Lecture 2 -Slides

1. RISK MANAGEMENT
LECTURE 2
a. Fundamentals of financial instruments
b. Futures and forwards
c. Options
1

2. Part 1
FUNDAMENTALS OF
FINANCIAL
INSTRUMENTS
a. Present and future value
b. Maturity
c. Duration
d. Convexity
2

3. 1. Present, future value
PV Present Value
FV Future Value FV
PV  T,i P
i interest rate (discount rate) (1  i )T
T number of periods
FV  PV (1  i )T T=f(Years,m(per
per year)
Part 2. Fundamentals of statistics
𝐵𝑒𝑐𝑎𝑢𝑠𝑒 𝑇 = 𝑚 ∗ 𝑌
Compound
i mY period
FV  PV (1  )
m
𝐷𝑒𝑓𝑖𝑛𝑒 𝑥 = 𝑚 𝑖
i xiY
FV  PV (1  )
x
1 x iY
Aggregation FV  PV [lim x (1  ) ]
x
(Present) Value of any investment
Lecture 2
is the sum total of all future FV  PV [e]iY
financial benefits T
CashFlow
P
1 (1  i )T 3

4. 2. Price sensitivity
There are four measures of bond price sensitivity
Maturity
Macaulay Duration (effective maturity)
Modified Duration
Convexity.
Maturity
The time left to maturity on a bond
Part 2. Fundamentals of statistics
The longer the time to maturity, the more sensitive a particular bond is
to changes in the rate of return.
FV
PV 
i
(1  ) mY
m
• Bond A matures in 10 years
and has a required rate of
return of 10%.
Lecture 2
10 year steaper 5 year • Bond B has a maturity of 5
Steaper years and also has a
required rate of return of
10% 4

5. 2. Price sensitivity
Duration (Macaulay)
But duration is a better measure of term than maturity
Relationship price - maturity is affected when considered non-zero coupon
bonds.: many of the cash flows occur before the actual maturity of the
bond and the relative timing of these cash flows will affect the pricing of
Part 2. Fundamentals of statistics
the bond.
T
Dm   t  wt  Period * Payments
t 1
PV(CFt ) CFt /(1  y )t
wt  
PV( Bond ) P
q
Lecture 2
w
t 1
t 1
5

6. 2. Price sensitivity
Duration (Macaulay) We find the
weighted
values
This bond is 6 Y
Part 2. Fundamentals of statistics
to maturity
The semi-annual
duration for this
bond is 10.014
semianual.
Annual: 5
Lecture 2
We find the value of the
bond by discounting each Duration is a
of the flows function of
cash flows
Summation of cash flows 6

7. 2. Price sensitivity
Modified Duration
More direct measure of the relationship between changes in interest rates and
changes in bond prices
Modified Duration, D, is defined as dP: change of price
1 𝜕𝑃 dY: change of rate of ret.
𝐷=− ∗
𝑃 𝜕𝑌 First derivative of the
Part 2. Fundamentals of statistics
bond price WR
discount factor
𝑇
𝜕𝑃 1 𝐶𝑡
=− ∗ 𝑡
𝜕𝑌 1+ 𝑦 (1 + 𝑦) 𝑡
1
𝑇 𝐶𝑡
1 (1 + 𝑦) 𝑡
𝐷= ∗ 𝑡
1+ 𝑦 𝑃
1
𝑇 𝐶𝑡 𝑇 𝐶𝑡
(1 + 𝑦) 𝑡 1 (1 + 𝑦) 𝑡
𝐷= 𝑡 𝑀𝑜𝑑. 𝐷 = ∗ 𝑡
Lecture 2
𝑃 1+ 𝑦 𝑃
1 1
7

8. 2. Price sensitivity
Modified Duration
So, at the end we have
• Macaulay Duration is an average or effective maturity.
Part 2. Fundamentals of statistics
• Modified Duration really measures how small changes in the yield to
maturity affect the price of the bond.
From the definition of Modified Duration we can write
% Change in bond price = - Mod. Duration times the change in yield to maturity
𝛻𝑃
= −𝐷 ∗ 𝛻𝑦
𝑃
How much in % will the price
change when the yield changes?
Lecture 2
8

9. 2. Price sensitivity
Convexity
Second derivative of price with respect to yield to maturity
• Measures how much a bond’s price-yield curve deviates from a
straight line
Notice the convex shape of price-yield relationship
Part 2. Fundamentals of statistics
Bond 1
Price
Bond 2
Yield
Lecture 2
Bond 1 is more convex than Bond 2
Price falls at a slower rate as yield increases
9

10. 2. Price sensitivity
Convexity
Second derivative of the
1 P2
bond price WR discount
Convexity 
P 2 y factor
Part 2. Fundamentals of statistics
2P 1 N
 Ct 

 2 y (1  y ) 2
  (1  y)t
t 1 
t (t  1)

 Ct 
N  
1 (1  y ) t This seems the w
2 
Convex  t (t  1) 
(1  y ) t 1  P 

 

Lecture 2
10

11. Lecture 2
Part 2. Fundamentals of statistics
Convexity
2. Price sensitivity
Period
11

12. 2. Price sensitivity
Convexity
Recall approximation using only duration:
P
  Dm  y
*
P
The predicted percentage price change accounting for convexity is:
Part 2. Fundamentals of statistics
P
 1
 
  Dm  y    Convexity  (y ) 2 
*
P 2 
Adding the convexity adjustment corrects for the fact that Modified Duration
understates the true bond price.
This is a really good approximation btw. change of yield and its effect on price
Lecture 2
12

13. Part 2
FUTURES AND
FORWARDS
a. Basics
b. The futures contract
c. Determinants of prices
d. Future prices V expected spot prices
13

14. 1. Basics
Definition
Financial contract obligating the buyer to purchase an asset (or the seller to sell
an asset) at a predetermined future date and price.
• Obligation !!
• Commitment today to make a transaction in the future
Practical example
Farmer Mill
• Sell a product • Buy a product. Only that product
• No diversification. (single product) • It is worried about the future price
Forward Contract
Agreed price
no real transaction (money)
• Deliver a product • Deliver the money
Part 2. Futures
• Get the money for sure • Get the product for sure
It is a zero sum game
Lecture 2
Each long position has a short position
Futures do not affect the market price
Can be seen as a Risk Management Technique
14

15. 1. Basics
The formalization of the forward contract is the futures market
Futures Forward
• Standardization
• Contracts more liquid No money changes until delivery
• Margin to market. Daily settling
up of gains and losses
• Margin account
Long position on a future Short position on a future
Buy a contract: Commitment to Sell a contract: Commitment to
purchase a product in the future deliver a product in the future
Price of the future
Price of the future
Profit
Profit
Value of the forward
Part 2. Futures
Lecture 2
Profits
can be
<0 Profit=Spot-F0
Loss=F0- Spot Profit=F0-Spot
Price Price 15

16. 1. Basics
Existing contracts Mechanisms of Futures
• Agricultural Commodities • Organized exchanges
• Metals and Minerals • Cash delivery instead product
• Foreign currencies • Standardization: specific contracts
• Financial: index and single and maturities
stocks • Clearing House: trading partner
for each trade (credibility)
• Marking to market: put positions
at market price
• Margins
Money Money
Part 2. Futures
Long Short
Clearing House
position position
Lecture 2
Commodity Commodity
16

17. 2.The futures contract
Mechanisms of Futures
Marking to market process
• At the beginning of the trade, each trader establishes a margin account
• Can be cash or near cash assets
• Both parties must give the margin
• 5% - 15% total value of contract
• Instead of waiting until the maturity date, the clearing house requires
traders to realize gains and losses in a daily basis
• The daily settling is called Marking to Market (MtM)
• When margin account falls below a maintenance margin, the trader
receives a margin call to give more money or close the operation
Convergence property (avoid arbitrage)
Futures price on delivery date and Silver is traded
Today MtM
Part 2. Futures
spot price must converge at maturity 14,10 per 5000 ounces
There are two sources of a commodity 1 14,20 0,10 500
2 14,25 0,05 250
: futures and spot, and both must be
Lecture 2
3 14,18 -0,07 -350
the same 4 14,18 0,00 0
Difference of values
times 5000 ounces 5 14,21 0,03 150
0,11
550 550 17

18. 3. Determination of future prices
Spot-Futures Parity Theorem
Futures can be used to hedge changes in the value
A perfect hedged portfolio should provide the risk free rate to avoid arbitrage
• SPX500 at 1500
• An investor has a position in an SPX500 indexed portfolio Long
• Future price of SPX500 is 1550
• The investor wants to hedge the market risk
HOW?
Short sell a future contract of SPX500 @ 1550
Final value of P 1510 1530 1550 1570 1590
Profit=F0-Spot Payoff of short 40 20 0 -20 -40 Convergence!!
Dividend 25 25 25 25 25
TOTAL 1575 1575 1575 1575 1575
Part 2. Futures
• Any increase in the value of the indexed portfolio is offset by an
equal decrease in the payoff of the position. 𝐹0 + 𝐷 − 𝑆0
Lecture 2
𝑟𝑓 =
• The final value is independent of the market price 𝑆0
• The rf (risk free rate) is 5% (1575-1500)/1500 𝐹0 = 𝑆0 (1 + 𝑟 𝑓 ) − 𝐷
𝑇
𝐹0 = 𝑆0 (1 + 𝑟 𝑓 − 𝑑)18
Any deviation from parity would give rise to arbitrage

19. 3. Determination of future prices
Spot-Futures Parity Theorem
Example
• rt becomes 4%
• F0=1535
• But the actual future price is 1550
What could be the strategy?
Short overpriced futures Will get
Buy the under-priced stock using money at 4% the future
price and
dividend
Initial Cash Cash Flow in
Flow 1 year
Borrow 1500 and repay with interest in 1 year 1500 -1560
Buy stock -1500 St+25
Part 2. Futures
Enter short future position 0 1550-St
0 15
Lecture 2
• Net initial investment is 0
• Cash flow in 1 year is 15 no matter the price of the stock
(riskless)
• When misprice, the market will equilibrate prices 19

20. 4. Future prices VS Expected spot prices
How well future price forecast the REAL spot price?
Three basic theories
Expectations Hypothesis
• Futures price equals the expected value of the future spot price of asset
𝐹0 = 𝐸(𝑃 𝑇 )
• Expected profit = 0
• Prices of goods at all future dates are known
• Resembles a market with no uncertainties
• Ignores risk premiums
Normal Backwardation
• Hedgers (Farmers) must give an expected profit to speculators to attract
their investments 𝐹 < 𝐸(𝑃 )
0 𝑇
• Expected profit: 𝐸 𝑃 𝑇 − 𝐹0
Part 2. Futures
Contango
• Purchasers of commodities need the product
Lecture 2
𝐹0 > 𝐸(𝑃 𝑇 )
𝐹0 − 𝐸 𝑃 𝑇
20

21. 4. Future prices VS Expected spot prices
How well future price forecast the REAL spot price?
Basic theories
F0
Contango 𝐹0 > 𝐸(𝑃 𝑇 )
𝐹0 = 𝐸(𝑃 𝑇 )
Expectations Hypothesis
𝐸(𝑃 𝑇 )
Part 2. Futures
Backwardation
Today’s price should be cheaper to
𝐹0 < 𝐸(𝑃 𝑇 )
Lecture 2
attract buyers
21

22. Part 3
OPTIONS
a. Definition
b. Values at expiration
c. Option strategies
d. Put-Call parity relation
e. Option valuation:
f. Exotic options
22

23. 1.The option contract
Definition
Are financial instruments that give to the holder the
• RIGHT (not an obligation) to buy or sell an asset
• at an specific time (depending on the option)
• at some specific price
• can be purchased or sold
Call option
Gives its holder the right to PURCHASE an asset
for a specific price (strike or exercise price)
Market Or on before some specific date
Strike • When it is not profitable, it expires
The value of the option is (St-K)
Stock price – Strike price
Part 3. Options
Market
C December call option @ 30
Lecture 2
Holder can buy C at a price of 30 if market is > 30
Holder does not have the obligation to exercise the call,
So he/she will exercise it only when it is profitable
23

24. 1.The option contract
Definition
Are financial instruments that give to the holder the
• RIGHT (not an obligation) to buy or sell an asset
• at an specific time
• at some specific price
• can be purchased or sold
Put option
Gives its holder the right to SELL an asset
for a specific price (strike or exercise price)
Or on before some specific date Market
Strike
When it is not profitable, it expires
The value of the option is (K-St)
Strike price - Stock price
Part 3. Options
Market
3M January put option @ 85
Holder can sell 3M at a price of 85 if market is < 85
Lecture 2
Holder does not have the obligation to exercise the put,
So he/she will exercise it only when it is profitable
24

25. 1.The option contract
Additional language
Option “in the money”: it is profitable
Option “out of the money”: it is unprofitable
Option “at the money”: S=K
American option: the right to buy(sell) at any time before expiration
European option: the right to buy(sell) at expiration
American option are more expensive than European options
Option on assets other than stocks are traded.
• Index options:
• Future options
• Foreign currency options: buy/sell a quantity of foreign currency for a specific
amount of local currency. (this is different that a currency future contract)
Part 3. Options
• Interest rate options
The premium: (cost of the option)
Lecture 2
• Upfront payment (unlike forwards)
• Purchase price of the option. Represents the compensation to have the
right to exercise the option
• This cost has to be included 25

26. 2.Values of options at expiration
Call option
Right to buy an asset St-K St>K
Payoff to call holder
0 otherwise
Profit
Price
Premium
K (Strike Price)
Limited risk
Part 3. Options
Lecture 2
Risk Management technique
26

27. 2.Values of options at expiration
Put option
Right to sell an asset 0 St>K
Payoff to put holder
K-St St<K
Profit
Price
Premium
K (Strike Price)
Part 3. Options
Limited risk
Lecture 2
27

28. 2.Values of options at expiration
Call option (writer)
Exposes the writer to losses when market falls
The writer will receive a call and will be -(St-K) St>K
obligated to deliver a stock worth St
Payoff to call writer 0 otherwise
Profit
Income
Price
K (Strike Price)
Part 3. Options
The income is
Unlimited risk given by the
premium, but
Lecture 2
there is unlimited
risk
28

29. 2.Values of options at expiration
Bullish strategy Bearish strategy
Bearish strategy Bullish strategy
Part 3. Options
Lecture 2
29

30. 2.Values of options at expiration
What is the difference among some portfolios?
We have $10.000 to spend
Portfolio A: Only stocks
Portfolio B: Only calls. K=100
Portfolio C: T+ 10% calls
Part 3. Options
1. While purchasing shares I can afford 100 units, by using calls I can have
Lecture 2
access to 1000 shares
2. Option offers leverage!!
3. Options’ return is 0% because I spent all the money in the premium
4. Consider combinations of financial assets 30

31. Very conservative investment strategy
3. Option strategies St ≤ K St > K
I have an obligation to give stocks, but I am long
Covered call St St
Payoff of stock
0 -(St-K)
Stock St K
XYZ is trading at $17.
Sell someone the right to purchase your
K XYZ stock for $17.50 for a premium of $2.
Write a
Call
(sell
someone
the right to
buy)
Part 3. Options
Lecture 2
Covered
call
31

32. 3. Option strategies
Covered call
Payoff of stock
Stock
K
If you are long, (very long) and
Write a
you have a target price to take
Call
(sell profits, you might want to write
someone a call.
the right to
buy)
Pension funds are always long,
Part 3. Options
but using a covered call, they
can hedge some positions.
Lecture 2
Covered
call Also you can get some cash
from premiums
32

33. 3. Option strategies
Product of the combination of options • Only stock seems risky
Protective Put
Payoff of stock St ≤ K St > K
St St
+
Stock K-St 0
K K St
• You bought 500 shares of stock XYZ
Long at $50, and it rises to $70. But, price
put could drops to $65…$60. Hmm.
ATM • When the price rises to $70, I can buy 5
puts (each put contract represents 100
Part 3. Options
shares of stock) at $2 per contract with
$65 strike price. Commissions are $8.20
New price : $50
Lecture 2
Protecti
ve put Exercise the put and sell at $65 when price
is $50.
Fees
33

34. 3. Option strategies
Product of the combination of options
Protective Put
Payoff of stock
Stock
K
Long
put
ATM
This is clearly a protective
Part 3. Options
portfolio strategy
Lecture 2
Protecti
ve put
Fees
34

35. • Call and put with same exercise
3. Option strategies price and same expiration day
• Ideal when prices will move a lot in
price, no matter the direction
Straddle (Long) • Are bets on volatility
Payoff of stock
• In no volatility, both premiums
are lost
St < K St ≥ K
Call 0 St-K
0 K-St 0
-C K
K-St St-K
Put XYZ stock is trading at $40.
Long straddle: buy put for $200 and a
call for $200. Cost, 400
0 If XYZ = $50
-P Put will expire
Part 3. Options
Call in the money. $1000.
Profit $600.
Lecture 2
Covered
If XYX= $40, both expire worthless
call and there is a loss of $400
35

36. 3. Option strategies
Straddle (Long)
Payoff of stock
Call
0
-C K
The point here is the cost of the
Put TWO premiums. This could be
expensive
0
-P
Part 3. Options
Investors who sell straddles are
betting on stability.
Lecture 2
Covered Nick Leeson is famous for that
call
36

37. • Same date
3. Option strategies • Same stock
Spreads: Product of the combination of options, puts or calls
Bullish spread • Different exercise prices
Payoff of stock • Three outcomes
• Holders profit when price increases
St ≤ K1 K1 < St ≤ K2 St ≤ K2
Long
call 0 St-K1 St-K1
K1 0 0 -(St-K2)
0 St-K1 K2-K1
Short XYZ at $42 could rally.
call Buying call for $300 at $40
Writing call for $100. at $45
Investment $200.
K2 XYZ rise and closes at $46 on expiration date.
Part 3. Options
Long call at 40: + $600
Short call at 45: - $100
Net: $500.
Lecture 2
Covered Net profit : $500-200=$300
call XYZ declined to $38, trader lose his entire
investment of $200, which is also his maximum
possible loss. 37

38. 4. Put-call parity relation
Definition:
• Establish a relationship between the prices of an European put and call
options of the same class
• Combinations of options can create positions that are the same as holding the
stock itself
First portfolio:
• Call option
• Risk free investment with face value = exercise price o a call
• Same expiration date
Part 3. Options
Lecture 2
St ≤ K St > K St ≤ K St > K St ≤ K St > K
+ K K K St
0 St-K
38

39. 4. Put-call parity relation
Second portfolio:
• Put option
• Long stock
• Must produce the same scenario
Therefore, the call+bond must cost the same than the put + stock to establish
Part 3. Options
Initial payoff must be the
price of the asset
Lecture 2
Put – call parity theorem
c + PV(x) = p + s
𝑐 + 𝐾𝑒 −𝑟𝑡 = 𝑝 + 𝑆0
39

40. 5. Option valuation
Determinants (in call option case)
• Stock price: direct relation
• Exercise price: inverse relation
• Volatility: direct relation
• K=30
• S1 has a volatility btw $10 and 50. EV: 6
• S2 has a volatility btw $20 and 40 EV:3
S1 10 20 30 40 50
Payoff 0 0 0 10 20
S2 20 25 30 35 40
Payoff 0 0 5 10 Each price has prob= 0.2
Part 3. Options
• Time to expiration: direct relation
• Interest rate: direct relation because the more r, the less PV of K
Lecture 2
• Dividend rate of stock: inverse relation.
When stocks pay out their dividends, the share price adjusts
downward to compensate for the pay-out
40

41. 5. Option valuation
• An additional relation: time VS price
• If S<K, the option is worthless?
• At expiration date: YES
• Before expiration date, always there is a chance that the option
becomes profitable
Payoff of stock Time Value
Most of an option’s time
value typically is a type
of volatility value
Long
call High volatility near K1
K1
Intrinsic Value
Part 3. Options
Payoff by immediate
exercise
Time Value
Lecture 2
The sensitivity of the option value to the amount of time to expiry is
known as the option's theta.
41

42. 5. Option valuation
Restrictions on the value of options
• European option < American option
• Price cannot be negative
• Call price low bound
𝑐 + 𝐾𝑒 −𝑟𝑡 = 𝑝 + 𝑆0
Put option
𝑐 − 𝑝 = 𝑆0 − +𝐾𝑒 −𝑟𝑡
𝑐 ≥ 𝑆0 − 𝐾𝑒 −𝑟𝑡 𝑝 ≥ 𝐾𝑒 −𝑟𝑡 - 𝑆0
• Call upper bound, is the stock price
𝑐 ≤ 𝑆0 𝑐 ≤ 𝐾0
Part 3. Options
Lecture 2
42

43. 5. Option valuation
Binomial option pricing model
• Proposed by Cox, Ross and Rubinstein. “Option Pricing: A
Simplified Approach”, Journal of Financial Economics, 1979, 7,
229-263.
• Replication principle: Two portfolios producing the
exact same future payoffs must have the same value.
• Otherwise, there will be opportunities for riskless
arbitrage.
• Use this model to price European call options.
Part 3. Options
Main idea:
construct a synthetic portfolio that replicate option’s payoffs using
Lecture 2
rF and stocks .
These portfolios SHOULD have the same return to avoid arbitrage
Find the value of that portfolio. That must be the price of the call
43

44. 5. Option valuation
Binomial option pricing model
S0=110 C1= 10
S0=100 K=100
S0= 90 C2= 0
• If the investors borrow money, the interest rate=6% for one year.
• What is the price of the European call option?
We can replicate the payment of the call by a
suitable portfolio : f ( underlying asset + risk free )
Stock Bond
Part 3. Options
C1 Payoff 10  N 110  B  (1  0.06)
Two eq = Two unk
C2 Payoff 0  N  90  B  (1  0.06)
Lecture 2
N  0.5
B  42.4528
44

45. 5. Option valuation
Binomial option pricing model
S=100, it will move to either 110 or 90 in one year
X=100, r=6%
Form a synthetic portfolio: short position in a bond (sell a
bond to borrow money) at $42.4528 and long position in ½
share of stock
after 1 year ST=110 ST=90
Synthetic portfolio stock 55 45
bond -45 -45
Net payoff 10 0
Part 3. Options
Lecture 2
Call Call 10 0
45

46. 5. Option valuation
Binomial option pricing model
Since the payoff (value) for the synthetic
portfolio is exactly the same as that for the Call
option in all circumstances, the price (initial value)
of the portfolio must be the same as that of the
Call.
C0  N  S0  B  0.5 100  42.4528  7.5472  7.55
Part 3. Options
Lecture 2
46

47. 5. Option valuation
Black – Scholes valuation
• Assumes that the price follow a Geometric Brownian Motion (GBM)
with constant drift and volatility.
𝑑𝑆
= 𝜇𝑑𝑡 + 𝜎𝑑𝑧
𝑆
• The model incorporates the
• constant price variation of the stock
• the time value of money
• the option's strike price
• the time to the option's expiry.
Assumptions
1. Stock pays no dividends
Part 3. Options
2. Option can only be exercised upon expiration (European)
3. Market direction cannot be predicted, hence "Random Walk."
4. No commissions, taxes are charged in the transaction.
Lecture 2
5. Short sales allowed
5. Interest rates remain constant.
6. Stock returns are normally distributed, thus volatility is constant over time.
47

48. 5. Option valuation
Black – Scholes valuation
Co call option value
So current stock price
N(d) cumulative distribution function of the standard normal distribution
T-t time to maturity
r risk free rate
𝜎 is the volatility of the underlying asset
Scenario 1: N(d) = 1
Part 3. Options
High probability the option will be exercised
Intrinsic value
Lecture 2
Scenario 2: N(d) = 0
No probability the option will be exercised
Scenario 3: N(d) = btw 0 and 1
Value depends on the call potential value (PV) 48

49. 6. Exotic Options
Asian Options
Payoff depends on the average price of the underlying asset over a certain
period of time as opposed to at maturity
Barrier Options
A type of option whose payoff depends on whether or not the underlying
asset has reached or exceeded a predetermined price.
Lookback option
Payoffs that depend in part on the minimum or maximum price of the
underlying asset during the life of the option.
• Payoff could be against the max or min instead of the final price
Part 3. Options
Lecture 2
49