Derivatives Markets: Pearson International Edition
This document contains everything that is discussed in the lectures of Financial Risk Management, including extra explanation. It also includes a detailed explanation of all the practice exercises discussed during the lectures.
Summary Financial Risk Management: All relevant formulas and step by step calculations
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Financial Risk Management (MANMEC024)
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Financial Risk Management Lectures
End-of-chapter questions (!) and pages to read from book: Robert L. McDonald (2013),
Derivative Markets, Pearson New International Edition (Third Edition). ISBN: 9781292021256
- 2 assignments (20% of grade).
Course objectives:
1. understand valuation principles, equity premium puzzle, and excess volatility; perform
empirical tests of the above concepts.
2. derive, discuss, and test empirically Capital Asset Pricing Model (CAPM); address the
limitation in the application of CAPM in practice; have an understanding of alternative
models of CAPM, such as consumption-based CAPM.
3. understand and assess factor models; statistically analyse real data using factor models.
4. understand the source of inter-temporal discounting; assess different models of inter-
temporal choices.
5. critically assess and discuss the most recent developments in financial economics, and
communicate those messages to others via a presentation.
Introduction
30-1, L0: Introduction
Course overview
- In this course, you learn about complex financial products, how to model asset/derivative
prices, and eventually how to price them.
- By the end of the course, you are able to apply concepts from probability theory and
(stochastic) calculus to:
* understand derivatives and how to use them
* value and hedge (financial) derivatives
- A final goal is the famous Black-Scholes formula.
- The course is divided in 4 large “Blocks”.
Block 1: Lognormal Distribution, Monte Carlo Simulation, Value at Risk, Brownian Motion,
and Ito’s Lemma
We start with mathematical tools. Things we consider:
- If Philips has lognormally distributed stock prices, what is the probability that St > K = 60?
* when is a certain stock price (S) at any time (t), larger than a certain value (K)?
* you can build a trading strategy on this
- How can we simulate stock prices?
- Using the new Ito calculus, we learn the Black-Scholes model for stock prices: dSt = αStdt +
σStdWt.
* W: Brownian motion, indicates the volatility of the stock
* α: average return
Block 2: Forwards and Options, Insurance and Collars (and Other
Strategies), Risk Management
- You become familiar with basic financial products: options (puts
,and calls), forwards, and futures.
* positions to take: short or long
* when do these products pay off?
- Options are actually insurance.
- How can we hedge the price of gold, being an input for a producer?
Picture (long put option): strike of 20, the put option makes a profit when the underlying
asset is below 20.
Block 3: Financial and Interest Rate Forwards and Futures (i.e., bonds), Swaps
- We study more complex financial products, e.g: currency forwards, interest rate swaps.
- You learn about bonds, interest rates, and forward rates: P(0,t) = 1 / [1 + r(0,t)]t
* how bond prices are inversely related to interest rates
- Example: can we find the value of a contract that fixes the price of gold 1
year from now for a period of 9 months? Can I do a similar thing with interest
rates?
Block 4: Put-Call Parity and Other Option Relationships, Binomial Option
Pricing, Black-Scholes Model, the “Greeks”
- We derive the price of an option using binomial pricing.
* start at the end: suppose we know what the stock price will be in 2 years, can we calculate
back the price of the option today
- You learn the Black-Scholes formula to price options: C(S,K, σ, r, T, δ) = Se−δT N(d1) − Ke−rT
N(d2).
* C (value call option) depends on S (current stock price), K (strike price), N (normal
distributions), r (interest rate), δ (delta, dividend yield), σ (sigma, volatility), T (time horizon)
* with d1 =
ln
S
K ( )( 1 2
+ r−δ + σ T
2 )
and d2 = d1 - σ √ T
σ √T
- You learn how to hedge via “The Greeks”, such as delta-hedging.
Block 1: Lognormal Distribution, Monte Carlo Simulation, Value at Risk,
Brownian Motion, and Ito’s Lemma.
30-1, L1: Introduction to Derivatives and Continuous Compounding
Book: Ch. 1 (introduction to derivatives) & Appendix B (continuous compounding)
Questions) Appendix B: continuous compounding. 1, 2, 3.
Ch. 1: Introduction to derivatives
What is a derivative?
- A financial instrument that has a value determined by the price of something else.
- Example: an agreement where you pay $1 if it rains tomorrow and receive $1 if it doesn’t.
- This contract can be used to speculate on rainfall or it can be used to reduce risk.
* e.g. useful for a farmer: when it doesn’t rain, crops don’t grow, but you still receive money
, It is not the contract itself, but how it is used, and who uses it, that determines whether
or not it is risk-reducing.
Why use derivatives?
1. Risk management: derivatives are a tool for companies and other users to reduce risks,
hedging.
2. Speculation: derivatives can serve as investment vehicles. Not to use as insurance but in
the hope to make profits (gamble).
3. Reduce transaction costs: sometimes derivatives provide a lower cost way to undertake a
particular financial transaction.
4.Regulatory arbitrage: it is sometimes possible to circumvent regulatory restrictions, taxes,
and accounting rules by trading derivatives.
Perspectives on derivatives
- End users (mostly used throughout course): corporations, investment managers, investors.
* they use derivatives for one of the four above reasons
* they have a goal and use derivatives to help them reach it
- Intermediaries: market-makers, traders.
* they exploit the Bid-Ask spread to make money: buy low, sell high
* bid price: price you can sell at
* ask price: price you can buy at
- Economic observers: regulators, researchers.
Short selling: when price of an asset is expected to fall (e.g., Bitcoin Bubble).
- Go to bank/broker to give you shares and immediately sell them to the market at higher
price.
- Now you have the money and you wait till the price drops.
- Use that money to buy the shares back at lower price.
- Pay back bank/broker money left over (profit).
Overview of financial market
The trading of a financial asset involves at least four discrete steps:
1. A buyer and a seller must locate one another and agree on a price.
2. The trade must be cleared: the obligations of each party are specified.
3. The trade must be settled: the buyer must deliver the cash and the seller the securities
necessary to satisfy their obligations in the required period of time.
4. Ownership records are updated after the trade.
Relevant market measures: volume, market value, notional value, open interest (amount of
settlements still needed to be done).
Over-the-counter (OTC)
- Over-the-counter (OTC) market: trading financial securities directly with a dealer,
bypassing organized exchanges.
- Exchange activity is public and highly regulated.
- OTC trading is not easy to observe or measure and therefore generally less regulated.
- For many categories of financial claims, the value of OTC trading is greater than the value
, traded on exchanges.
* less regulation, easier to trade large quantities, different financial claims at once, custom
claims
Financial engineering
- The construction of a financial product from other products: from already complex
derivatives to even more complex ones.
- New securities can be designed by using existing securities.
Financial engineering principles:
- Facilitate hedging of existing positions.
- Allow for creation of customized products.
- Enable understanding of complex positions.
- Render regulation less effective.
Appendix B: Continuous compounding.
Effective annual rate (EAR) (discrete compounding)
- Invest x0 today, then you will have xn = x0(1 + rear)n , n years later.
- This implies: rear = (xn/x0)1/n – 1.
* when you invest x0 and earn xn, n years later, you get the implied EAR
- Q: Invest 100 today at rear = 0.08. What do you have in 3 years?
100 · 1.083 = 125.97
- Q: Invest 100 today and earn 130 in 4 years. What is the effective annual rate?
(130/100)1/4 – 1 = 0.0678 = 6.78%
Continuous compounding
- Invest x0 today, then you will have xn = x0ercoco · n, n years later.
- This implies (annual): rcoco = ln (xn/x0) / n.
- Q: Invest 100 today at rcoco = 0.08. What do you have in 3 years?
100e0.08 · 3 = 127.12
- Q: Invest 100 today and earn 130 in 4 years. What is the “coco” rate?
ln(130/100) / 4 = 0.0656 = 6.56%
Be able to derive the EAR or COCO from the xn formula. Also be able to rewrite to derive xn,
x0, or n.
Interest rates: per period, over 1 year
You can also compound more times per year instead of only
annually (as in the above examples).
Interest rates: per period, over multiple years
Can you also compound over multiple years and multiple times within a year?
- What’s the value in n years?
- Suppose the interest rate is r per annum. What is the future value xn of your initial capital
x0?
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