Long form:

1) Buy a stock at price S₀ = $45.00/sh

2) Write a call for price C = $1.91/sh having strike price X = $50.00/sh and time to expiry T = 1 year

3) Buy a put for price P = $1.20/sh having strike price X = $50.00/sh and time to expiry T = 1 year

4) It’s going to cost S + P - C = $44.29 to enter into the position, so you finance this by getting a loan in that amount at the risk-free rate r_f.

5) At time T, close the position and use the money to pay back the loan.

Create the payoff diagram.

1 year from now, on the expiry date, the price of the stock will be either higher or lower than $50.

If Higher:

· The holder of the call option that you wrote will EXECUTE forcing you to SELL your stock for $50, even though its market price is (higher/lower).

· You do so, but you still make $50-$45 = $5 from the stock sale and ______ from writing the call & buying the put, so your total profit is $5.71

· Your put option expires worthless.

If Lower:

· The call option that you wrote expires worthless

· You exercise the put option, and force someone BUY from you at $50 even though its market price is price is (higher/lower).

· You still make $5 from the stock sale and $0.71, = $5.71 total from writing call and buying put

What is the equivalent continuously compounded rate of return that would have gotten you this same profit, if you could have just put the $44.29 in a bank account earning that rate?

è ln($50/$44.29) = 12% per year

That’s a really high interest rate – much higher than the risk-free rate that can be gotten for an asset like a savings account or a government bond.

The No-Arbitrage principle says that as long as these opportunities exist, the forces of supply and demand will drive the prices of S, P, and C into equilibrium with the future value as follows:

S + P – C = X /e^rt

Another way of understanding this:

“Everyone knows” that “stocks grow at the risk free rate” (recall, from Black Scholes, Binomial Discussions). Therefore, the market consensus of the future value S₁ of a stock that is worth S₀ now should be S₀e^rt = S₁.

S₁ is a FUTURE CASH FLOW that we get from executing an option at strike price X.

So, the relationship relating present cash value to future is: S₀ = X/e^rt.

If puts and calls are added to the starting principal to get S₀ + P – C = X/e^rt, then the No-Arbitrage principle holds that traders will quickly purchase stocks, puts, and calls to create their own risk-free positions in excess of the prevailing risk-free rate if they possibly can. As they do this, market forces (supply and demand) move the three prices back into alignment as the risk-free opportunities are taken up by traders.

This is called being “arbitraged away”.

No-Arbitrage does not say that risk-free opportunities never exist – only that they never exist FOR VERY LONG and that they tend towards the No-Arbitrage Relationship in the long-term.

Very analogous to how long-term alpha = 0, etc.

From Reading:

Re-read these pages - no calculation, understand at a conceptual level.

è No Early Exercise of American Options (on non-dividend-paying stocks) [Benninga, p. 628-629]

è Option Prices are always Convex (convex/smile – assuming same underlying asset, same expiration date, different exercise prices); No Arbitrage Demonstration of Convexity of Call Prices [Benninga, p. 633-636]

Black-Scholes

· Basic pricing formula

· Underlying assumptions: constant vol, risk-free rate drift, etc

· Understand the Black-Scholes inputs: S, X, T, r, and σ

· Using the Excel NORMSDIST(Z) function

· Using Solver to “back out” the implied fifth input if given the other four

· We have dealt with European Options on Non-Dividend paying stocks (Black-Scholes can be adjusted for American Options and stocks that pay dividends)

· Volatilities:

o Historical – the “Statistical” volatility that is observed on data (calculated with STDEV.P)

o Forecast – The expected volatility for a future time period.

§ Can assume historical volatility = forecast volatility if we don’t expect the volatility to change.

o Implied – The volatility that a trader assumes by making a bet

The Greeks

Delta

· The change of an option’s price relative to the underlying

· Opposite signs for calls and puts because:

o If the price of the stock goes up, then the right to buy that stock at a price x becomes more valuable, so dC/dS is positive.

o If the price of the stock goes up, then the right to sell that stock at a price x becomes less valuable, so dP/dS is negative.

Gamma

· The rate of change of delta

· i.e., the derivative of delta

· i.e., the second order derivative of P or C with respect to the price of the underlying

· The closer the price of the underlying gets to X-price and the closer that option is to expiry in time, the more sensitive Delta becomes to the price of the underlying.

Vega

· Change of an option’s price with respect to a change in volatility

· If the price of the underlying is near the strike price, and there is a ______(long/short) time to expiry, than a little change in volatility will greatly affect the price of an option because there is more room for the stock price to move between now and expiry.

Theta

· Change in the price of a call or put with respect to time

· Time decay – loss in value of the “time value” of an option, which is the chance that it could be worth something at expiration.

Rho

· Sensitivity of price of an option to change in risk-free rate

· If the risk-free rate goes up, Black Scholes predicts that the future price of the stock will go up correspondingly (S₀e^rt = S₁), therefore the right to lock in a future buy price (call) becomes more valuable the farther away (in time) the date of expiry is.

· Reverse for puts.

Binomial Option Pricing

· Calculating Hedge Ratios

· Basic stochastic assumption (aka Weiner Process, aka Brownian Motion, aka Random Walk) that a stock’s price – about which we have no other information – rises at the risk-free rate.

· Using hedge ratio to calculate the final price f of an option, and discount it to present

· Binomial models exist because they can help you visualize and calculate probable price paths

· Calculating the probability of an up- or down- movement using the binomial equation: