- Количество слайдов: 161
Risk Management Dr. Keith M. Howe Summer 2008
Definition Risk and uncertainty Risk management Risk aversion The process of formulating the benefit-cost tradeoffs of risk reduction and deciding on the course of action to take (including the decision to take no action at all).
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.
Figure 1. 2. Gains and losses from buying shares and a call option on Risky Upside Inc.
Panel B. Forward contract payoff.
Panel C. Hedged firm income.
Panel D. Comparison of income with put contract and income with forward contract.
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.
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.
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
Bankruptcy costs and costs of financial distress
• 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.
• Direct bankruptcy costs • • Indirect bankruptcy costs • • Average ratio of direct bankruptcy costs to total assets: 2. 8% Many of these indirect costs start accruing as soon as a firm’s 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.
Figure 3. 1. Cash flow to shareholders and operating cash flow.
Figure 3. 2. Creating the unhedged firm out of the hedged firm.
Figure 3. 3. Cash flow to claimholders and bankruptcy costs.
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
Taxes and risk management
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
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.
Optimal capital structure and risk management
• 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 likely. • As a result, it can take on more debt.
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
Figure 3. 6. Firm after-tax cash flow and debt issue.
Risk management process Risk identification Risk assessment Review Selection of risk-mgt techniques Implementation
The rules of risk management Risk Management • There is no return without risk • Be transparent • Seek experience • Know what you don’t know • Communicate • Diversify • Show discipline • Use common sense • Get a Risk. Grade Source: Riskmetrics Group (www. riskmetrics. com)
Types of risks firms face Market risk - interest rate - foreign exchange - commodity price Operational risk - industry sectors - geographical regions Hazard risk - physical damage - liabilities - business interruption Strategic risk - competition - reputation - investor support
Assignment of risk responsibilities CEO Strategic risk management CRO Market risk management Hazard risk management Operational risk management Hedgeable Insurable Diversifiable
Three dimensions of risk transfer • Hedging • Insuring • Diversifying
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
Example of VAR • The US bank J. P. Morgan states in its 2000 annual report that its aggregate VAR is about $22 m. • 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 $22 m on a given day.
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.
Instruments used in risk management • • Forward contracts Futures contracts Hedging Interest rate futures contracts Duration hedging Swap contracts Options
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.
Example Suppose S&P 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.
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 > $1020, the long makes money and the short loses money. If the index price < $1020, the long loses money and the short makes money. S&R Index S&R Forward in 6 months long short 900 -$120 950 -70 70 1000 -20 20 1020 0 0 1050 30 -30 1100 80 -80
Problem: The current S&P 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: F 0=1100 S 1=1000*1. 07=1070 Payoff: 1070 -1100= - $30.
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?
Valuing a forward contract Strategy 1 ---- the forward contract Today Buy a forward contract One year from now Buy stock at a price of $100. Sell the share for cash at market Strategy 2 ---- the portfolio strategy Today Buy stock today Sell short $100 in face value of 1 -year zero-coupon bonds One year from now Sell the stock Buyback the zero-coupon bonds of $100
Valuing a forward contract Cost Today Strategy 1 Strategy 2 ? $97 -$92 Cash flow one year from now S 1 - $100 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 $5.
Valuing a forward contract The no-arbitrage value of a forward contract on a share of stock (the obligation to buy a share of stock at a price of K, T years in the future), assuming the stock pays no dividends prior to T, is where S 0 = current price of the stock = the current market price of a default-free zero-coupon bond paying K, T years in the future At no arbitrage:
Currency Forward Rates • Currency forward rates are a variation on forward price of stock. • In the absence of arbitrage, the forward currency rate F 0 (for example, Euros/dollar) is related to the current exchange rate (or spot rate) S 0, by the equation • where r = the return (unannualized) on a domestic or foreign risk-free security over the life of the forward agreement, as measured in the respective country's currency
Forward Currency Rates Example: The Relation Between Forward Currency Rates and Interest Rates Assume that six-month LIBOR on Canadian funds is 4 percent and the US$ 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 Can$1. 25/US$? Assume that six months from now is 182 days.
Currency Forward Rates Answer: (LIBOR is a zero-coupon rate based on an actual/360 day count. ) So Canada Six-month interest Rate (unannualized): The forward rate is United States
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.
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.
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 $1=¥ 150.
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:
Daily Resettlement: An Example If tomorrow, the futures rate closes at $1 = ¥ 149, then your position’s 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.
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.
Selected Futures Contracts
Futures Markets • The Chicago Mercantile Exchange (CME) is by far the largest. • Others include: • The Philadelphia Board of Trade (PBOT) • The Mid. America Commodities Exchange • The Tokyo International Financial Futures Exchange • The London International Financial Futures Exchange
The Chicago Mercantile Exchange • Expiry cycle: March, June, September, December. • Delivery date 3 rd Wednesday of delivery month. • Last trading day is the second business day preceding the delivery day. • CME hours 7: 20 a. m. to 2: 00 p. m. CST.
CME After Hours • Extended-hours trading on GLOBEX runs from 2: 30 p. m. to 4: 00 p. m dinner break and then back at it from 6: 00 p. m. to 6: 00 a. m. CST. • Singapore International Monetary Exchange (SIMEX) offer interchangeable contracts. • There’s other markets, but none are close to CME and SIMEX trading volume.
Wall Street Journal Futures Price Quotes Highest price that day Highest and lowest prices over the lifetime of the contract. Opening price Closing price Lowest price that day Expiry month Daily Change Number of open contracts
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.
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
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
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.
Interest Rate Futures Contracts
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 … 0 1 2 3 2 T Value of the T-bond under a flat term structure = PV of face value + PV of coupon payments
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 … 0 1 2 3 2 T = PV of face value + PV of coupon payments
Pricing of Forward Contracts An N-period forward contract on that T-Bond … 0 N N+1 N+2 N+3 N+2 T Can be valued as the present value of the forward price.
Futures Contracts • • The pricing equation given above will be a good approximation. The only real difference is the daily resettlement.
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. • It’s likely to be easier and cheaper to use interest rate futures contracts however.
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.
Duration Hedging • Duration measures the combined effect of maturity, coupon rate, and YTM on bond’s price sensitivity • Measure of the bond’s effective maturity • Measure of the average life of the security • Weighted average maturity of the bond’s cash flows
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?
Calculating Duration is expressed in units of time; usually years.
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
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.
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.
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 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.
An Example of an Interest Rate Swap • Firm B is a BBB-rated U. S. company. It needs $10, 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 floatingrate notes at LIBOR + ½ percent. • Firm B would prefer to borrow at a fixed rate.
An Example of an Interest Rate Swap The borrowing opportunities of the two firms are:
An Example of an Interest Rate Swap 10 3/8% Bank LIBOR – 1/8% Bank A 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
An Example of an Interest Rate Swap ½% of $10, 000 = $50, 000. That’s quite a cost savings per year for 5 years. 10 3/8% Swap Bank LIBOR – 1/8% Bank 10% A Here’s what’s 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.
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 10 ½% LIBOR – ¼% Company B
An Example of an Interest Rate Swap Here’s what’s in it for B: Swap Bank They can borrow externally at LIBOR + ½ % and have a net ½ % of $10, 000 = $50, 000 that’s quite a cost savings per year for 5 years. 10 ½% LIBOR – ¼% borrowing position of 10½ + (LIBOR + ½ ) - (LIBOR - ¼ ) = 11. 25% which is ½% better than they can borrow floating. Company B LIBOR + ½%
An Example of an Interest Rate Swap The swap bank makes money too. Swap 10 3/8% Bank 10 ½% LIBOR – 1/8% Bank A ¼% of $10 million = $25, 000 per year for 5 years. LIBOR – ¼% LIBOR – 1/8 – [LIBOR – ¼ ]= 1/8 10 ½ - 10 3/8 = 1/8 ¼ Company B
An Example of an Interest Rate Swap The swap bank makes ¼% Swap 10 3/8% Bank LIBOR – 1/8% 10 ½% LIBOR – ¼% Bank Company A B A saves ½% B saves ½%
An Example of a Currency Swap • Suppose a U. S. MNC wants to finance a £ 10, 000 expansion of a British plant. • They could borrow dollars in the U. S. where they are well known and exchange for dollars for pounds. • • This will give them exchange rate risk: financing a sterling project with dollars. They could borrow pounds in the international bond market, but pay a premium since they are not as well known abroad.
An Example of a Currency Swap • • If they can find a British MNC with a mirrorimage financing need they may both benefit from a swap. If the spot exchange rate is S 0($/£) = $1. 60/£, the U. S. firm needs to find a British firm wanting to finance dollar borrowing in the amount of $16, 000.
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 other’s country of the same size. Their borrowing opportunities are given in the table below.
An Example of a Currency Swap Bank $8% £ 11% $8% $9. 4% £ 12% Firm A B £ 12%
An Example of a Currency Swap A’s net position is to borrow at £ 11% Swap Bank $8% £ 11% $8% $9. 4% £ 12% Firm A B A saves £. 6% £ 12%
An Example of a Currency Swap B’s net position is to borrow at $9. 4% Swap Bank $8% £ 11% $8% $9. 4% £ 12% Firm A B £ 12% B saves $. 6%
An Example of a Currency Swap The swap bank makes money too: Swap Bank $8% £ 11% $8% Firm A 1. 4% of $16 million financed with 1% of £ 10 million per year for 5 years. $9. 4% £ 12% Firm £ 12% At S 0($/£) = $1. 60/£, that is a gain of $124, 000 per B year for 5 years. The swap bank faces exchange rate risk, but maybe they can lay it off (in another swap).
Variations of Basic Swaps • Currency Swaps • • fixed for floating • floating for floating • • fixed for fixed amortizing Interest Rate Swaps • • • zero-for floating Exotica • For a swap to be possible, two humans must like the idea. Beyond that, creativity is the only limit.
Risks of Interest Rate and Currency Swaps • Interest Rate Risk • • Basis 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. 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.
Risks of Interest Rate and Currency Swaps • Credit Risk • This is the major risk faced by a swap dealer—the risk that a counter party will default on its end of the swap. • Mismatch Risk • It’s 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.
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.
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.
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. •
Options Contracts: Preliminaries • Exercising the Option • • 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 act of buying or selling the underlying asset through the option contract. 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.
Options Contracts: Preliminaries • In-the-Money • • At-the-Money • • The exercise price is less than the spot price of the underlying asset. 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.
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
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.
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. Ca. T = Ce. T = Max[ST - E, 0] Where ST is the value of the stock at expiry (time T) E is the exercise price. Ca. T is the value of an American call at expiry Ce. T is the value of a European call at expiry
Call Option Payoffs 60 Option payoffs ($) 40 Buy a call 20 0 0 10 20 30 40 50 60 70 80 90 100 Stock price ($) -20 -40 -60 Exercise price = $50
Call Option Payoffs 60 Option payoffs ($) 40 20 0 -20 0 10 20 30 40 50 60 70 80 Stock price ($) Write a call -40 -60 Exercise price = $50 90 100
Call Option Profits 60 Option profits ($) 40 Buy a call 20 0 -20 0 10 20 30 40 50 60 70 80 90 Stock price ($) Write a call -40 -60 Exercise price = $50; option premium = $10 100
Put Options • Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset on or before some time in the future, at prices agreed upon today. • When exercising a put, you “put” the asset to someone.
Basic Put Option Pricing Relationships at Expiry • • • At expiry, an American put option is worth the same as a European option with the same characteristics. If the put is in-the-money, it is worth E - ST. If the put is out-of-the-money, it is worthless. Pa. T = Pe. T = Max[E - ST, 0]
Put Option Payoffs 60 Option payoffs ($) 40 Buy a put 20 0 -20 0 10 20 30 40 50 60 70 80 90 100 Stock price ($) -40 -60 Exercise price = $50
Put Option Payoffs 60 Option payoffs ($) 40 20 0 0 10 20 30 -20 -40 40 50 60 70 80 90 100 Stock price ($) write a put -60 Exercise price = $50
Option profits ($) Put Option Profits 60 40 20 10 0 -10 -20 Write a put 0 10 20 30 Stock price ($) 40 50 60 70 80 Buy a put 90 -40 -60 Exercise price = $50; option premium = $10 100
Selling Options The seller (or writer) of an option has an obligation. Option profits ($) • • The purchaser of an option has an option. 60 40 20 10 0 -10 -20 -40 -60 Buy a call Write a put 0 10 20 30 Stock price ($) 40 50 60 70 80 Buy a put Write a call 90 100
Reading The Wall Street Journal
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
Reading The Wall Street Journal This makes a call option with this exercise price in-themoney by $3. 25 = $138¼ – $135. Puts with this exercise price are out-of-the-money.
Reading The Wall Street Journal On this day, 2, 365 call options with this exercise price were traded.
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.
Reading The Wall Street Journal On this day, 2, 431 put options with this exercise price were traded.
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.
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 client’s needs.
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 $50 Value of stock at expiry
Protective Put Strategy Profits Value at expiry $40 Buy the stock at $40 Protective Put strategy has downside protection and upside potential $0 $40 $50 -$40 Buy a put with exercise price of $50 for $10 Value of stock at expiry
Covered Call Strategy Value at expiry $40 Buy the stock at $40 Covered call $10 $0 Value of stock at expiry $30 $40 $50 -$30 -$40 Sell a call with exercise price of $50 for $10
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 -$20 $30 $40 $50 $60 Buy a put with an exercise price of $50 $70 for $10 A Long Straddle only makes money if the stock price moves $20 away from $50. Value of stock at expiry
Short Straddle: Sell a Call and a Put Value at expiry $20 $10 $0 A Short Straddle only loses money if the stock price moves $20 away from $50. Sell a put with exercise price of $50 for $10 Value of stock at expiry -$30 -$40 $30 $40 $50 $60 $70 Sell a call with an exercise price of $50 for $10
Long Call Spread Value at expiry Buy a call with an exercise price of $50 for $10 $5 $0 -$5 -$10 long call spread $50 $60 Value of stock at expiry $55 Sell a call with exercise price of $55 for $5
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. Value at expiry Buy a call option with an exercise price of $40 Buy the stock at $40 financed with some debt: FV = $X Sell a put with an exercise price of $40 $0 -[$40 -P 0] -$40 $40 -P 0 $40 Value of stock at expiry
Valuing Options • The last section • This section considers the value of an option prior to the expiration date. • A much more interesting question. concerned itself with the value of an option at expiry.
Option Value Determinants 1. 2. 3. 4. 5. Stock price Exercise price Interest rate Volatility in the stock price Expiration date + Call Put + – – + + + + The value of a call option C 0 must fall within max (S 0 – E, 0) < C 0 < S 0. The precise position will depend on these factors.
Market Value, Time Value and Intrinsic Value for an American Call Profit The value of a call option C 0 must fall within max (S 0 – E, 0) < C 0 < S 0. ST Ca. T > Max[ST - E, 0] Market Value Time value loss Out-of-the-money ST -E Intrinsic value E In-the-money ST
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 realworld option valuation.
Binomial Option Pricing Model Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. S 0= $25 today and in one year S 1 is either $28. 75 or $21. 25. The risk-free rate is 5%. What is the value of an at-the-money call option? S 0 S 1 $28. 75 $21. 25
Binomial Option Pricing Model 1. 2. 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. S 0 S 1 C 1 $28. 75 $3. 75 $21. 25 $0 $25
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 option’s payoff so the portfolio is worth twice the call option value. S 0 ( S 1 - debt ) = portfolio C 1 $28. 75 - $21. 25 = $7. 50 $3. 75 $21. 25 - $21. 25 = $0 $0
Binomial Option Pricing Model The levered equity portfolio value today is today’s value of one share less the present value of a $21. 25 debt: S 0 ( S 1 - debt ) = portfolio C 1 $28. 75 - $21. 25 = $7. 50 $3. 75 $21. 25 - $21. 25 = $0 $0
Binomial Option Pricing Model We can value the option today as half of the value of the levered equity portfolio: S 0 ( S 1 - debt ) = portfolio C 1 $28. 75 - $21. 25 = $7. 50 $3. 75 $21. 25 - $21. 25 = $0 $0
The Binomial Option Pricing Model If the interest rate is 5%, the call is worth: S 0 ( S 1 - debt ) = portfolio C 1 $28. 75 - $21. 25 = $7. 50 $3. 75 $21. 25 - $21. 25 = $0 $0
The Binomial Option Pricing Model If the interest rate is 5%, the call is worth: S 0 C 0 ( S 1 - debt ) = portfolio C 1 $28. 75 - $21. 25 = $7. 50 $3. 75 $2. 38 $21. 25 - $21. 25 = $0 $0
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.
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
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 S(D), V(D) underlying asset today. 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.
The Risk-Neutral Approach to Valuation S(U), V(U) q S(0), V(0) 1 - q S(D), V(D) • 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:
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: q $28. 75, C(D) $25, C(0) 1 - q $21. 25, C(D)
Example of the Risk-Neutral Valuation of a Call: The next step would be to compute the risk neutral probabilities 2/3 $28. 75, C(D) $25, C(0) 1/3 $21. 25, C(D)
Example of the Risk-Neutral Valuation of a Call: After that, find the value of the call in the up state and down state. 2/3 $28. 75, $3. 75 $25, C(0) 1/3 $21. 25, $0
Example of the Risk-Neutral Valuation of a Call: Finally, find the value of the call at time 0: 2/3 $28. 75, $3. 75 $25, $2. 38 $25, C(0) 1/3 $21. 25, $0
Risk-Neutral Valuation and the Replicating Portfolio This risk-neutral result is consistent with valuing the call using a replicating portfolio.
The Black-Scholes Model is Where C 0 = 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.
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 $10 —our answer must be at least that amount.
The Black-Scholes Model Let’s try our hand at using the model. If you have a calculator handy, follow along. First calculate d 1 and d 2 Then,
The Black-Scholes Model N(d 1) = N(0. 52815) = 0. 7013 N(d 2) = N(0. 31602) = 0. 62401
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-themoney call, they will not pay the bondholders (i. e. the shareholders will declare bankruptcy) and let the call expire.
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-ofthe-money put, they will not exercise the option (i. e. NOT declare bankruptcy) and let the put expire.
Stocks and Bonds as Options • It all comes down to put-call parity. Value of a call on the firm Value of a Value of = the firm + put on the – firm Stockholder’s position in terms of call options Stockholder’s position in terms of put options Value of a risk-free bond
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.
Balance Sheet for a Company in Distress Assets BV Cash $200 Fixed Asset Total $600 MV $200 $400 $200 Liabilities LT bonds $0 Equity Total $600 BV $300 $200 MV ? ? What happens if the firm is liquidated today? The bondholders get $200; the shareholders get nothing.
Selfish Strategy 1: Take Large Risks (Think of a Call Option) The Gamble Win Big Lose Big Probability 10% 90% Payoff $1, 000 $0 Cost of investment is $200 (all the firm’s cash) Required return is 50% Expected CF from the Gamble = $1000 × 0. 10 + $0 = $100
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.
Mergers and Options • This is an area rich with optionality, both in the structuring of the deals and in their execution.
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.
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
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. 1. 2. 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.