Risk Management

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Risk Management

• Dr. Keith M. Howe
• Summer 2008

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Definition
Risk and uncertainty
Risk management
Risk aversion
The process of formulating the benefit-cost
trade-offs of risk reduction and deciding on the
course of action to take (including the decision
to take no action at all).
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Two more definitions

• Derivatives
• financial assets (e.g., stock option, futures,
  forwards, etc) whose values depend upon the value
  of the underlying assets.
• Hedge
• the use of financial instruments or of other
  tools to reduce exposure to a risk factor.

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Figure 1.2. Gains and losses from buying shares
and a call option on Risky Upside Inc.
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Panel B. Forward contract payoff.
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Panel C. Hedged firm income.
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Panel D. Comparison of income with put contract
and income with forward contract.
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Risk management irrelevance proposition

• Bottom line hedging a risk does not increase
  firm value when the cost of bearing the risk is
  the same whether the risk is borne within the
  firm or outside the firm by the capital markets.
• This proposition holds when financial markets are
  perfect.

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Risk management irrelevance proposition

• Allows us to find out when homemade risk
  management is not equivalent to risk management
  by the firm.
• This is the case whenever risk management by a
  firm affects firm value in a way that investors
  cannot mimic.
• For risk management to increase firm value, it
  must be more expensive to take a risk within the
  firm than to pay the capital markets to take it.

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Role of risk management

• Risk management can add value to the firm by
• Decreasing taxes
• Decreasing transaction costs (including
  bankruptcy costs)
• Avoiding investment decision errors

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Bankruptcy costs and costs of financial distress
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• Costs incurred as a result of a bankruptcy filing
  are called bankruptcy costs.
• The extent to which bankruptcy costs affect firm
  value depends on their extent and on the
  probability that the firm will have to file for
  bankruptcy.
• The probability that a firm will be bankrupt is
  the probability that it will not have enough cash
  flow to repay the debt.

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• Direct bankruptcy costs
• Average ratio of direct bankruptcy costs to total
  assets 2.8
• Indirect bankruptcy costs
• Many of these indirect costs start accruing as
  soon as a firms financial situation becomes
  unhealthy, called costs of financial distress
• Managers of a firm in bankruptcy lose control of
  some decisions. They might not allowed to
  undertake costly new projects, for example.

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Figure 3.1. Cash flow to shareholders and
operating cash flow.
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Figure 3.2. Creating the unhedged firm out of the
hedged firm.
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Figure 3.3. Cash flow to claimholders and
bankruptcy costs.
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Analysis of decreasing transaction cost by
hedging
Value of firm unhedged PV (C Bankruptcy
costs) PV (C) PV (Bankruptcy costs)
value of firm without bankruptcy costs
PV (bankruptcy costs)
Gain from risk management value of firm hedged
value of firm unhedged PV( bankruptcy costs)
Value of firm unhedged gain from risk
management value of firm hedged value of
firm without bankruptcy costs
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Taxes and risk management
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• Tax rationale for risk management If it moves a
  dollar away from a possible outcome in which the
  taxpayer is subject to a high tax rate and shifts
  it to a possible outcome where the taxpayer
  incurs a low tax rate, a firm or an investor
  reduces the present value of taxes to be paid. It
  applies whenever income is taxed differently at
  different levels.

- Carrybacks and carryforwards - Tax shields -
Personal taxes
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Example
The firm pays taxes at the rate of 50 percent on
cash flow in excess of 300 per ounce. For
simplicity, the price of fold is either 250 or
450 with Equal probability. The forward price is
350.
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Optimal capital structure and risk management
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• In general, firms cannot eliminate all risk, debt
  is risky.
• By having more debt, firms increase their tax
  shield from debt but increase the present value
  of costs of financial distress.
• The optimal capital structure of a firm
• Balances the tax benefits of debt against the
  costs of financial distress.
• Through risk management
• A firm can reduce the present value of the costs
  of financial distress by making financial
  distress less likely.
• As a result, it can take on more debt.

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Should the firm hedge to reduce the risk of large
undiversified shareholders?

• Large undiversified shareholders can increase
  firm value
• Risk and the incentives of managers
• Large shareholders, managerial incentives, and
  homestake

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Figure 3.6. Firm after-tax cash flow and debt
issue.
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Risk management process
Risk identification
Risk assessment
Review
Selection of risk-mgt techniques
Implementation
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The rules of risk management
Risk Management

• There is no return without risk

• Be transparent

• Seek experience

• Know what you dont know

• Communicate

• Diversify

• Show discipline

• Use common sense

• Get a RiskGrade

Source Riskmetrics Group (www.riskmetrics.com)
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Types of risks firms face
Hazard risk - physical damage - liabilities -
business interruption
Market risk - interest rate - foreign exchange
- commodity price
Strategic risk - competition - reputation -
investor support
Operational risk - industry sectors -
geographical regions
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Assignment of risk responsibilities
CEO Strategic risk management
CRO
Hazard risk management
Operational risk management
Market risk management
Hedgeable
Insurable
Diversifiable
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Three dimensions of risk transfer

• Hedging
• Insuring
• Diversifying

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A new concept of risk management (VAR)

• Value-at-risk (VAR) is a category of risk
  measures that describe probabilistically the
  market risk of mostly a trading portfolio.
• It summarizes the predicted maximum loss (or
  worst loss) over a target horizon within a given
  confidence interval.
• If the portfolio return is normally distributed,
  has zero mean, and has volatility s over the
  measurement period, the 5 percent VAR of the
  portfolio is

VAR 1.65 X s X Portfolio value
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Example of VAR

• The US bank J.P. Morgan states in its 2000 annual
  report that its aggregate VAR is about 22m.
• The bank, one of the pioneers in risk management,
  may say that for 95 percent of the time it does
  not expect to lose more than 22m on a given day.

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More on VAR

• The main appeal of VAR was to describe risk in
  dollars - or whatever base currency is used -
  making it far more transparent and easier to
  grasp than previous measures.
• VAR also represents the amount of economic
  capital necessary to support a business, which is
  an essential component of economic value added
  measures.
• VAR has become the standard benchmark for
  measuring financial risk.

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Instruments used in risk management

• Forward contracts
• Futures contracts
• Hedging
• Interest rate futures contracts
• Duration hedging
• Swap contracts
• Options

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Forward Contracts

• A forward contract specifies that a certain
  commodity will be exchanged for another at a
  specified time in the future at prices specified
  today.
• Its not an option both parties are expected to
  hold up their end of the deal.
• If you have ever ordered a textbook that was not
  in stock, you have entered into a forward
  contract.

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Example
Suppose SP index price is 1050 in 6 months. A
holder who entered a long position at a forward
price of 1020 is obligated to pay 1020 to
acquire the index, and hence earns 1050 - 1020
30 per unit of the index. The short is
likewise obligated to sell for 1020, and thus
loses 30.
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Payoff after 6 months
If the index price in 6 months 1020, both the
long and short have a 0 payoff. If the index
price gt 1020, the long makes money and the short
loses money. If the index price lt 1020, the long
loses money and the short makes money.
SR Index SR
Forward in 6 months long
short 900
-120 120 950
-70 70 1000
-20 20 1020
0
0 1050 30
-30 1100
80 -80
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Problem The current SP index is 1000. You have
just purchased a 6- month forward with a price of
1100. If the index in 6 months has appreciated
by 7, what is the payoff of this position?
Solution F01100 S110001.071070
Payoff 1070-1100 - 30.
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Example Valuing a Forward Contract on a Share of
Stock

• Consider the obligation to buy a share of
  Microsoft stock one year from now for 100.
  Assume that the stock currently sells for 97 per
  share and that Microsoft will pay no dividends
  over the coming year. One-year zero-coupon bonds
  that pay 100 one year from now currently sell
  for 92. At what price are you willing to buy or
  sell this obligation?

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Valuing a forward contract
Strategy 1---- the forward contract
One year from now
Today
Buy stock at a price of 100. Sell the share for
cash at market
Buy a forward contract
Strategy 2 ---- the portfolio strategy
Today
One year from now
Buy stock today Sell short 100 in face value of
1-year zero-coupon bonds
Sell the stock Buyback the zero-coupon bonds of
100
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Valuing a forward contract
Cost Today
Cash flow one year from now
? 97-92
S1- 100 S1- 100
Strategy 1 Strategy 2
Since strategies 1 and 2 have identical cash
flows in the future, they should have the same
cost today to prevent arbitrage.
? 97 - 92 5 In
strategy 1, the obligation to buy the stock for
100 one year from now, should cost 5.
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Valuing a forward contract
The no-arbitrage value of a forward contract on a
share of stock (the obligation to buy a share of
stock at a price of K, T years in the future),
assuming the stock pays no dividends prior to T,
is where S0 current price of the stock
the current market price of a
default-free zero-coupon bond paying K, T years
in the future
At no arbitrage
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Currency Forward Rates

• Currency forward rates are a variation on forward
  price of stock.
• In the absence of arbitrage, the forward currency
  rate F0 (for example, Euros/dollar) is related to
  the current exchange rate (or spot rate) S0, by
  the equation
• where r the return (unannualized) on a domestic
  or foreign risk-free security over the life of
  the forward agreement, as measured in the
  respective country's currency

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Forward Currency Rates

• Example The Relation Between Forward
  Currency Rates and Interest Rates
• Assume that six-month LIBOR on Canadian funds is
  4 percent and the US Eurodollar rate (six-month
  LIBOR on U.S. funds) is 10 percent and that both
  rates are default free. What is the six-month
  forward Can/US exchange rate if the current
  spot rate is Can1.25/US? Assume that six months
  from now is 182 days.

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Currency Forward Rates
Answer (LIBOR is a zero-coupon rate based on an
actual/360 day count.) So
Canada United
States Six-month interest Rate (unannualized)
The forward rate is
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Futures Contracts Preliminaries

• A futures contract is like a forward contract
• It specifies that a certain commodity will be
  exchanged for another at a specified time in the
  future at prices specified today.
• A futures contract is different from a forward
• Futures are standardized contracts trading on
  organized exchanges with daily resettlement
  (marking to market) through a clearinghouse.

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Futures Contracts Preliminaries

• Standardizing Features
• Contract Size
• Delivery Month
• Daily resettlement
• Minimizes the chance of default
• Initial Margin
• About 4 of contract value, cash or T-bills held
  in a street name at your brokerage.

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Daily Resettlement An Example

• Suppose you want to speculate on a rise in the
  / exchange rate (specifically you think that
  the dollar will appreciate).

Currently 1 140.
The 3-month forward price is 1150.
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Daily Resettlement An Example

• Currently 1 140 and it appears that the
  dollar is strengthening.
• If you enter into a 3-month futures contract to
  sell at the rate of 1 150 you will make
  money if the yen depreciates. The contract size
  is 12,500,000
• Your initial margin is 4 of the contract value

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Daily Resettlement An Example

• If tomorrow, the futures rate closes at 1
  149, then your positions value drops.
• Your original agreement was to sell 12,500,000
  and receive 83,333.33

But 12,500,000 is now worth 83,892.62
You have lost 559.28 overnight.
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Daily Resettlement An Example

• The 559.28 comes out of your 3,333.33 margin
  account, leaving 2,774.05
• This is short of the 3,355.70 required for a new
  position.

Your broker will let you slide until you run
through your maintenance margin. Then you must
post additional funds or your position will be
closed out. This is usually done with a reversing
trade.
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Selected Futures Contracts
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Futures Markets

• The Chicago Mercantile Exchange (CME) is by far
  the largest.
• Others include
• The Philadelphia Board of Trade (PBOT)
• The MidAmerica Commodities Exchange
• The Tokyo International Financial Futures
  Exchange
• The London International Financial Futures
  Exchange

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The Chicago Mercantile Exchange

• Expiry cycle March, June, September, December.
• Delivery date 3rd Wednesday of delivery month.
• Last trading day is the second business day
  preceding the delivery day.
• CME hours 720 a.m. to 200 p.m. CST.

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CME After Hours

• Extended-hours trading on GLOBEX runs from 230
  p.m. to 400 p.m dinner break and then back at it
  from 600 p.m. to 600 a.m. CST.
• Singapore International Monetary Exchange (SIMEX)
  offer interchangeable contracts.
• Theres other markets, but none are close to CME
  and SIMEX trading volume.

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Wall Street Journal Futures Price Quotes
Highest price that day
Highest and lowest prices over the lifetime of
the contract.
Opening price
Closing price
Daily Change
Lowest price that day
Number of open contracts
Expiry month
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Basic Currency Futures Relationships

• Open Interest refers to the number of contracts
  outstanding for a particular delivery month.
• Open interest is a good proxy for demand for a
  contract.
• Some refer to open interest as the depth of the
  market. The breadth of the market would be how
  many different contracts (expiry month, currency)
  are outstanding.

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Hedging

• Two counterparties with offsetting risks can
  eliminate risk.
• For example, if a wheat farmer and a flour mill
  enter into a forward contract, they can eliminate
  the risk each other faces regarding the future
  price of wheat.
• Hedgers can also transfer price risk to
  speculators and speculators absorb price risk
  from hedgers.
• Speculating Long vs. Short

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Hedging and Speculating Example

• You speculate that copper will go up in price, so
  you go long 10 copper contracts for delivery in 3
  months. A contract is 25,000 pounds in cents per
  pound and is at 0.70 per pound or 17,500 per
  contract.
• If futures prices rise by 5 cents, you will gain
• Gain 25,000 .05 10 12,500
• If prices decrease by 5 cents, your loss is
• Loss 25,000 -.05 10 -12,500

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Hedging How many contacts?

• You are a farmer and you will harvest 50,000
  bushels of corn in 3 months. You want to hedge
  against a price decrease. Corn is quoted in cents
  per bushel at 5,000 bushels per contract. It is
  currently at 2.30 cents for a contract 3 months
  out and the spot price is 2.05.
• To hedge you will sell 10 corn futures contracts

Now you can quit worrying about the price of corn
and get back to worrying about the weather.
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Interest Rate Futures Contracts
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Pricing of Treasury Bonds

• Consider a Treasury bond that pays a semiannual
  coupon of C for the next T years
• The yield to maturity is r

Value of the T-bond under a flat term structure
PV of face value PV of coupon payments
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Pricing of Treasury Bonds

• If the term structure of interest rates is not
  flat, then we need to discount the payments at
  different rates depending upon maturity

PV of face value PV of coupon payments
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Pricing of Forward Contracts

• An N-period forward contract on that T-Bond

Can be valued as the present value of the forward
price.
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Futures Contracts

• The pricing equation given above will be a good
  approximation.
• The only real difference is the daily
  resettlement.

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Hedging in Interest Rate Futures

• A mortgage lender who has agreed to loan money in
  the future at prices set today can hedge by
  selling those mortgages forward.
• It may be difficult to find a counterparty in the
  forward who wants the precise mix of risk,
  maturity, and size.
• Its likely to be easier and cheaper to use
  interest rate futures contracts however.

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Duration Hedging

• As an alternative to hedging with futures or
  forwards, one can hedge by matching the interest
  rate risk of assets with the interest rate risk
  of liabilities.
• Duration is the key to measuring interest rate
  risk.

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Duration Hedging

• Duration measures the combined effect of
  maturity, coupon rate, and YTM on bonds price
  sensitivity
• Measure of the bonds effective maturity
• Measure of the average life of the security
• Weighted average maturity of the bonds cash flows

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Duration Formula
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Calculating Duration
Calculate the duration of a three-year bond that
pays a semi-annual coupon of 40, has a 1,000
par value when the YTM is 8 semiannually?
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Calculating Duration
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Duration

• The key to bond portfolio management
• Properties
• Longer maturity, longer duration
• Duration increases at a decreasing rate
• Higher coupon, shorter duration
• Higher yield, shorter duration
• Zero coupon bond duration maturity

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Swaps Contracts Definitions

• In a swap, two counterparties agree to a
  contractual arrangement wherein they agree to
  exchange cash flows at periodic intervals.
• There are two types of interest rate swaps
• Single currency interest rate swap
• Plain vanilla fixed-for-floating swaps are
  often just called interest rate swaps.
• Cross-Currency interest rate swap
• This is often called a currency swap fixed for
  fixed rate debt service in two (or more)
  currencies.

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The Swap Bank

• A swap bank is a generic term to describe a
  financial institution that facilitates swaps
  between counterparties.
• The swap bank can serve as either a broker or a
  dealer.
• As a broker, the swap bank matches counterparties
  but does not assume any of the risks of the swap.
• As a dealer, the swap bank stands ready to accept
  either side of a currency swap, and then later
  lay off their risk, or match it with a
  counterparty.

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An Example of an Interest Rate Swap

• Consider this example of a plain vanilla
  interest rate swap.
• Bank A is a AAA-rated international bank located
  in the U.K. and wishes to raise 10,000,000 to
  finance floating-rate Eurodollar loans.
• Bank A is considering issuing 5-year fixed-rate
  Eurodollar bonds at 10 percent.
• It would make more sense to for the bank to issue
  floating-rate notes at LIBOR to finance
  floating-rate Eurodollar loans.

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An Example of an Interest Rate Swap

• Firm B is a BBB-rated U.S. company. It needs
  10,000,000 to finance an investment with a
  five-year economic life.
• Firm B is considering issuing 5-year fixed-rate
  Eurodollar bonds at 11.75 percent.
• Alternatively, firm B can raise the money by
  issuing 5-year floating-rate notes at LIBOR ½
  percent.
• Firm B would prefer to borrow at a fixed rate.

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An Example of an Interest Rate Swap

• The borrowing opportunities of the two firms are

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An Example of an Interest Rate Swap
The swap bank makes this offer to Bank A You pay
LIBOR 1/8 per year on 10 million for 5 years
and we will pay you 10 3/8 on 10 million for 5
years
Swap Bank
Bank A
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An Example of an Interest Rate Swap
½ of 10,000,000 50,000. Thats quite a cost
savings per year for 5 years.
Heres whats in it for Bank A They can borrow
externally at 10 fixed and have a net borrowing
position of -10 3/8 10 (LIBOR 1/8) LIBOR
½ which is ½ better than they can borrow
floating without a swap.
Swap Bank
Bank A
10
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An Example of an Interest Rate Swap
The swap bank makes this offer to company B You
pay us 10½ per year on 10 million for 5 years
and we will pay you LIBOR ¼ per year on 10
million for 5 years.
Swap Bank
Company B
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An Example of an Interest Rate Swap
Heres whats in it for B
½ of 10,000,000 50,000 thats quite a cost
savings per year for 5 years.
Swap Bank
They can borrow externally at LIBOR ½ and
have a net borrowing position of 10½ (LIBOR
½ ) - (LIBOR - ¼ ) 11.25 which is ½ better
than they can borrow floating.
Company B
LIBOR ½
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An Example of an Interest Rate Swap
The swap bank makes money too.
¼ of 10 million 25,000 per year for 5 years.
Swap Bank
Company B
Bank A
LIBOR 1/8 LIBOR ¼ 1/8 10 ½ - 10 3/8
1/8 ¼
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An Example of an Interest Rate Swap
The swap bank makes ¼
Swap Bank
Company B
Bank A
B saves ½
A saves ½
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An Example of a Currency Swap

• Suppose a U.S. MNC wants to finance a 10,000,000
  expansion of a British plant.
• They could borrow dollars in the U.S. where they
  are well known and exchange for dollars for
  pounds.
• This will give them exchange rate risk financing
  a sterling project with dollars.
• They could borrow pounds in the international
  bond market, but pay a premium since they are not
  as well known abroad.

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An Example of a Currency Swap

• If they can find a British MNC with a
  mirror-image financing need they may both benefit
  from a swap.
• If the spot exchange rate is S0(/) 1.60/,
  the U.S. firm needs to find a British firm
  wanting to finance dollar borrowing in the amount
  of 16,000,000.

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An Example of a Currency Swap

• Consider two firms A and B firm A is a
  U.S.based multinational and firm B is a
  U.K.based multinational.
• Both firms wish to finance a project in each
  others country of the same size. Their borrowing
  opportunities are given in the table below.

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An Example of a Currency Swap
Swap Bank
Firm B
Firm A
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An Example of a Currency Swap
As net position is to borrow at 11
Swap Bank
9.4
Firm B
Firm A
8
12
A saves .6
90
An Example of a Currency Swap
Bs net position is to borrow at 9.4
Swap Bank
9.4
Firm B
Firm A
8
12
B saves .6
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An Example of a Currency Swap
The swap bank makes money too
1.4 of 16 million financed with 1 of 10
million per year for 5 years.
Swap Bank
9.4
Firm B
Firm A
8
At S0(/) 1.60/, that is a gain of 124,000
per year for 5 years.
12
The swap bank faces exchange rate risk, but maybe
they can lay it off (in another swap).
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Variations of Basic Swaps

• Currency Swaps
• fixed for fixed
• fixed for floating
• floating for floating
• amortizing
• Interest Rate Swaps
• zero-for floating
• floating for floating
• Exotica
• For a swap to be possible, two humans must like
  the idea. Beyond that, creativity is the only
  limit.

93
Risks of Interest Rate and Currency Swaps

• Interest Rate Risk
• Interest rates might move against the swap bank
  after it has only gotten half of a swap on the
  books, or if it has an unhedged position.
• Basis Risk
• If the floating rates of the two counterparties
  are not pegged to the same index.
• Exchange Rate Risk
• In the example of a currency swap given earlier,
  the swap bank would be worse off if the pound
  appreciated.

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Risks of Interest Rate and Currency Swaps

• Credit Risk
• This is the major risk faced by a swap dealerthe
  risk that a counter party will default on its end
  of the swap.
• Mismatch Risk
• Its hard to find a counterparty that wants to
  borrow the right amount of money for the right
  amount of time.
• Sovereign Risk
• The risk that a country will impose exchange rate
  restrictions that will interfere with performance
  on the swap.

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Pricing a Swap

• A swap is a derivative security so it can be
  priced in terms of the underlying assets
• How to
• Plain vanilla fixed for floating swap gets valued
  just like a bond.
• Currency swap gets valued just like a nest of
  currency futures.

96
Options

• Many corporate securities are similar to the
  stock options that are traded on organized
  exchanges.
• Almost every issue of corporate stocks and bonds
  has option features.
• In addition, capital structure and capital
  budgeting decisions can be viewed in terms of
  options.

97
Options Contracts Preliminaries

• An option gives the holder the right, but not the
  obligation, to buy or sell a given quantity of an
  asset on (or perhaps before) a given date, at
  prices agreed upon today.
• Calls versus Puts
• Call options gives the holder the right, but not
  the obligation, to buy a given quantity of some
  asset at some time in the future, at prices
  agreed upon today. When exercising a call option,
  you call in the asset.
• Put options gives the holder the right, but not
  the obligation, to sell a given quantity of an
  asset at some time in the future, at prices
  agreed upon today. When exercising a put, you
  put the asset to someone.

98
Options Contracts Preliminaries

• Exercising the Option
• The act of buying or selling the underlying asset
  through the option contract.
• Strike Price or Exercise Price
• Refers to the fixed price in the option contract
  at which the holder can buy or sell the
  underlying asset.
• Expiry
• The maturity date of the option is referred to as
  the expiration date, or the expiry.
• European versus American options
• European options can be exercised only at expiry.
• American options can be exercised at any time up
  to expiry.

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Options Contracts Preliminaries

• In-the-Money
• The exercise price is less than the spot price of
  the underlying asset.
• At-the-Money
• The exercise price is equal to the spot price of
  the underlying asset.
• Out-of-the-Money
• The exercise price is more than the spot price of
  the underlying asset.

100
Options Contracts Preliminaries

• Intrinsic Value
• The difference between the exercise price of the
  option and the spot price of the underlying
  asset.
• Speculative Value
• The difference between the option premium and the
  intrinsic value of the option.

Option Premium
Intrinsic Value
Speculative Value


101
Call Options

• Call options gives the holder the right, but not
  the obligation, to buy a given quantity of some
  asset on or before some time in the future, at
  prices agreed upon today.
• When exercising a call option, you call in the
  asset.

102
Basic Call Option Pricing Relationships at Expiry

• At expiry, an American call option is worth the
  same as a European option with the same
  characteristics.
• If the call is in-the-money, it is worth ST - E.
• If the call is out-of-the-money, it is worthless.
• CaT CeT MaxST - E, 0
• Where
• ST is the value of the stock at expiry (time T)
• E is the exercise price.
• CaT is the value of an American call at expiry
• CeT is the value of a European call at expiry

103
Call Option Payoffs
60
40
Buy a call
20
0
Option payoffs ()
100
90
80
70
60
0
10
20
30
40
50
Stock price ()
-20
-40
-60
Exercise price 50
104
Call Option Payoffs
Write a call
Exercise price 50
105
Call Option Profits
Buy a call
Write a call
Exercise price 50 option premium 10
106
Put Options

• Put options gives the holder the right, but not
  the obligation, to sell a given quantity of an
  asset on or before some time in the future, at
  prices agreed upon today.
• When exercising a put, you put the asset to
  someone.

107
Basic Put Option Pricing Relationships at Expiry

• At expiry, an American put option is worth the
  same as a European option with the same
  characteristics.
• If the put is in-the-money, it is worth E - ST.
• If the put is out-of-the-money, it is worthless.
• PaT PeT MaxE - ST, 0

108
Put Option Payoffs
60
40
Buy a put
20
0
Option payoffs ()
100
90
80
70
60
0
10
20
30
40
50
Stock price ()
-20
-40
-60
Exercise price 50
109
Put Option Payoffs
60
40
20
0
Option payoffs ()
100
90
80
70
60
0
10
20
30
40
50
Stock price ()
-20
-40
write a put
-60
Exercise price 50
110
Put Option Profits
60
Option profits ()
40
20
Write a put
10
0
100
90
80
70
60
0
10
20
30
40
50
-10
Buy a put
Stock price ()
-20
-40
-60
Exercise price 50 option premium 10
111
Selling Options

• The seller (or writer) of an option has an
  obligation.

• The purchaser of an option has an option.

112
Reading The Wall Street Journal
113
Reading The Wall Street Journal
This option has a strike price of 135
a recent price for the stock is 138.25
July is the expiration month
114
Reading The Wall Street Journal
This makes a call option with this exercise price
in-the-money by 3.25 138¼ 135.
Puts with this exercise price are
out-of-the-money.
115
Reading The Wall Street Journal
On this day, 2,365 call options with this
exercise price were traded.
116
Reading The Wall Street Journal
The CALL option with a strike price of 135 is
trading for 4.75.
Since the option is on 100 shares of stock,
buying this option would cost 475 plus
commissions.
117
Reading The Wall Street Journal
On this day, 2,431 put options with this exercise
price were traded.
118
Reading The Wall Street Journal
The PUT option with a strike price of 135 is
trading for .8125.
Since the option is on 100 shares of stock,
buying this option would cost 81.25 plus
commissions.
119
Combinations of Options

• Puts and calls can serve as the building blocks
  for more complex option contracts.
• If you understand this, you can become a
  financial engineer, tailoring the risk-return
  profile to meet your clients needs.

120
Protective Put Strategy Buy a Put and Buy the
Underlying Stock Payoffs at Expiry
Value at expiry
Protective Put strategy has downside protection
and upside potential
50
Buy the stock
Buy a put with an exercise price of 50
0
Value of stock at expiry
50
121
Protective Put Strategy Profits
Value at expiry
Buy the stock at 40
40
Protective Put strategy has downside protection
and upside potential
0
Buy a put with exercise price of 50 for 10
40
50
-40
Value of stock at expiry
122
Covered Call Strategy
Value at expiry
Buy the stock at 40
40
Covered call
10
0
Value of stock at expiry
40
30
50
Sell a call with exercise price of 50 for 10
-30
-40
123
Long Straddle Buy a Call and a Put
Value at expiry
Buy a call with an exercise price of 50 for 10
40
30
0
-10
Buy a put with an exercise price of 50 for 10
-20
40
50
60
30
70
Value of stock at expiry
A Long Straddle only makes money if the stock
price moves 20 away from 50.
124
Short Straddle Sell a Call and a Put
Value at expiry
A Short Straddle only loses money if the stock
price moves 20 away from 50.
20
Sell a put with exercise price of 50 for 10
10
0
Value of stock at expiry
40
50
60
30
70
-30
Sell a call with an exercise price of 50 for 10
-40
125
Long Call Spread
Value at expiry
Buy a call with an exercise price of 50 for 10
5
long call spread
0
-5
-10
Value of stock at expiry
50
60
55
Sell a call with exercise price of 55 for 5
126
Put-Call Parity
In market equilibrium, it mast be the case that
option prices are set such that
Otherwise, riskless portfolios with positive
payoffs exist.
Buy the stock at 40
Value at expiry
Buy the stock at 40 financed with some debt FV
X
Buy a call option with an exercise price of 40
Sell a put with an exercise price of 40
0
Value of stock at expiry
40
-40-P0
40-P0
-40
127
Valuing Options

• The last section concerned itself with the value
  of an option at expiry.

• This section considers the value of an option
  prior to the expiration date.
• A much more interesting question.

128
Option Value Determinants

• Call Put
• Stock price
• Exercise price
• Interest rate
• Volatility in the stock price
• Expiration date
• The value of a call option C0 must fall within
• max (S0 E, 0) lt C0 lt S0.
• The precise position will depend on these factors.

129
Market Value, Time Value and Intrinsic Value for
an American Call
The value of a call option C0 must fall within
max (S0 E, 0) lt C0 lt S0.
Profit
ST
ST - E

• CaT gt MaxST - E, 0

Market Value
Time value
Intrinsic value
ST
E
loss
Out-of-the-money
In-the-money
130
An Option-Pricing Formula

• We will start with a binomial option pricing
  formula to build our intuition.

• Then we will graduate to the normal approximation
  to the binomial for some real-world option
  valuation.

131
Binomial Option Pricing Model

• Suppose a stock is worth 25 today and in one
  period will either be worth 15 more or 15 less.
  S0 25 today and in one year S1is either 28.75
  or 21.25. The risk-free rate is 5. What is the
  value of an at-the-money call option?

S0
S1
28.75
25
21.25
132
Binomial Option Pricing Model

• A call option on this stock with exercise price
  of 25 will have the following payoffs.
• We can replicate the payoffs of the call option.
  With a levered position in the stock.

S0
S1
C1
28.75
3.75
25
21.25
0
133
Binomial Option Pricing Model

• Borrow the present value of 21.25 today and buy
  1 share.
• The net payoff for this levered equity portfolio
  in one period is either 7.50 or 0.
• The levered equity portfolio has twice the
  options payoff so the portfolio is worth twice
  the call option value.

S1
S0
debt
portfolio
C1
( - )
- 21.25
7.50
28.75

3.75
25
21.25
- 21.25
0

0
134
Binomial Option Pricing Model

• The levered equity portfolio value today is
  todays value of one share less the present value
  of a 21.25 debt

S1
S0
debt
portfolio
C1
( - )
- 21.25
7.50
28.75

3.75
25
21.25
- 21.25
0

0
135
Binomial Option Pricing Model

• We can value the option today as half of the
  value of the levered equity portfolio

S1
S0
debt
portfolio
C1
( - )
- 21.25
7.50
28.75

3.75
25
21.25
- 21.25
0

0
136
The Binomial Option Pricing Model

• If the interest rate is 5, the call is worth

S1
S0
debt
portfolio
C1
( - )
- 21.25
7.50
28.75

3.75
25
21.25
- 21.25
0

0
137
The Binomial Option Pricing Model

• If the interest rate is 5, the call is worth

S1
S0
debt
portfolio
C1
( - )
- 21.25
7.50
28.75

3.75
25
21.25
- 21.25
0

0
138
Binomial Option Pricing Model
The most important lesson (so far) from the
binomial option pricing model is

• the replicating portfolio intuition.

Many derivative securities can be valued by
valuing portfolios of primitive securities when
those portfolios have the same payoffs as the
derivative securities.
139
The Risk-Neutral Approach to Valuation
S(U), V(U)
q
S(0), V(0)
1- q
S(D), V(D)

• We could value V(0) as the value of the
  replicating portfolio. An equivalent method is
  risk-neutral valuation

140
The Risk-Neutral Approach to Valuation
S(U), V(U)
q
q is the risk-neutral probability of an up move.
S(0), V(0)
1- q

• S(0) is the value of the underlying asset today.

S(D), V(D)
S(U) and S(D) are the values of the asset in the
next period following an up move and a down move,
respectively.
V(U) and V(D) are the values of the asset in the
next period following an up move and a down move,
respectively.
141
The Risk-Neutral Approach to Valuation

• The key to finding q is to note that it is
  already impounded into an observable security
  price the value of S(0)

A minor bit of algebra yields
142
Example of the Risk-Neutral Valuation of a Call

• Suppose a stock is worth 25 today and in one
  period will either be worth 15 more or 15 less.
  The risk-free rate is 5. What is the value of an
  at-the-money call option?
• The binomial tree would look like this

28.75,C(D)
q
25,C(0)
1- q
21.25,C(D)
143
Example of the Risk-Neutral Valuation of a Call

• The next step would be to compute the risk
  neutral probabilities

28.75,C(D)
2/3
25,C(0)
1/3
21.25,C(D)
144
Example of the Risk-Neutral Valuation of a Call

• After that, find the value of the call in the up
  state and down state.

28.75, 3.75
2/3
25,C(0)
1/3
21.25, 0
145
Example of the Risk-Neutral Valuation of a Call

• Finally, find the value of the call at time 0

25,2.38
146
Risk-Neutral Valuation and the Replicating
Portfolio

• This risk-neutral result is consistent with
  valuing the call using a replicating portfolio.

147
The Black-Scholes Model

• The Black-Scholes Model is

Where C0 the value of a European option at
time t 0
r the risk-free interest rate.
N(d) Probability that a standardized, normally
distributed, random variable will be less than or
equal to d.
The Black-Scholes Model allows us to value
options in the real world just as we have done in
the 2-state world.
148
The Black-Scholes Model

• Find the value of a six-month call option on the
  Microsoft with an exercise price of 150
• The current value of a share of Microsoft is 160
• The interest rate available in the U.S. is r
  5.
• The option maturity is 6 months (half of a year).
• The volatility of the underlying asset is 30 per
  annum.
• Before we start, note that the intrinsic value of
  the option is 10our answer must be at least
  that amount.

149
The Black-Scholes Model

• Lets try our hand at using the model. If you
  have a calculator handy, follow along.

First calculate d1 and d2
Then,
150
The Black-Scholes Model
N(d1) N(0.52815) 0.7013 N(d2) N(0.31602)
0.62401
151
Another Black-Scholes Example
Assume S 50, X 45, T 6 months, r 10,
and ? 28, calculate the value of a call and a
put.
From a standard normal probability table, look
up N(d1) 0.812 and N(d2) 0.754 (or use
Excels normsdist function)
152
Stocks and Bonds as Options

• Levered Equity is a Call Option.
• The underlying asset comprise the assets of the
  firm.
• The strike price is the payoff of the bond.
• If at the maturity of their debt, the assets of
  the firm are greater in value than the debt, the
  shareholders have an in-the-money call, they will
  pay the bondholders and call in the assets of
  the firm.
• If at the maturity of the debt the shareholders
  have an out-of-the-money call, they will not pay
  the bondholders (i.e. the shareholders will
  declare bankruptcy) and let the call expire.

153
Stocks and Bonds as Options

• Levered Equity is a Put Option.
• The underlying asset comprise the assets of the
  firm.
• The strike price is the payoff of the bond.
• If at the maturity of their debt, the assets of
  the firm are less in value than the debt,
  shareholders have an in-the-money put.
• They will put the firm to the bondholders.
• If at the maturity of the debt the shareholders
  have an out-of-the-money put, they will not
  exercise the option (i.e. NOT declare bankruptcy)
  and let the put expire.

154
Stocks and Bonds as Options

• It all comes down to put-call parity.

Stockholders position in terms of call options
Stockholders position in terms of put options
155
Capital-Structure Policy and Options

• Recall some of the agency costs of debt they can
  all be seen in terms of options.
• For example, recall the incentive shareholders in
  a levered firm have to take large risks.

156
Balance Sheet for a Company in Distress

• Assets BV MV Liabilities BV MV
• Cash 200 200 LT bonds 300 ?
• Fixed Asset 400 0 Equity 300 ?
• Total 600 200 Total 600 200
• What happens if the firm is liquidated today?

The bondholders get 200 the shareholders get
nothing.
157
Selfish Strategy 1 Take Large Risks (Think of a
Call Option)

• The Gamble Probability Payoff
• Win Big 10 1,000
• Lose Big 90 0
• Cost of investment is 200 (all the firms cash)
• Required return is 50
• Expected CF from the Gamble 1000 0.10 0
  100

158
Selfish Stockholders Accept Negative NPV Project
with Large Risks

• Expected cash flow from the Gamble
• To Bondholders 300 0.10 0 30
• To Stockholders (1000 - 300) 0.10 0
  70
• PV of Bonds Without the Gamble 200
• PV of Stocks Without the Gamble 0
• PV of Bonds With the Gamble 30 / 1.5 20
• PV of Stocks With the Gamble 70 / 1.5 47

The stocks are worth more with the high risk
project because the call option that the
shareholders of the levered firm hold is worth
more when the volatility is increased.
159
Mergers and Options

• This is an area rich with optionality, both in
  the structuring of the deals and in their
  execution.

160
Investment in Real Projects Options

• Classic NPV calculations typically ignore the
  flexibility that real-world firms typically have.
• The next chapter will take up this point.

161
Summary and Conclusions

• The most familiar options are puts and calls.
• Put options give the holder the right to sell
  stock at a set price for a given amount of time.
• Call options give the holder the right to buy
  stock at a set price for a given amount of time.
• Put-Call parity

162
Summary and Conclusions

• The value of a stock option depends on six
  factors
• 1. Current price of underlying stock.
• 2. Dividend yield of the underlying stock.
• 3. Strike price specified in the option contract.
• 4. Risk-free interest rate over the life of the
  contract.
• 5. Time remaining until the option contract
  expires.
• 6. Price volatility of the underlying stock.
• Much of corporate financial theory can be
  presented in terms of options.
• Common stock in a levered firm can be viewed as a
  call option on the assets of the firm.
• Real projects often have hidden option that
  enhance value.