FINANCE 2
BLOCK 3 2023
INSTRUCTOR: E. Smailbegovic
smajlbegovic@ese.eur.nl
LECTURE 1
WHAT IS A DERIVATIVE?
Derivative is a financial instrument that has a value determined by the price of something else
(underlying offer of a derivative)
Example: An agreement where you pay $1 if the price of corn is greater than $3 and receive $1 if
the price of corn is less than $1 is a derivative
-
This contract can be used to speculate on the price of corn or it can be used to reduce
risk. It is not the contract itself, but how it is used, and who uses it that determines
whether or not it is risk-reducing.
Interested are the people who are speculators
If the farmer enters an agreement like this, he can reduce the risk of reduction of the price of
corn. This can happen if the price of corn reduces, so the farmer earns less, and because he is in
this agreement, he will receive $1 for the reduction of the price of corn. This way this agreement
compensates the reduction of income of the farmer.
HISTORICAL EXAMPLES
1
-
Thales of Milet (6th century a.d.) expected a rich olive harvest and agreed with the owners
of olive mills to rent their mills long term, fixing their (low) rent today. During the
harvesting season he sub-rented those mills and made a fortune
-
During the tulip-mania in the 17th century in the Netherlands, there was an active
derivatives market for tulip bulbs
-
The Dojima Rice Exchange was the first organized futures exchange (est. 1697)
DERIVATIVES MARKETS
The introduction of derivatives in a market often coincides with an increase in price risk in that
market.
-
-
Currencies were permitted to float in 1971 when the gold standard was officially
abandoned. The modern market in financial derivatives began in 1972, when the Chicago
Mercantile Exchange started trading futures contracts on seven currencies
OPEC’s 1973 reduction in the supply of oil was followed by high and variable oil prices
US interest rates became more volatile following inflation and recessions in the 1970s
The market for natural gas has been deregulated gradually since 1978, resulting in a
volatile market in recent years
The deregulation of electricity behan during the 1990s
THE USES OF DERIVATIVES
1. Risk Management: Derivatives are a tool for companies and other users to reduce risk
2. Speculation: Derivatives can serve as investment vehicles
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
EXAMPLES OF DERIVATIVES APPLICATIONS
-
Transnational companies buy FX-derivatives to hedge their currency exposure
Airlines enter futures contracts in order to be less vulnerable to oil price changes
A goldmine can use derivatives to lock in the current gold price
Bank can use derivatives to manage interest rate and default risks
AND THOSE THAT WENT WRONG…
-
2
Barings Bank (1995): Nick Leeson engages in arbitrage business using derivatives. To
recover his losses, he bets on a quick recovery of the Asian economy after the Kobe
earthquake by investing in futures.
-
-
LTCM (1998): Nobel prize winner advised fund speculates using OTC derivatives. During
the Russian crises the counterparties of hedges default. LTCM cannot settle its positions
due to illiquidity
Societe Generale (2008): Jerome Kerviel nearly brings down the bank by massive
fraudulent speculation in equity index futures
USA (2007-2009): Role of derivatives in financial crises is debated
PERSPECTIVES ON DERIVATIVES AND THE TRADING PROCESS
End Users:
-
Corporations
Investment managers
Investors
Intermediaries:
-
Market-makers
Traders
Economic observers:
-
Regulators
Researchers
EXCHANGE-TRADED DERIVATIVES MARKETS
-
Much trading of financial claims takes place on organized exchanges.
In the past, the exchange was solely a physical location where traders would buy and sell.
Such in-person venues have largely been replaced by electronic networks that provide a
virtual trading venue.
After a trade has taken place, a clearinghouse matches the buyers and sellers, keeping
track of their obligations and payments.
To facilitate these payments and to help manage credit risk, a derivatives clearinghouse
typically imposes itself in the transaction, becoming the buyer to all sellers and the seller
to all buyers.
OVER-THE-COUNTER MARKET
It is possible for large traders to trade many financial claims directly with a dealer bypassing
organized exchanges. Such trading is said to occur in the over-the-counter (OTC) market
Exchange activity is public and highly regulated
Over-the-counter trading is not easy to observe or measure and is generally less regulated
3
For many categories of financial claims, the value of OTC trading is greater than the value traded
on exchanges
INTRODUCTION TO FORWARDS, FUTURES AND OPTIONS
Basic derivatives contracts:
-
Forward contracts
Call options
Put options
Types of positions:
-
Long position
Short position
Graphical representation:
-
Payoff diagrams
Profit diagrams
FORWARD CONTRACTS
Forward contract is a binding agreement (obligation) to buy/sell an underlying asset in the
future at a price set today
Futures contracts are the same as forwards in principle except for some institutional and pricing
differences
A forward contract specifies:
-
The features and quantity of the asset to be delivered
The delivery logistics, such as time, date and place
The price the buyer will pay at the time of delivery
READING PRICE QUOTES
Index future price listings:
4
Expiration either June or September
Open: the open price
-
High/low: lifetime high or low
Settle: Settlement price
Chg: daily change
THE PAYOFF ON A FORWARD CONTRACT
Payoff for a contract is its value at expiration
Payoff for:
-
Long forward = spot price at expiration - forward price
Short forward = forward price - spot price at expiration
Example: Today S&R 500 index: spot price = $1,000, 6-month forward price = $1,020
In six months at contract expiration: spot price = $1,050
-
Long position payoff: $1,050 - $1,020 = $30
Short position payoff: $1,020 - $1,050 = -$30
PAYOFF DIAGRAM FOR FORWARDS
Long and short forward positions on the S&R 500 index:
ADDITIONAL CONSIDERATIONS
Type of settlement:
-
Cash settlement: less costly and more practical
Physical delivery: often avoided due to significant costs
Credit risk of the counterparty:
-
5
Major issue for over-the-counter contracts
-
- Credit check, collateral, bank letter of credit
Less severe for exchange-traded contracts
- Exchange guarantees transactions, requires collateral
CALL OPTIONS
Call Option is a non-binding agreement (right but not an obligation) to buy an asset in the
future, at a price set today
Preserves the upside potential, while at the same time eliminating the unpleasant downside (for
the buyer)
The seller of a call option is obligated to deliver if asked
Why would anyone agree to be on the seller side? The seller receives an insurance, an option
premium
DEFINITION AND TERMINOLOGY
A call option gives the owner the right but not the obligation to buy the underlying asset at a
predetermined price during a predetermined time period
Strike (or exercise) price: the amount paid by the option buyer for the asset if he/she decides to
exercise
Exercise: the act of paying the strike price to buy the asset
Expiration: the date by which the option must be exercised or becomes worthless
Exercise style: specifies when the option can be exercised
-
European style: can be exercised only at expiration date
American style: can be exercised at any time before expiration
Bermudan style: can be exercised during specified periods (used for speculative purposes)
PAYOFF/PROFIT OF A PURCHASED OPTION
Payoff = max [0, spot price at expiration - strike price]
Profit = payoff - future value of option premium
6
Example: S&R index 6-month call option
-
Strike price = $1,000, premium = $93.81, 6-month risk-free rate = 2%
- If index value in six months = $1,100
- Payoff = max [0,$1,100 - $1,000] = $100
- Profit = $100 - ($93.81 x 1,02) = $4.32
- If index value in six months = $900
- Payoff = max [0, $900 - $1000] = 0
- Profit = $0 -($93.81 x 1,02) = - $95.68
DIAGRAM FOR PURCHASED CALL
Payoff at expiration:
With blue: payoff of the buyer of the option
With red: payoff of the seller of the option
With pink: profit of the buyer of the option (lower by the
premium)
Profit at expiration:
PAYOFF/PROFIT OF A WRITTEN CALL
7
Payoff = - max [0, spot price at expiration - strike price]
Profit = payoff + future value of option premium
Example: S&R index 6-month call option
-
Strike price = $1,000, premium = $93.81, 6-month risk-free rate = 2%
- If index value in six months = $1,100
- Payoff = - max [0,$1,100 - $1,000] = -$100
- Profit = - $100 + ($93.81 x 1.02) = -$4.32
- If index value in six months = $900
- Payoff = - max [0, $900 - $1,000] = $0
- Profit = - $0 + ($93.81 x 1.02) = $95.68
PUT OPTION
Put option gives the owner the right but not the obligation to sell the underlying asset at a
predetermined price during a predetermined time period
The seller of a put option is obligated to buy if asked
Payoff/profit of a purchased (i.e. long) put:
-
Payoff = max [o, strike price - spot price at expiration]
Profit = payoff - future value of option premium
Payoff/profit of a written (i.e. short) put:
-
Payoff = - max [0, strike price - spot price at expiration]
Profit = payoff + future value of option premium
Example: S&R index 6-month put option
-
Strike price = $1,000, premium = $74.20, 6-month risk-free rate = 2%
- If index value in six months = $1,100
- Payoff = max [0, $1,000 - $1,100] = $0
- Profit = $0 - ($74.20 x 1.02) = - $75.68
- If index value in six months = $900
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- Payoff = max [0, $1,000 - $900] = $100
- Profit = $100 - ($74.20 x 1.02) = $24.32
PROFIT FOR A LONG PUT POSITION
Profit diagram:
FEW ITEMS TO NOTE
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A call option becomes more profitable when the underlying asset appreciates in value
A put option becomes more profitable when the underlying asset depreciates in value
Moneyness:
-
In-the-money option: positive payoff if exercised immediately
At-the-money option: zero payoff if exercised immediately
Out-of-money option: negative payoff if exercised immediately
OPTION AND FORWARD POSITIONS: A SUMMARY
9
MINI CASE: BANK OF AMERICA MITTS
Questions:
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What does the payoff from an investment in 1,000 MITTS look like?
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How can we decompose the product in terms of options?
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How could the issuing bank hedge the risk associated with the issuance of the MITTS?
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How does the bank make a profit?
LECTURE 2
INTEREST RATES AND BOND BASICS
NOTATIONS
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rt (t1, t2): interest rate from time t1 to t2 prevailing at time t
r0 (0,t2): zero rate or spot rate
A zero rate (or spot rate), for maturity t2 is the rate of interest earned on today’s investment
that provides a payoff only at time t2. Usually expressed in annual terms.
-
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Pt0 (t1, t2): price of a bond quoted at t = t0 to be purchased at t = t1 maturing at t = t2 and a
$1 payoff only at time t2
-
Yield to maturity/ bond yield: percentage increase in dollars earned from the bond
MEASURING INTEREST RATES AND IMPACT OF COMPOUNDING
The compounding frequency used for an interest rate is the unit of measurement
The difference between quarterly and annual compounding is analogous to the difference
between miles and kilometers
-
When we compound n time per year at an annual rate r(0,1) an amount A grows to
A(1+r(0,1)/n)^n in one year.
Annual (n=1) => 100 * [(1 +10%)/1]^1
Semiannual (n=2) => 100 * [1+10%)/2]^2
Quarterly (n=4) => 100 * [1+10%/4]^4
CONTINUOUS COMPOUNDING
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In the limit as we compound more and more frequently we obtain continuously
compounded interest rates: (compounding goes to almost infinity)
-
$100 grows to $100e^r (0,1)T when invested at a continuously compounded rate r(0,1) for
time T
$100 received at time T discounts to $100e^−r (0,1)T at time zero when the continuously
compounded discount rate is r(0,1)
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CONVERSION FORMULAS
Define:
rc: continuously compounded rate (to compound)
11
rn: same rate with compounding n times per year (discrete interest rate + already compounded)
Rates used in option pricing are usually expressed with continuous compounding
BOND PRICE
Zero-coupon bond price that pays CT at T:
-
Discrete:
-
Continuous:
-
General:
Coupon bonds: The price at time of issue of t of a bond maturing at time T that pays n coupons
of size c and maturity payment of CT at T:
Where ti = t + i (T - t)/n
BOND YIELD (Not part of the exam)
The bond yield is the discount rate that makes the present value of the cash flows on the bond
equal to the market price of the bond
The bond yield at t = 0 is given by solving:
for the value y (this is the continuous version)
EXAMPLE: ZERO CURVE AND BOND YIELD
You open the financial section of your daily newspaper and find the following table with different
bonds, their main characteristics and prices.
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1. Given the bond prices above, please draw the zero curve for the next two years
(continuously compounded rate)
Bond A: 100 * e^-r*0.25 = 97.5 => r = 10.13%
Bond B: 100 * e^-r*0.5 = 94.9 => r = 10.469%
Bond C: r(0,1) = 10.536%
Bond D: c * e^-r (0,0.5)*0.5 + c * e^-r (0,1)*1 + (100 + c) * e^-r(0,1.5)*1.5 = 96
=> 4 * e^-0.10469*0.5 + 4 * e^-0.10536*1 + 104 * e^r(0,1.5) *1.5 = 96
r(0,1.5) = 10.681%
Bond E: r(0,2) = 10.808%
2. You are interested in a long-term investment in a safe bond. However, before investing in
Bond E, you would like to know its yield to maturity
YTM = 6 * e^-y*0.5 + 6 * e^-y*1 + 6 * e^-y*1.5 +106 * e ^-y *2 = 101.6
(Can only be solved with computer)
FORWARD RATES
The forward rate is the future zero rate implied by today’s term structure of interest rates
FORMULA FOR FORWARD RATES
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Suppose that the zero rates for time periods T1 and T2 are r(0,T1) and r(0,T2) with both rates
continuously compounded
The continuously compounded forward rate for the period between times T1 and T2 is:
This formula is only approximately true when rates are not expressed with continuous
compounding
The discrete forward rate for the period between T1 and T2 is:
UPWARD VS DOWNWARD SLOPING YIELD CURVE
-
For an upward sloping yield curve:
For a downward sloping yield curve:
forward rate > zero rate (developing economy)
zero rate > forward rate (recessions)
FINANCIAL FORWARDS AND FUTURES
INTRODUCTION
Financial futures and forwards:
-
On stocks and indexes
On currencies
On interest rates
How are they used?
How are they priced?
How are they hedged?
ALTERNATIVE WAYS TO BUY A STOCK
Four different payment and receipt timing combinations:
1)
2)
3)
4)
Outright purchase: ordinary transaction
Fully leveraged purchase: investor borrows the full amount
Prepaid forward contract: pay today, receive the share later
Forward contract: agree on price now, pay/receive later
Payments, receipts and their timing:
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Four different ways to buy a share of stock that has price S0 at time 0. At time 0 you agree to a
price, which is paid either today or at time T. The shares are received either at 0 or T. The interest
rate is r.
PRICING PREPAID FORWARDS
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If we can price the prepaid forward (FP), then we can calculate the price for a forward
contract:
F = future value of F
Three possible methods to price prepaid forwards:
1. Pricing by analogy
2. Pricing by arbitrage
3. Pricing by discounted present value (not discussed)
For now, assume that there are no dividends
PRICING BY ANALOGY
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In the absence of dividends, the timing of delivery is irrelevant
Price of the prepaid forward contract same as current stock price
Where the asset is bought at t = 0, delivered at t = T
PRICING BY ARBITRAGE
-
Arbitrage: a situation in which we can generate positive cash flow by simultaneously
buying and selling related assets, with no net investment and with no risk => free money
If at time t = 0 , the prepaid forward price somehow exceeded the stock price, i.e.
, an arbitrageur could do the following:
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-
Since, this sort of arbitrage profits are traded away quickly, and cannot persist, at
equilibrium we can expect:
PRICING PREPAID FORWARDS WITH DIVIDENDS
What if there are dividends? Is
-
still valid?
No, because the holder of the forward will not receive dividends that will be paid to the
holder of the stock =>
- For discrete dividends Dti at times ti = 1,...,n
- The prepaid forward price is:
- For continuous dividends with an annualized yield δ:
- The prepaid forward price is:
PRICING PREPAID FORWARDS: TWO EXAMPLES
Example 1: XYZ stock costs $100 today and is expected to pay a quarterly dividend of $1.25. If the
risk free rate of 10% compounded continuously, how much does a 1-year prepaid forward cost?
FP = $100 - (1.25*e^-0.0.25*1 +1.25*e^-0.05*1 +1.25*e^-0.075*1 +1.25*e^-0.1*1 ) = $95.30
Example 2: The index is $125 and the dividend yield is 3% continuously compounded. How much
does a 1-year prepaid forward cost?
FP = $125 e^-0.03*1 = $121.31
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PRICING FORWARDS ON STOCK
Forward price is the future value of the prepaid forward price
-
No dividends:
-
Continuous dividends:
Forward premium: The difference between current forward price and stock price
Forward premium = F0,T / S0
Annualized forward premium =
-
Can be used to infer the current stock price from forward price
CREATING A SYNTHETIC FORWARD
One can offset the risk of a forward by creating a synthetic forward to offset a position in the
actual forward contract
How can we do this? (Assume continuous dividends at rate δ)
17
-
Recall the long forward payoff at expiration: ST - F0,T
Borrow and purchase shares as follows:
-
Note that the total payoff at expiration is the same as forward payoff
The idea of creating synthetic forward leads to following:
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Forward = stock - zero coupon bond
Stock = forward + zero coupon bond
Zero coupon bond = stock - forward
Cash-and-carry arbitrage: buy the index, short the forward
NOTE: Cash-and-carry arbitrage with transaction costs: Trading fees, bid-ask spreads, different
borrowing/lending rates, the price effect of trading in large quantities, make arbitrage harder
OTHER ISSUES IN FORWARD PRICING
Does the forward price predict the future price?
-
According to the formula, the forward price conveys no additional information beyond
what S0, r and δ provides
Moreover, the forward price underestimates the future stock price
Forward pricing formula and cost of carry:
-
Forward price = spot price + interest to carry the asset - asset lease rate
FUTURES CONTRACTS
18
-
Exchange-traded “forward contracts”
-
Typical features of futures contracts:
a. Standardized, with specific delivery dates, locations, procedures
b. A clearinghouse
i.
Matches the buy and sell orders
ii.
Keeps track of members’ obligations and payments
iii.
After matching the trades, becomes counterparty
-
Differences from forward contracts:
- Settled daily through the mark-to-market process => low credit risk
- Highly liquid => easier to offset an existing position
- Highly standardized structure => harder to customize
EXAMPLE: S&P 500 FUTURES
-
Notional value: $250 x index
Cash - settled contract
Open interest: total number of buy/sell pairs
Margin and mark-to-market
a. Initial margin
b. Maintenance margin (70 - 80% of initial margin)
c. Margin call
d. Daily mark-to-market
Mark-to-market proceeds and margin balance for 8 long futures contracts:
Notional value of one contract is 250x $1100 = $275,000
We want to enter 2.2 million / $275,000 = 8 contracts
=> 8 x 250 x $1100 = 2000 x $1100 =$ 2.2 million
r = 6% and margin is 10%
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Week
Multiplier
Future Price
Price Change
Margin Balance
0
2000
$1100
-
1
2000
$1027.99
-72.01
a)
2
2000
$1037.88
9.89
b)
2000
$1011,65
2.2m x 10% = $220,000
…
10
44,990.57
1) Payoff forward: ST=10 - F0,T=10) x multiplier => (1011.65 - 1100) x 2000 = - $176,700
WEEK 1: Price change = - 72.01
MARGIN ACCOUNT)
=> 2000 x (-72.01) = - $144,020 (DEDUCTED FROM THE
a) New margin balance: 220,000 x e^0.06*1/52 - $144,020 = $76,233.32
WEEK 2: Price change = 9.85
=> 2000 x (9,85) = $19,780
b) new margin balance: $76,233.32 x e ^0.06*1/52 + $19,780 = $96,102.01
Futures payoff:
(-$176,700)
$44,990.57 - $220,000xe^0.06*10/52 = -$177,562.60 < Forward payoff
Futures prices vs Forward prices:
-
The differences negligible especially for short-lived contracts
Can be significant for long-lived contracts and/or when interest rates are correlated with
the price of the underlying assets
USES OF INDEX FUTURES
Why buy an index futures contract instead of synthesizing it using the stocks in the index? Lower
transaction costs
-
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Asset allocation: switching investments among asset classes
Example: Invested in the S&P 500 index and wish to temporarily invest in bonds instead of index.
What to do?
-
Alternative 1: Sell all 500 stocks and invest in bonds
Alternative 2: Take a short forward position in S&P 500 index
CURRENCY CONTRACTS
Currency prepaid forward
-
Widely used to hedge against changes in exchange rates
Suppose you want to purchase ұ1 one year from today using $x0 =>
Where x0 is current ($/ ұ) exchange rate and ry is the
yen-denominated interest rat
Why? By deferring delivery of the currency one loses interest income from bonds
denominated in that currency
Currency forward:
-
r is the $ - denominated domestic interest rate
(domestic risk-free rate exceeds foreign risk-free rate)
CURRENCY CONTRACTS: PRICING
Example 1: Y-denominated interest rate is 2% and current ($/Y) exchange rate is 0.009. To have
Y1 in one year one needs to invest today:
0.009*e^-0.02*1 = 0.00882
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Example 2: Y-denominated interest rate is 2% and $-denominated rate is 6%. The current ($/Y)
exchange rate is 0.009. The 1-year forward rate is:
0.009*e^(0.06-0.02)*1= 0.009367
SUMMARY
-
Understanding the term structure of interest rates is essential for derivatives markets
The value of a forward contract (or any other derivative) is based on the non-arbitrage
condition
The payoff of a forward contract can be replicated using the underlying and a risk-free
investment
LECTURE 3
FORWARD RATE AGREEMENTS
FRAs are over-the-counter contracts that guarantee a borrowing or lending rate on a given
notional principal amount
-
Can be settled at maturity (in arrears) or the initiation of the borrowing or lending
transaction
Forward price = Implied forward rate
-
FRA settlement in arrears: (r - rFRA) x Notional principal
At the time of borrowing: notional principal x (r - rFRA) / (1 + r)
FRAs can be synthetically replicated using zero-coupon bonds
INTRODUCTION TO COMMODITY FORWARDS
Financial forward prices can be described by the formula:
-
A commodity forward contract is a “different
Three examples of futures contracts:
1) E-mini S&P 500 Futures
3) Copper Futures
animal”
2) Corn Futures
The set of prices for different expiration dates for a given commodity is called the forward curve
(or the forward strip) for that date.
If on a given date the forward curve is upward sloping, then the market is in contango.
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If the forward curve is downward sloping, the market is in backwardation
-
NOTE: that forward curves can have portions in backwardation and portions in contango
EQUILIBRIUM PRICING OF COMMODITY FORWARDS
Different commodities have their distinct forward curves, reflecting different properties of:
-
Storability
Storage costs
Production
Demand
Seasonality
SHORT-SELLING AND THE LEASE RATE
For a commodity owner who lends the commodity, the lease rate is like a dividend
-
With the stock, the dividend yield, δ, is an observable characteristic of the stock
With a commodity, the lease rate, δI, is income earned only if the commodity is loaned
The lease rate has to be consistent with the forward price
Therefore, when we observe the forward price, we can infer what the lease rate would have to be
if a lease market existed
The annualized lease rate:
NO ARBITRAGE PRICING INCORPORATING STORAGE COSTS
The cost of storing a physical item such as corn or copper can be large relative to its value
Moreover, some commodities deteriorate over time, which is also a storage cost
We can view storage costs as a negative dividend
THE CONVENIENCE YIELD
Some holders of a commodity receive benefits from physical ownership (e.g. a commercial user).
This benefit is called the commodity’s convenience yield.
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If there is a continuously compounded convenience yield, c, and interest rates, r, and storage
costs, λ, are paid continuously and are proportional to the value of the commodity then:
A user who buys and stores the commodity will be compensated for interest and physical
storage costs less a convenience yield.
The commodity lease rate will be: δI = c - λ
HEDGING RISK AND THE CASE OF METALLGESELLSCHAFT AG I
Futures are very useful in hedging commodity price exposure (e.g. risk of changing prices in the
future)
However, in many cases futures do not represent exactly, what is being hedged (basis risk)
Two common types of basis risk are:
1. Cross hedging:
- Airline companies often use crude oil futures to hedge jet fuel price risk
- Issues: 1) Estimation error and 2) Hedge ratios may change over time
- Time series: 1) Crude Oil Prices (WTI)
2) Kerosene-Type Jet Fuel Prices
2. Stack and roll:
- Sometimes hedgers need to hedge distant obligations with near-term futures
Example: An oil producer agreed to deliver 100,000 barrels of oil each month at a
fixed price for one year.
-
Natural way to hedge this contract is to enter 12 futures contracts for each of the
months (strip hedge)
Issue: Distant futures contracts are illiquid or unavailable at that point in time
Alternative: Stack hedge and roll
A stack hedge entails going long 12x100,000 barrels of oil using the next month’s
futures contract
At maturity (one month), the hedger re-establishes the stack hedge with the next
month’s futures contract (rolling)
However, at each maturity date, the hedger makes a profit or loss, depending on
the term structure of forward prices
24
Ideally, any potential profits, losses will be offset by the payoff from the fixed price
contract at the end
Metallgesellschaft AG: Very prominent case of “bad” hedging
-
MG sold a huge volume of 10-year oil fixed-price supply contracts
These contracts were hedged with short-term futures contracts that were rolled forward
Issue: At this time period the oil price fell dramatically causing margin calls
These additional costs could not be offset by the short term profits from the fixed-price
contracts
In long-term these losses would have been offset, but the short-term losses were so
severe that all contracts were closed with a loss of USD 1.33 billion
INTRODUCTION TO SWAPS
A swap is a contract calling for an exchange of payments, on one or more dates, determined by
the difference in two prices.
A swap provides a means to hedge a steam of risky payments.
A single-payment swap is the same thing as a cash-settled forward contract.
AN EXAMPLE OF A COMMODITY SWAP
An industrial producer, IP Inc., needs to buy 100,000 barrels of oil 1 year from today and 2 years
from today.
-
The forward prices for delivery in 1 year and 2 years are $110 and $111/barrel
The 1- and 2-year zero-coupon bond yields are 6% and 6.5% respectively.
IP can guarantee the cost of buying oil for the next 2 years by entering into long forward
contracts for 100,000 barrels in each of the next 2 years. The PV of this cost per barrel is:
Thus, IP could pay an oil supplier $201.638 and the supplier would commit to delivering one
barrel in each of the next two years.
A prepaid swap is a single payment today to obtain multiple deliveries in the future.
With a prepaid swap, the buyer might worry about the resulting credit risk.
25
Therefore, a more attractive solution is to defer payment until the oil is delivered, while still fixing
the total price.
Any payments that have a present value of $201.638 are acceptable.
Typically, a swap will call for equal payments in each year.
-
For example, the payment per year per barrel, x, will have to be $110.483 to satisfy the
following equation:
We then can say that the 2-year swap price is $110.483.
PHYSICAL VS FINANCIAL SETTLEMENT
Physical settlement of the swap:
Financial settlement of the swap:
-
The oil buyer, IP, pays the swap counterparty the difference between $110.483 and the
spot price, and the oil buyer then buys oil at the spot price.
If the difference between $110.483 and the spot price is negative, then the swap
counterparty pays the buyer
Whatever the market price of oil, the net cost to the buyer is the swap price, $110.483:
The results for the buyer are the same whether the swap is settled physically or financially.
In both cases, the net cost to the oil buyer is $110.483:
26
THE MARKET VALUE OF A SWAP
The market value of a swap is zero at interception.
Once the swap is struck, its market value will generally no longer be zero because:
-
The forward prices for oil and interest rates will change over time
Even if prices do not change, the market value of swaps can change over time due to the
implicit borrowing and lending
A buyer wishing to exit the swap could negotiate terms with the original counterparty to
eliminate the swap obligation or enter into an offsetting swap with the counterparty offering the
best price
The market value of the swap is the difference in the PV of payments between the original and
new swap rates
COMPUTING THE SWAP RATE
Notation:
-
Suppose there are n swap settlements, occurring on dates ti, i = 1, …, n
The forward prices on these dates are given by F0, ti
The price of a zero-coupon bond maturing on date ti is P (0,ti)
The fixed swap rate is R
If the buyer at time zero were to enter into forward contracts to purchase one unit on each of the
n dates, the present value of payments would be the present value of the forward prices, which
equals the price of the prepaid swap:
We determine the fixed swap price, R, by requiring that the present value of the swap payments
equal the value of the prepaid swap:
27
Above equation can be rewritten as:
We can rewrite the above equation to make it easier to interpret:
Thus, the fixed swap rate is as a weighted average of the implied forward rates, where
zero-coupon bond prices are used to determine the weights
GENERAL PRICING EQUATION
The swap formulas in different cases all take the same general form
Let f0 (ti) denote the forward price for the floating payment in the swap.
Then, the fixed swap payment is:
The following table summarizes the substitutions to make in the above equation to get various
swap formulas:
28
SUMMARY
-
Commodity forwards slightly more complex in pricing because of the underlyings’
properties
Forward rate agreements are essential to protect against increases in the cost of
borrowing
Swap contract = collection of forward contracts
LECTURE 4
OPTION TYPES AND PAYOFFS (RECAP)
Payoff and profit for a long call position:
-
Payoff = max [0, spot price at expiration - strike price]
-
Profit = payoff - future value of option premium
Payoff and profit for a short call position:
29
-
Payoff = - max [0, spot price at expiration - strike price]
-
Profit = payoff + future value of option premium
Payoff and profit for a long put position:
-
Payoff = max [0, strike price - spot price at expiration]
-
Profit = payoff - future value of option premium
Payoff and profit for a short put position:
-
Payoff = - max [0,strike price - spot price at expiration]
-
Profit = payoff + future value of option premium
PROPERTIES OF STOCK OPTIONS
-
Which variables affect option prices and how?
-
Differences in European and American option price
-
Lower bounds
-
Put-call parity
-
Early exercise of American options
NOTATION
30
EFFECT OF VARIABLES ON OPTION PRICES
The price of an option is determined by the price of the underlying asset, the strike price, time to
maturity, the volatility of the underlying asset, dividend payouts/yield and the risk-free rate.
EUROPEAN VS AMERICAN OPTIONS
Since an American option can be exercised at any time, whereas a European option can only be
exercised at expiration,
An American option must always be at least as valuable as an otherwise identical European
option:
LOWER BOUND OF CALL OPTION PRICES
1)
Is there an arbitrage opportunity, if we assume:
31
-
CEUR= 3
-
S0 = 20
-
T=1
-
r = 10%
-
K =18
-
Div= 0
CEUR >= max [0, PV (Forward price) - PV(Strike Price)]
CEUR >= max[ 0, S0, - K*e^(-rT) ]
CEUR >= max [0, 20 - 18*e^-0.1*1]
CEUR >= max [0, 20 - 16.2871]
CEUR >= 3.71
If call price lower than lower bound for a call price => arbitrage exists in this example (3 < 3.71)
t=0
t=T
S < K (no exercise of the
S > K (exercise of the option)
option)
Buy option
-3
0
ST - K
Short Stock
+20
- ST
- ST
Invest at r
-17
17e^r(0,1)T
17e^r(0,1)T
0
18.788 - ST
18.788 - K
> 0 because ST < K
> 0 because K =18
TOTAL
Zero investment but positive cash flow at T irrespective of value of ST => Arbitrage
Alternative strategy to show arbitrage:
t=0
t=T
S < K (no exercise of the option)
S > K (exercise of the option)
Buy option
-3
0
ST - K
Short forward
0
F0,T - ST = S0e^r(0,1)T - ST
F0,T - ST = S0e^r(0,1)T - ST
Borrow at r
+3
TOTAL
0
32
-3e^r(0,1)T
-3e^r(0,1)T
18.788 - ST
18.788 - K
>0
>0
LOWER BOUND OF PUT OPTION PRICES
2)
Is there an arbitrage opportunity, if we assume:
-
PEUR = 1
-
S0 = 37
-
T = 0.5
-
r = 5%
-
K = 40
-
Div= 0
PEUR >= max [0, PV (Strike Price) - PV (Forward Price)]
PEUR >= max [0, K*e^-r(0,T)T - S0]
PEUR >= max [0, 40*e^-0.05*0.5 - 37]
PEUR >= max [0, 39.01 - 37]
PEUR >= 2.01
If put price lower than lower bound for a call price => arbitrage exists in this example (1 < 2.01)
t=0
Buy put option
Buy stock
Borrow at r
TOTAL
-1
-S0 = -37
+38
0
t=T
S < K (exercise of the option)
S > K (no exercise of the option)
K - ST
0
+ST
+ST
-38e^0.05*0.5
K - 38.962
ST - 38.962
>0
>0
Investment at t = 0 of zero dollars, but profit of > 0 => Arbitrage
33
-38e^0.05*0.5
Alternative strategy to show arbitrage:
t=0
t=T
S < K (exercise of the option)
S > K (no exercise of the option)
Buy put option
-1
K - ST
0
Buy forward
0
ST - S0e^rT
ST - S0e^rT
Borrow at r
+1
-1*e^rT
-1*e^rT
TOTAL
0
K - 38.962
ST - 38.962
>0
>0
PROPERTIES OF OPTION PRICES
Maximum and minimum option prices
-
Call price cannot:
1) Be negative
2) Exceed stock price
3) Be less than the present value of the difference between the forward price
and the strike price:
-
Put price cannot:
1) Be negative
2) Be more than the strike price
3) Be less than the present value of the difference between the strike price
and the forward price:
34
PUT-CALL PARITY
For European options with the same strike price and time to expiration, the parity relationship is:
Call - Put = PV (Forward price - Strike price)
Or:
Intuition:
Buying a call and selling a put with the strike equal to the forward price (F0,T = K) creates a
synthetic forward contract and hence must have a zero price
PARITY FOR OPTIONS ON STOCKS
If underlying asset is a stock and PV0,T (Div) is the present value of the dividends payable over the
life of the option, then:
Therefore:
For index options,
Therefore:
Equation:
helps us to construct synthetic positions in options, stocks or zero-coupon bonds
Synthetic security creation using parity:
35
-
Synthetic stock: buy call, sell put, lend PV of strike and dividends
-
Synthetic zero-coupon bond: buy stock, sell call, buy put (conversion)
-
Synthetic call: buy stock, buy put, borrow PV of strike and dividends
-
Synthetic put: sell stock, buy call, lend PV of strike and dividends
SUMMARY OF PARITY RELATIONSHIPS
PROPERTIES OF OPTION PRICES
Early exercise for American Options
-
An American call option on a non-dividend-paying stock should not be exercised early,
because:
CAMER >= CEUR >= ST - K
-
That means, one would lose money by exercising early instead of selling the option
-
If there are dividends, it may be optimal to exercise early, just prior to a dividend
-
It may be optimal to exercise a non-dividend-paying put option early if the underlying
stock price is sufficiently low
Example:
Call on non-dividend-paying stock:
CEUR = S4 + PEUR - Ke^(-r
36
CAMER > S1 - K
INTRODUCTION TO BINOMIAL OPTION PRICING
The binomial option pricing model enables us to determine the price of an option, given the
characteristics of the stock or other underlying asset
The binomial option pricing model assumes that the price of the underlying asset follows a
binomial distribution - that is, the asset price in each period can move only up or down by a
specified amount
The binomial model is often referred to as the “Cox–Ross-Rubinstein pricing model”
A ONE-PERIOD BINOMIAL TREE
Example:
Consider a European call option on the stock XYZ, with a $40 strike and 1 year to expiration.
-
XYZ does not pay dividends and its current price is $41
-
The continuously compounded risk-free interest rate is 8%
-
The following figure depicts possible stock prices over 1 year, i.e. a binomial tree:
COMPUTING THE OPTION PRICE
Next consider two portfolios:
-
37
Portfolio A: buy one call option
-
Portfolio B: buy ⅔ shares of XYZ and borrow $18.462 at the risk-free rate
Costs:
-
Portfolio A: the call premium, which is unknown
-
Portfolio B: ⅔ x $41 - $18.462 = $8.871
Payoffs:
Portfolios A and B have the same payoff. Therefore:
-
Portfolios A and B should have the same cost. Since Portfolio B costs $8.871, the price of
one option must be $8.871
-
There is a way to create the payoff to call by buying shares and borrowing. Portfolio B is a
synthetic call
-
One option has the risk of ⅔ shares. The value ⅔ is the delta (Δ) of the option: the number
of shares that replicates the option payoff
THE BINOMIAL SOLUTION
How do we find a replication portfolio consisting of Δ shares of stock and a dollar amount of B in
lending, such that the portfolio imitates the option whether the stock rises or falls?
-
Suppose that the stock has a continuous dividend yield of δ, which is reinvested in the
stock. Thus, if you buy one share at time t, at time t + h you will have e^δh shares
38
-
If the length of a period is h, the interest factor per period is e^rh
-
uS denotes the stock price when the price goes up, and dS denotes the stock price when
the price goes down
Stock price tree:
Corresponding tree for the value of the option:
Note that u (d) in the stock price tree is interpreted as one plus the rate of capital gain (loss) on
the stock if it goes up (down)
The value of the replicating portfolio at time h, with stock price Sh is:
At the prices Sh = uS and Sh = dS, a successful replicating portfolio will satisfy:
Solving for Δ and B gives:
The cost of creating the option is the net cash flow required to buy the shares and bonds. Thus,
the cost of the option is ΔS + B:
39
The no-arbitrage condition is:
RISK-NEUTRAL PRICING
We can interpret the terms
as
probabilities
Let:
Then the equation of the cost of the option ΔS + B can then be written as:
We call p* the risk-neutral probability of an increase in the stock price
CONSTRUCTING u AND d
In the absence of uncertainty, a stock must appreciate at the risk-free rate less the dividend yield.
Thus, from time t to time t + h, we have:
The stock price next period equals the forward price
With uncertainty, the stock price evolution is:
40
Where σ is the annualized standard deviation of the continuously compounded return, and
is standard deviation over period of length h
We can also rewrite the stock price evolution as:
We refer to a tree constructed using the above equations as a “forward tree”
SUMMARY
In order to price an option, we need to know:
-
Stock price
-
Strike price
-
Standard deviation of returns on the stock
-
Dividend yield
-
Risk-free rate
Using the risk-free rate and σ, we can approximate the future distribution of the stock by
creating a binomial tree using equations for the stock price evolution
Once we have the binomial tree, it is possible to price the option using the equation of ΔS + B
ONE-PERIOD EXAMPLE WITH A FORWARD TREE
Consider a European call option on a stock, with a $40 strike and 1 year to expiration. The stock
does not pay dividends, and its current price is $41. Suppose the volatility of the stock is 30%.
41
The continuously compounded risk-free interest rate is 8%
S = 41, r = 0.08, Div = 0, σ = 0.30 and h = 1
Use these inputs to:
-
Calculate the final stock prices
-
Calculate the final option values
-
Calculate Δ and B
-
Calculate the option price
Calculate the final stock prices:
Calculate the final option values
Calculate Δ and B
Calculate the option price
42
The following figure depicts the possible stock prices and option prices over 1 year, i.e. a
binomial tree
LECTURE 5
A TWO-PERIOD EUROPEAN CALL
We can extend the previous example to price a 2-year option, assuming all inputs are the same
as before
Note that an up move by the stock followed by a down move (Sud) generates the same stock
price as a down move followed by an up move (Sdu). This is called a recombining tree.
Otherwise, we would have a non recombining tree
Sud = Sdu = u x d x $41 = e^(0.08+0.3) x e^(0.08-0.3) x $41 = $48.114
43
PRICING THE CALL OPTION
To price an option with two binomial periods, we work backwards through the tree
-
Year 2, stock price = $87.669: since we are at expiration, the option value is max(0, S-K) =
$47.669
-
Year 2, stock price = $48.114: similarly, the option value is $8.114
-
Year 2, stock price = $26.405: since the option is out of money, the value is 0
-
Year 1, stock price = $59.954: at this node, we compute the option value using the
equation C = ΔS + B, where uS is $87.669 and dS is $48.114
-
Year 1, stock price = $32.903: again using the equation C = ΔS + B, the option value is
$3.187
-
Year 0, stock price = $41: similarly the option value is computed to be $10.737
Notice that:
-
The option price is greater for the 2-year than for the 1-year option
-
The option was priced by working backwards through the binomial tree
-
The option’s Δ and B are different at different nodes. At a given point in time, Δ increases
to 1 as we go further into the money
-
Permitting early exercise would make no difference. At every node prior to expiration, the
option price is greater than S-K; hence, we would not exercise even if the option had been
American
MANY BINOMIAL PERIODS
Dividing the time to expiration into more periods, allows us to generate a more realistic tree with
a larger number of different values at expiration
-
44
Consider the previous example of the 1-year European call option
-
Let there be 3 binomial periods. Since it is a 1-year call, this means that the length of a
period is h = ⅓
-
Assume that other inputs are the same as before (so, r=0.08 and σ=0.3)
The stock price and option price tree for this option
Note that since the length of the binomial period is shorter, u and d are smaller than before:
u=1.2212 and d=0.8637 (as opposed to 1.46 and 0.803 with h=1)
-
The second-period nodes are computed as follows:
-
The remaining nodes are computed similarly
Analogous to the procedure for pricing the 2-year option, the price of the three-period option is
computed by working backwards using equation C = ΔS + B
-
The option price is $7.074
PUT OPTIONS
45
We compute put option prices using the same stock price tree and in almost the same way as call
option prices
The only difference with a European put option occurs at expiration
-
Instead of computing the price as max(0, S-K, we use max (0, K-S)
A binomial tree for a European put option with 1-year to expiration:
BINOMIAL PRICING OF AMERICAN OPTIONS
The value of the option if it is left “alive” (i.e unexercised) is given by the value of holding it for
another period (equation C = ΔS + B)
The value of the option if it is exercised is given by max (0, S-K) if it is a call and max(0, K-S) if it is
a put
For an American call/put option, the value of the option at a node is given by:
46
At each node, we check for early exercise
If the value of the option is greater when exercised, we assign that value to the node. Otherwise,
we assign the value of the option unexercised
We work backward through the tree as usual
HOW REALISTIC IS THE CRR OPTION PRICING FRAMEWORK?
Assumption of only two possible states is clearly unrealistic
But: One period can be split into many periods
Nevertheless, we assume:
-
Volatility is constant
-
“Large” stock price movements do not occur
-
Returns are independent over time
OPTIONS ON DIFFERENT UNDERLYINGS
Pricing options with different underlying assets requires adjusting the risk-neutral probability for
the borrowing cost or lease rate of the underlying asset
Thus, we can use the formula for pricing an option on a stock with an appropriate substitution
for the dividend yield
47
BLACK-SCHOLES FORMULA FOR STOCKS
-
The Black-Scholes formula is a limiting case of the binomial formula (indefinitely many
periods) for the price of a European option
-
Consider an European call (or put) option written on a stock
-
Assume that the stock pays dividend at the continuous rate δ
Call option price:
Put option price:
Where:
THE N(x) FUNCTION
N(x) is the probability that a normally distributed variable with a mean of zero and a standard
deviation of 1 is less than x
48
UNDERSTANDING BLACK-SCHOLES
Call option price:
BLACK-SCHOLES ASSUMPTIONS
Assumptions about stock return distribution:
-
Continuously compounded returns on the stock are normally distributed and
independent over time (no “jumps”)
-
The volatility of continuously compounded returns is known and constant
-
Future dividends are known, either as dollar amount or as a fixed dividend yield
Assumptions about the economic environment:
49
-
The risk-free rate is known and constant
-
There are no transaction costs or taxes
-
It is possible to short-sell costlessly and to borrow at the risk-free rate
APPLYING THE FORMULA TO ALL OTHER ASSETS
OPTIONS ON STOCKS WITH DISCRETE DIVIDENDS
The prepaid forward price for stock with discrete dividends is:
Example:
-
S = $41, K = $40, σ = 0.3, r = 8%, T =0.25, Div= $3 in one month
-
PV(Div) = $3 * e^(-0.08*1/12) = $2.98
-
Use $41 - $2.98 = $38.02 as the stock price in the Black-Scholes formula
Example:
Compare to European call on stocks without dividends: $3.399
d1 =[ ln( S/Ke^(-rT)) + 1/2σ^2T ]/ [σ*sqr(T)] = 0.3725
d2 = 0.2229
N(d1) = N(0.3725) = N(0.37) + 0.25 * (N(0.37) - N(0.38)) = 0.6443 + 0.25*(0.6443 - 0.6480) = 0.6434
N(d2) = N(0.2229) = 0.5882
50
C = 41 * 0.6434 - 40* 0.5882 = 3.3553
OPTIONS ON CURRENCIES
The prepaid forward price for the currency is:
Where x0 is domestic spot rate and rf is foreign
interest rate
OPTION ON FUTURES
The prepaid forward price for a futures contract is the PV of the futures prices
Therefore:
Where:
OPTION GREEKS
What happens to the option price when one and only one input changes?
-
Delta (Δ): change in option price when stock price increases by $1
-
Gamma: (Γ): change in delta when option price increase by $1
-
Vega: change in option price when volatility increases by 1%
-
Theta (θ): change in option price when time to maturity decreases by 1 day
-
Rho (ρ): change in option price when interest rate increases by 1%
Greek measures for portfolios:
-
The Greek measure of a portfolio is weighted average of Greeks of individual portfolio
components:
51
DELTA
-
Definition: The number of shares in the portfolio that replicates the option
-
Stock with continuous dividend yield:
-
Holding Δ in shares and borrow Ke^(-rT)N(d2) costs Se^(-δT)N(d1) - Ke^(-rT)N(d2)
-
As Δ changes with the stock price, replicating portfolio changes and must be adjusted
dynamically
SUMMARY
-
The binomial option pricing framework is a simple, yet rich approach to value option
prices
-
The Black-Scholes model provides us with the most-widely used formula to calculate the
theoretical value of European-style options
-
Greeks measure the sensitivity of the option prices with respect to a number of
determinants
52
LECTURE 6
IMPLIED VOLATILITY
-
Volatility is unobservable
-
Option prices, particularly for near-the-money options, can be quite sensitive to volatility
-
One approach is to compute historical volatility using the history of returns
-
A problem with historical volatility is that expected future volatility can be different from
historical volatility
-
Alternatively, we can calculate implied volatility, which is the volatility that when put into
a pricing formula (typically Black-Scholes), yields the observed option price
In practice, we extract the value of σ that fits the observed option price and other determinants
in the BS formula
Note that implied volatilities of in-, at-, and out-of the money options are generally different
-
A volatility smile refers to when volatility is symmetric, with volatility lowest for
at-the-money options and high for in-the-money and out-of-the-money
-
A difference in volatilities between in-the-money and out-of-the-money is referred to as a
volatility skew
CBOE VOLATILITY INDEX, VIX
It is a measure of expected volatility of the US stock market, derived from real-time, mid-quote
prices of S&P 500 index call and put options
It is one of the most recognized measures of volatility - widely reported by financial media and
closely followed by a variety of market participants as a daily market indicator
53
PRACTICAL USE OF IMPLIED VOLATILITY
Some practical uses of implied volatility include:
-
Implied volatility serves as an important financial market indicator
-
Use the implied volatility from an option with an observable price to calculate the price of
another option on the same underlying assets
-
Use implied volatility as a quick way to describe the level of options prices on a given
underlying asset: you could quote option prices in terms of volatility, rather than as a
dollar price
-
Checking the uniformity of implied volatilities across various options on the same
underlying assets allows one to verify the validity of the pricing model in pricing these
options
PERPETUAL AMERICAN OPTIONS
Because of the possibility of early exercise, it is sensible to define American options with infinite
maturity
They are called perpetual options or expiration less options
Because there are no obstacles produced by the finite horizon (maturity time), the valuation
formula is available for these options
To price the options, we need to start by establishing the conditions for optimal early exercise:
Perpetual American options are optimally exercised when the underlying asset reaches the
optimal exercise barrier HC or HP
Perpetual American option (options that never expire) are optimally exercised when the
underlying asset ever reaches the optimal exercise barrier (HC for a call and HP for a put)
For a perpetual call option the optimal exercise barrier and prices are:
54
REAL OPTIONS
Real options is the application of derivatives theory to the operation and valuation of real
investment projects
-
A call option is the right to pay a strike price to receive the present value of a stream of
future cash flows
-
An investment project is the right to pay an investment cost to receive the present value
of a stream of future cash flows
INVESTMENT AND THE NPV RULE
NPV rule:
-
Compute the NPV by discounting expected cash flows at the opportunity cost of capital
-
Accept a project if and only if its NPV is positive and it exceeds the NPV of all mutually
exclusive alternative projects
INVESTMENT UNDER CERTAINTY
55
Perpetuity: is a constant stream of identical cash flows until infinity
PV = C / (1 + r) + C / (1+r)^2 + … + C/ (1+r)^T = C / r
Perpetuity of constant growth: is a constant stream of growing cash flows (growing at g) until
infinity
PV = C / (1 + r) + C (1+g) / (1+r)^2 + … + C (1+g)/ (1+r)^T = C / (r - g)
Example: Invest in a $10 machine, that will produce one widget a year forever at a cost of $0.90
per widget
The price of the widget will be $0.55 next year and will increase at 4% per year. The risk-free rate
is 5% per year. We can invest, at any time, in one such machine. Should we invest and if so,
when?
-
Static NPV:
A natural question that arises from the calculation above: What is the optimal time to wait with the
investment?
Solution: Treat the project as an option
The decision to invest is analogous to the decision to exercise an American option early
-
Exercise price ~ investment cost
-
Underlying asset price ~ value of the project
Trade-off between 3 factors:
1) Dividends foregone by not exercising: cash flow from selling widgets
2) Interest saved by deferring the payment of exercise price: the value of delaying the
marginal widget cost is interest
56
3) Value of the insurance lost by exercising (the implicit put option): since no uncertainty,
there is no insurance value
Formulas:
S = CF / (r-g)
K = Investment cost + Marginal cost/ r
r = ln (1+r)
δ = ln (1+r) - ln (1+g)
We are using the Perpetual American options method to calculate the price of the
option/Investments (C):
Example (continues):
Invest when the widget price equals the investment trigger price of $1.472. We reach this price
after about 24.32 years
SUMMARY
57
-
Implied volatility is an important indicator variable in financial economics
-
Option pricing models help us to extract implied volatility from observed prices
-
Different assets yield different patterns in implied volatility
-
Option pricing formulas are also essential for investment decisions of companies