UE 2 - Capital Markets and Financial Instruments PDF

Part 4: fixed income Part 4.1: fixed income instruments Fixed income: refers to a class of investments that pay fixed interest or dividend payments until maturity. At maturity, the principal (the original amount invested) is returned to the investor à Bonds (government, corporate, municipal), Treasury bills, certificates of deposit (CDs), and fixed-rate preferred stocks à Remaining risks can be better identified and managed à Fixed income security markets: used to determine interest rates and price other securities Characteristics: Regular Income: investors receive a ﬁxed or variable interest payment at regular intervals (monthly, semi-annually, annually) Principal Repayment: the face value of the instrument is repaid at maturity Lower Risk: compared to equities, ﬁxed-income investments are typically less risky because they provide steady cash ﬂows and have priority in case of liquidation Credit Risk: the risk that the issuer may default on its obligations. Higher credit quality (like government bonds) typically means lower risk but also lower returns Risks associated: Interest Rate risk: value of a ﬁxed-income investment will decline due to rising interest rates. When interest rates rise, existing bonds with lower rates become less attractive, causing their prices to drop Credit risk: the issuer will default on interest payments or fail to return the principal Inﬂation risk: inﬂation will erode the purchasing power of future interest payments and principal repayments Reinvestment risk: when interest or principal payments are reinvested, the returns will be lower than the original investment’s yield Active Management: involves trying to outperform the market through strategies like duration management, credit selection, and yield curve positioning. Passive Management: aims to replicate the performance of a fixed-income index, emphasizing low cost and broad exposure Instruments of Fixed Income Government Bonds: Treasury Bonds (T-Bonds): Long-term bonds (10–30 years) Treasury Notes (T-Notes): Medium-term bonds (2, 5, or 10 years) that pay a ﬁxed interest rate every six months until maturity. Treasury Bills (T-Bills): Short-term debt securities (4 weeks to 1 year) issued at a discount and maturing at face value. These do not pay periodic interest; the di`erence purchase price and the face value is the yield. Inﬂation-Protected Bonds (TIPS) adjust their Bonds principal value based on inﬂation, providing Most common fixed-income protection against inﬂation risk. instruments. They are essentially Municipal Bonds: local governments or loans made by investors to municipalities. They often provide tax beneﬁts, issuers who promise to pay as interest earned may be exempt from federal interest and repay the principal at and, state and local taxes. maturity Corporate Bonds: Issued by corporations to raise capital. Higher yields to compensate for higher credit risk. à Investment-Grade Bonds: Issued by companies with higher credit ratings (BBB and above). Lower default risk and lower yields à High-Yield Bonds (Junk Bonds): Issued by companies with lower credit ratings (BB and below). Higher yields but increased risk of default à Supranational Bonds: international organizations to fund global development projects. Low-risk Agency Bonds: government-affiliated organizations, They may offer higher yields than Treasury bonds but still carry some government support. Treasury and Government Securities Sovereign Bonds: issued by national governments in currencies other than their own. These may carry higher risks and yields due to currency and credit factors. Money Market Instruments Commercial Paper: Short-term unsecured These are short-term debt promissory notes issued by corporations to finance instruments (typically less than short-term liabilities. Mature within 270 days and are one year) and are considered issued at a discount. highly liquid and low-risk Certificates of Deposit (CDs): Offered by banks, these pay a fixed interest rate for a specified term (a few months to several years). Very safe Bankers’ Acceptances: Short-term debt instruments guaranteed by a bank, used in international trade. They are sold at a discount and redeemed at face value. Repurchase Agreements (Repos): Short-term borrowing, usually overnight, where securities are sold with an agreement to repurchase them at a higher price Financial institutions to manage liquidity. Mortgage-Backed Securities (MBS): Securities backed by a pool of mortgages. Investors receive periodic payments based on the mortgage payments made by homeowners. Structured Fixed-Income Products Asset-Backed Securities (ABS): Similar to MBS, but These are complex instruments backed by other types of loans (auto loans, credit card designed to meet specific debt, or student loans). investor needs Collateralized Debt Obligations (CDOs): A complex structured product that pools various types of debt (corporate bonds, ABS) and then divides them into tranches with different risk levels and yields Preferred shares pay fixed dividends, making them Preferred Stocks similar to bonds. Predictable income streams These are corporate bonds that can be converted into a predetermined number of the company’s equity Convertible Bonds shares. They offer the fixed income and lower risk of bonds Bonds whose principal and interest payments are adjusted for inflation, such as TIPS. These provide a Inflation-Linked Bonds hedge against inflation by ensuring that returns are adjusted to maintain purchasing power. Bonds with variable interest rates that are tied to a benchmark rate (LIBOR or SOFR). The coupon Floating Rate Notes (FRNs) payments adjust periodically, making them less sensitive to interest rate risk Foreign Bonds: Issued in a domestic market by a foreign entity Foreign Bonds and Eurobonds Eurobonds: Bonds issued in a currency other than the currency of the country where it is issued Issued at a discount to their face value and mature at Zero-Coupon Bonds par. The investor's return is the difference between the Bonds that do not pay coupons purchase price and the amount received at maturity Pricing conventions: Price & coupons listed as percentage of par: prices & coupons are always on base 100 Coupon rates are annualized, even if paid n times per year. E.g., 5% coupon, semiannual, means 2.5% of face value paid every 6 months Unsecured: Short-term loans, mostly Money market between banks, uncollateralized Collateralized loans. Formally, a repurchase agreement (repo) is an agreement to sell securities to another party, and to buy them back at a fixed date, for a fixed amount (loan + repo rate) A reverse repo is an agreement to buy Repo market securities now and resell at a fixed date for a fixed price Repo rates close to zero may lead to many failures to deliver: the “keeper” of the security does not hand them back at maturity. Regulation to prevent failures may artificially increase bond prices. U.S. Treasury Securities T-bills zero-coupon bonds with maturity up to 1 year pay coupons semi-annually T-notes coupon bonds with maturity up to 10 years pay coupons semi-annually T-bonds coupon bonds with longer maturities, up to 30 years pay coupons semi-annually The bond’s price is determined by the market: Primary or secondary Organized as auction, continuous market, over the counter What should a bond’s price be? Bond : series of cash ﬂows Present value of cash ﬂows is bond’s “value” today: bond’s fair price Discounting cash ﬂows requires using a cost of capital (discount rate, interest rate) Face Value: value at maturity 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒 𝑃𝑟𝑖𝑐𝑒 = (1 + 𝑟)! Yield to maturity: when we consider a fixed rate with which all cash flows of a bond are discounted, we talk about the bond’s yield to maturity à Required return by investors given available investment opportunities at the same risk à Fixed rate of return implicit given a bond’s cash flows and current market price à YTM is always annualized The yield to maturity equals the expected fixed rate of return for a bond If the bond is held until its maturity, If there are no defaults on payments If all coupons are reinvested at the same rate 1 𝑍(𝑡, 𝑇𝑗) = 𝑌𝑇𝑀 !∗($%&') (1 + 𝑛 ) Dirty price: the price we just computed is the full price or dirty price of the bond at date t, between coupons. This price appropriately considers that the buyer of the bond today will cash in ALL current and ensuing coupons. Therefore, this buyer should pay the present value of all these coupons Clean price: when bond prices are listed, bonds that are “between coupons”, are listed with their clean price. The clean price is as if the seller had pocketed her share of the next coupon (accrued interest) and the buyer would pocket only his share (remaining interest) Part 4.2: interest-rate risk The higher the required YTM, the lower the price à Moreover, coupon-YTM relation: If coupon < YTM, price < 100 If coupon > YTM, price > 100 If coupon = YTM, price = 100 The smaller coupon bond is more sensitive (in percentage) to changes in YTM Why: High coupon bond: big part of investment’s cash ﬂows arrives every six months Low coupon bond: bulk of the investment’s cash ﬂows arrives at maturity Change in YTM: change in time value of money à mainly a`ects cash ﬂows arriving far in time The coupon bond with longer maturity is more sensitive to changes in YTM Why: Change in YTM: change in time value of money à a`ects cash ﬂows arriving far in time the most Long maturity: more cash ﬂows arriving far in time à Price–YTM relationship is downward sloping, with the exact shape being specific to the bond: coupon and maturity matter à Fix the coupon rate. Bonds with longer maturity have greater price volatility à Fix the time to maturity. Bonds with a lower coupon rate have greater price volatility à Since the price of a bond changes depending on the YTM, the holder of a bond is exposed to YTM-risk, or interest-rate risk Remember that a bond’s price varies differently with yield depending on its coupon and time to maturity MacD is a weighted average of the times to maturity of each cashflow of the bond à Larger coupons decrease the Macaulay duration of a bond à Longer time to maturity increases the Macaulay duration of a bond Part 4.3: varying discount factors Interest rates linked to loans of different maturity are different: the US Treasury offers you a higher rate of return for lending them money for 1 semester than 1 quarter à The YTM of zero-coupon bonds is equivalent to the interest rate from today to the date of the bond’s maturity The yield curve or term structure of interest rates shows interest rates for each maturity for a given class of bonds. If ZCB available for all maturities, these would be the YTMs implied by ZCB prices. The yield curve can be used to price other bonds of the same or similar class If discount rates are determined from market information, you learn the yield curve YTM interpreted as the implied return from a given investment After you know the price of a bond, you can determine its YTM (= its internal rate of return if held until maturity) When yield curve is used for pricing, each cash flow of the bond (coupon, principal) is discounted with a different rate Notation: we use 𝑟m (𝑡, 𝑇) to denote an m-compounding interest rate, and 𝑟 (t, 𝑇) to denote a continuously-compounding interest rate Part 4.4: floating rate bonds A floating rate bond or “floater” is defined by: type LIBOR A ﬁxed maturity at date 𝑇, Fixed coupon payment dates, called reset dates, 𝑇1 , 𝑇2 , ⋯ , 𝑇n A variable part of the coupon, which changes, but is known one period in advance A ﬁxed part of the coupon, called spread (2bps = +0.02%) à When we buy a floating-rate bond, we only know the value of the first coupon. Remaining cash flows are unknown. This formula give the price of the bond when this isn’t a reset date: At Reset Date: The bond price will generally be close to par value since the coupon is reset to match market interest rates. Between Reset Dates: The bond price reflects the accrued interest and adjusts according to the market rate environment. The formula incorporates the fraction of the coupon period that has passed. Coupon value: if semi annual Pt= (100 + (3 months rate – spread bps/2) x e(-r(0,T)xT) PV (n cash ﬂows)= spread bps/2 x (e(-r(0,T1)xT1) + … + e(-r(0,Tn)xTn)) Price of the bond = Pt + PV(n cash ﬂows) The only “interest-rate risk” you are exposed to is that affecting the next coupon payment à In the near future, the interest rate risk of all other cash flows doesn’t matter, since those cash flows will vary with interest rates! à Only the present value of the cash flow that was already fixed at the last reset date will be affected by changing interest rates If the floater has a spread, we treat it as a portfolio à The duration of the portfolio is the weighted sum of each component’s duration à We find duration for the spread (a series of cash flows) and the zero-spread floater separately, then sum (with weights) Part 4.5: term structure of interest rates We already saw that interest rates for cash flows arriving at different points in time are different The term spread is the difference between long-term interest rates (e.g., 10-year rate) and short-term interest rates (e.g., 3-months rate) The term spread is often positive. It depends on, among others, expected future inflation, expected future growth of the economy… The term structure of interest rates can be used to deduce investors’ expectations on inflation, economic growth… The term structure of interest rates, or spot rate curve, or yield curve, at a certain time 𝑡, gives the relation between interest rates and their time to maturity, 𝑇 − 𝑡. à Government bonds are very liquid and come in many forms ; it helps construct a reliable yield curve Interest rates or, equivalently, bond prices, depend on the choices made by bond market participants Lenders: investors, many institutional Borrowers: governments, corporations If investors/borrowers didn’t care about maturity, the yield curve would be flat, but it almost never is! Why care about maturity? Anticipated changes in the economy (inﬂation, growth, etc.) and rates Attitudes toward risk and perceived risk The nature (depreciation, use, riskiness…) of assets held by market participants When yield curve is (very) upward-sloping, on average future rates are lower than predicted à Excess return from holding long-term bonds! There are several reasons why investors prefer investing in short-term than in long-term securities Preference for Liquidity: You want to be sure you can convert your investment in cash if necessary Risk Aversion: You want to be sure your investment will pay what it promised Preference for liquidity implies preference for short-term bonds because: Markets for recently-issued bonds are more liquid than for stale bonds Short-term bonds pay back their principal in cash soon Risk aversion implies preference for short-term bonds because: Long-term bonds have higher duration – they are more exposed to interest rate risk Long-term bonds are exposed to credit risk for longer – higher total risk Pure preference-for-short term models imply an increasing yield curve even if one expects future short-term rates to remain unchanged à Forward rates are larger than the future spot rates they predict The markets for investments of different maturities are segmented Inhabited by investors with di`erent investment needs Demand-supply in each segmented mkt. determines interest rates for each maturity Contradict pure expectation and preference for short term models Investors for maturity T ignore other strategies than buying T-maturity bonds. They do not participate in the short-term market à forward rates are meaningless Markets for di`erent maturities are not connected! Any shape of the yield curve can arise In its purest form, it assumes investors are very myopic True that some investors mostly invest in certain maturities (e.g., pension funds) But arbitrageurs could bridge interest rate mismatch! Milder form: some arbitrageurs, but insu`icient to fully connect di`erent maturities Term structure of interest rates at a time t can be used to Gauge information about the economy (expectations, investment, for example) Correlation (especially in US) between term spread (increasing) and recessions Timing of investments and portfolio management (used to determine durations!) Price other bonds of a similar class as those used in the considered yield curve The information contained in the yield curve depends on the model used to interpret it! à For example, if expectations models are correct, an increasing curve suggests an increase of interest rates in the future à If preference-for-short term models are correct, we know a premium should be expected for long-term investments à Given empirical findings, if curve is very steeply increasing, this premium is very high The yield curve changes in time à must use up-to-date information! Remember that we must use DIRTY prices to construct the term structure of interest rates If the bond you’re using for calculations paid a coupon today, dirty price = clean price If you are “between coupons”, dirty price = clean price + accrued interest Part 4.6: duration, convexity and portfolios à Construct “good” portfolios of fixed income securities à Immunization vs. cash flow matching à Other strategies can be implemented using duration, but can be improved by using convexity: notion of convexity and hedging with duration and convexity How does a change in interest rate affect different sources of income from a bond? The resale price: when will we sell it? The coupon values: how many coupons before we resell? The interest obtained from reinvesting coupons at the new rate: for how long? Clearly, the effect of a change in interest rate depends on when we look at the sources of income from investment Depending on time frame of investment, a change of YTM can make the investment more or less profitable than expected We cannot protect an investment from changes of YTM or interest rate for all investment time frames, but we can for a specific time frame! There are two types of interest rate risk that move in opposite directions: Market price risk: higher interest rate à lower resale price Coupon reinvestment risk: higher interest rate à higher income from coupons reinvested à At different points in time, these two risks carry different weights At date = bond’s Macaulay duration, For small changes of rate, the two risks exactly oRset each other For large changes of rate, the two risks almost oRset each other Two types of dedication strategies insure income needs at a fixed date against any change in YTM (or parallel shift of the yield curve) Cash ﬂow matching: buy securities with maturity at the date when the money is needed Immunization: create a portfolio insured against interest rate risk à a portfolio with Macaulay duration equal to the date when the money is needed (or duration of ﬂows that are needed) An immunized portfolio insures investment return at a given point in time against changes in the term structure of interest rates. Classical immunization uses Macaulay duration and provides insurance against parallel shifts of yield curve à Other forms of immunization focus on key interest rate changes à use key rate duration Cash flow matching is safe, but… Not always possible to ﬁnd ﬁxed income securities with the right maturity for our cash needs May require “expensive” investments and unnecessary commitments. Examples: Cash Flow Matching The corporation who must pay for the new machine in 3 years, can buy a single ZCB with maturity 3 years, to make sure the cash for payment will be available. The pension fund needing to make monthly payments to its investors for 30 years, can buy 360 ZCB, each with the appropriate maturity, to match the payments required every month. You match duration of liabilities to duration of assets If liability is a single payment, MacD = maturity If liability is a series of payments, determine this set of cashﬂows’ MacD, invest in a portfolio that matches it Examples: The corporation who must pay for Immunization the new machine in 3 years, can buy a combination of coupon and ZCB with portfolio duration equal 3 years The pension fund needing to make monthly payments to its investors for 30 years, needs a portfolio that has the same duration as her liabilities (a 30-years monthly annuity) à Must match duration AND PV of liabilities and assets Immunization is a dynamic strategy that requires updating: Duration of coupon bonds changes after coupon payments Duration of coupon bonds changes each time the term structure of interest rates changes Portfolio is immunized up to next coupon payment and next actual change of yield curve Must adjust portfolio after each coupon payment and each change of yield curve A bank is highly invested in mortgages and provides mainly current-account services to consumers Asset-liability management consists of matching duration of assets and liabilities, so that duration of equity is zero (or almost). An investment fund takes a highly profitable position that they plan to rebalance soon through retrading. They wish to protect the value at resale (prices) against changes of interest rate. A portfolio is hedged against interest-rate risk if it is sure to maintain its value (price) at current conditions, even if interest rates change The convexity of a bond is the percent sensitivity of the bond’s price’s slope to a small parallel shift of the yield curve 1 𝑑) 𝑃 𝐶= ∗ 𝑃 𝑑𝑌𝑇𝑀) Like MacD was a weighted sum of coupon maturities, convexity is a weighted sum of coupon maturities squared 𝑃& + 𝑃* − (2 ∗ 𝑃) 𝐴𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒 𝐶𝑜𝑛𝑣𝑒𝑥𝑖𝑡𝑦 = (𝑑𝑌𝑇𝑀)) ∗ 𝑃 𝑃& + 𝑃* − (2 ∗ 𝑃) 𝐸𝑓𝑓𝑒𝑡𝑖𝑣𝑒 𝐶𝑜𝑛𝑣𝑒𝑥𝑖𝑡𝑦 = (𝑑𝑟)) ∗ 𝑃 P- is the price computed using current YTM or rates minus dYTM or dr P+ is the price computed using current YTM or rates plus dYTM or dr Part 5: derivatives Part 5.1: forwards, futures and swaps Derivative security: is a financial security (asset, claim) whose value is derived from other, more primitive, variables such as: Stock prices Exchange rates Interest rates Commodity prices Futures and Forwards are agreements where two parties agree to a specified trade at a specified point in the future Swaps are similar to Forwards except that the parties commit to multiple exchanges at different points in time Options are contracts in which two parties agree to a trade in the future, but one party retains in the right to opt out of the trade Forward contract: an agreement, negotiated directly, between a buyer and a seller to trade in a specified quantity of a specified good at a specified date in the future at a specified price The buyer is said to have a long position, the seller has a short position The speciﬁed good is known as the underlying asset The speciﬁed date is known as the expiration or maturity date The speciﬁed price is known as the delivery price Default risk for both parties Currencies, commodities and interest rates Allow investors to lock–in a price for the transaction: they facilitate hedging, the reduction of risk in cash ﬂows associated with market commitments A zero–sum game: ex–post, the buyer’s gains are the seller’s losses, and vice versa Forward price: by convention, the delivery price in a forward contract is chosen so that the contract has zero value to both parties à It follows that at the time the contract is entered into, the forward price and the delivery price are equal à As time passes, the forward price is liable to change, whereas the delivery price remains fixed through the life of the contract Futures contract: is similar to a forward contract but is traded on an organized exchange Differences: Buyers and sellers deal through the exchange, not directly Contract terms are standardized: promoting liquidity and improving quality of hedge (quantity, quality, delivery arrangements) Either party can reverse its position at any time by closing out its contract Default risk is borne by exchange, not by individual parties “Margin accounts” are used to reduce default risk Cash vs. Physical Settlement: the majority of financial futures do not lead to the actual delivery of the underlying asset, but are cash–settled à Instead of the long position receiving the underlying, it just receives its value (net of agreed delivery price) based on a pre-specified reliable price quote on the settlement day à Holders of futures contracts can unilaterally reverse Reversal of position is important because standardization of delivery dates creates “delivery basis risk”: delivery dates on the contract may not match the market commitment dates of the hedger à elimination of part of delivery basic risk Hedging with delivery basis risk: a company that knows it is due to buy (sell) an asset at a particular time in the future can hedge by taking a long (short) futures position Price goes up: company takes a loss from the purchase (gains from the sale) but makes a gain on the long (takes a loss on the short) futures position Price goes down: company makes a gain from the purchase (loss from the sale) but takes a loss on the long (makes a gain on the short) futures position The hedge requires the futures contract to be closed out before its expiration date Since buyers and sellers do not interact directly, there is an incentive for either party to default if prices move adversely à Default risk can render market illiquid (asymmetric information) à To inhibit default, futures exchanges use margin accounts The level at which margins are set is crucial for liquidity. High levels eliminate default, but inhibit market participation However, too low levels increase default risk à In practice, margin levels are not set very high The margining procedure amount initially deposited by investor into Initial margin a margin account daily adjustment of the customer’s margin Marking-to-market account reflects gains/losses from futures price movements over the day floor level of margin account: If balance falls below this, Maintenance margin customer receives margin call If the margin call is not met, account is closed out immediately Futures price: is defined in the same way as the forward price à It is the delivery price that will make the contract have zero value to both parties à However, there is an important difference between futures and forward contracts: Unlike a forward contract, a futures contract is marked–to–market daily Therefore, the value of a futures contract is reset to zero every day Swap: is an agreement between two parties to exchange two streams of same or different assets over regular intervals until a terminal date à Most follow the International Swaps and Derivatives Association (ISDA) Master Agreement Assets to be exchanged Dates of exchange Delivery instructions for both parties Close-out and netting Events of default and events of termination As in forwards and futures, swap terms are set such that neither party pays the other to enter a swap. A swap can be treated as a portfolio of forward contracts. An interest rate swap is an agreement between two parties A and B, where: A pays to B a ﬁxed rate on a notional principal for a number of years B pays to A a ﬂoating rate on the same principal for the same period of time (LIBOR type: average rate of interest o`ered by one major global bank on deposits by another one ; 5 currencies and 7 maturities) Suppose the floating rate specified in the swap is the six-month LIBOR: then at each payment date, the floating rate paid is the six-month LIBOR rate which prevailed six- months before the payment date à There is uncertainty about the second and subsequent payments (not the first) because they are determined by the LIBOR rate six months before the payment is made Currency rate swap is an agreement between two parties A and B, where: A makes a series of payments in one currency to B for a number of years. B makes a series of payments in another currency to A for the same period of time Commodity swap is an agreement between two parties A and B, where: A makes a series of ﬁxed payments to B for a number of years B makes a series of payments dependent on the price of an underlying commodity to A for the same period of time. A commodity swap is usually used to hedge against price swings in the market for a commodity, such as oil and livestock Part 5.2: options Call option: right to purchase an asset (the underlying asset) for a given price (exercise price), on or before a given date Put option: right to sell an asset for a given price, on or before a give date European Options: Gives the owner the right to exercise the option only at the expiration date American Options: Gives the owner the right to exercise the option on or before the expiration date The payoff of an option on the expiration date is determined by the price of the underlying asset at that date. The payoff of an option is never negative. Sometimes it is positive. The actual payoff depends on the price of the underlying asset. Options can in many situations be used strategically to modify ones exposure to risk, creating portfolios Put-call parity: useful relation between the price of an European put and that of an European call Options can in many situations be used strategically to modify ones exposure to risk, creating portfolios Since the two portfolios have identical payoffs at the future date T, they must have the same value now (at date t) Strategy 1: optimism à reasons to be optimistic and believe that the price of a stock will substantially increase from its present level à Creating a “bull-spread”: buying one call at a strike price K1 and selling one call at a strike price K2, greater than K1 à Profit if positive price deviation à Pessimism: reverse “bear-spread” Strategy 2: speculating on volatility à Creating a “straddle”: buying one call and buying one put with a same strike price K, close to the underlying asset price S à Profit if substantial deviations occur, whatever way it goes à Conversely, if you anticipate a stable stock price, creating a “short (naked) straddle”: selling one call and selling one put with a same strike price K , close to the underlying asset price S à Profit if not much deviation Strategy 3: hedging downside while keeping upside à One can hedge against the downside risk of an asset: buying a stock and buying a put The higher the interest rate, the lower the present value of the strike price the call buyer has contracted to pay to exercise, which is the cost of exercising. Therefore, an increase in interest rate increases the price of a call option, and decreases the price of a put option Dividends: when dividends are paid-out the stock price falls. Thus, their rise affects option prices as a fall in the stock price The owner of call option benefits from an increase in the stock price but faces no downside risk if the stock price decreases (rises) à The option is worth more under the high volatility scenario Part 5.3: option pricing Option pricing: price follows a binomial process à The resulting model is known as the Binomial Option Pricing Model Crucial Trick: we can form a portfolio of shares and treasury bonds that gives identical payoffs as the call, at maturity à Then the present value of the call must equal the current cost of this replicating portfolio à If we construct a portfolio of the stock and bond which replicates the value of the option next period, then the cost of this replicating portfolio must equal the options present value Replicating strategy gives payoff identical to those of the call in every possible circumstance à Initial cost of the replicating strategy, must equal the call price, otherwise there would be an arbitrage opportunity Part 5.4: historical, implied volatility and option greeks Volatility: this parameter affects very substantially the price of the option one is trying to calculate à Difficult to assess Historical volatility: past observations of the stock price Implied volatility: calculating The implied volatility is the volatility that is implied by an option price observed in the market at the time. à Used as an estimate of the volatility one needs as input to calculate the price of another option on the same underlying stock Composite implied volatility: this is a weighted average of individual implied volatilities, where the weights reflect the sensitivity of the option price to the volatility. Sensitivities/Greeks of option prices Impact of small change in underlying asset price Delta becomes steeper around the strike Delta price (maturity date) If uncertain, position delta = 0 The delta of an In-The-Money call option (St ≫X ) increases as time elapses. The delta of an Out-of-The-Money call option (St ≪X ) decreases as time elapses Impact of large change in underlying asset price Gamma Gamma is largest for At-The-Money options (St ≃X ). In particular at maturity Impact of passage of time If, while the stock price remains unchanged, decreasing time to expiration decreases (increases) the Theta value of the option, we say that the option has negative (positive) time bias Theta is most substantial for At-The- Money options (St ≃X) Theta of In-The-Money put options (St ≪X) can be positive Impact of change in volatility Vega peaks around the strike price Vega Vega is largest for At-The-Money options (St ≃X ) Impact of change in interest rate Rho Rho is largest for In-The-Money options (St ≫X for calls and St ≪X for puts) Position value: is the sum of the value of the two associated securities, each weighted by the number bought or sold Position delta: is then the sum of the weighted deltas of its constituent securities The neutral position ratio can be determined by setting the position delta to zero Position gamma: is the sum of the weighted gammas of its constituent securities The absolute magnitude of the position gamma, measured at the target delta position ratio, indicates how fast changes in the stock price will push the position delta past the critical distance and force revision of the position ratio Position theta: measures how much the position value will change as time to expiration decreases, other things being equal Part 5.5: options in corporate securities and exotic options Zero-coupon bond: is a promise to receive a single payment, P (principal), at a future maturity date Warrant: gives the owner the right to buy a fixed number of newly created shares at a specified price, by a given date à Similar to call option à When warrant exercised, new shares created so value is diluted Convertible bonds: combine many of the features of ordinary bonds and warrants Like ordinary bonds, they are entitlements to ﬁxed coupon and principal payments, and have priority over the stock in the event of bankruptcy Like warrants, they can be exchanged for a speciﬁed number of newly issued shares Whereas warrants also require the payment of an exercise price when the exchange is made, convertible bonds rarely do Most convertible bonds can be converted any time before the maturity date. However in the absence of anticipated dividend payments, as for American call options, early conversion is never optimal Exotic option: it refers to derivatives whose payoffs are more intricate than combinations of American or European calls and puts à Classifying them is sometimes difficult a derivative such that part of the option contract is triggered if and only if, any time prior to expiry, the underlying asset value, hits some pre-established barrier 4 types: Down-and-out: the option becomes worthless if a barrier, S, is hit from above Up-and-out: the option becomes worthless if a barrier, S, is hit from below Down-and-in: the option remains worthless unless a barrier, S, is hit from above Up-and-in: the option remains worthless unless a barrier, S, is hit Barrier option from below Reaching this barrier can either terminate (out) or initiate (in) the existence of the option à The barrier can only be smaller or greater than the current value of the underlying, hence it can only be hit either form above (down) or from below (up) à The structure of barrier options can involve up to two barriers: a lower barrier and an upper barrier Rebate: a lump-sum to be paid if the barrier is reached (not reached) in the case of out-barriers (in-barriers) The value of such options evolve only with St and t. à The valuation of a barrier option is only a weakly path-dependent problem A derivative whose payoff depends on the maximum (or minimum) value realized by the underlying asset over the life of the option The payoff at expiry is the difference between the maximum value achieved by Lookback option the underlying asset during the life of the option, and its final value, i.e. the payoff of the option at expiry Therefore a lookback option with a continuously sampled maximum (or minimum) is not really an option, as exercise is certain A derivative whose payoff depends on a period of the history of the value of the underlying asset process, via some sort of average (arithmetic or geometric ; may involve weighted average) Its payoff is the difference, if positive, Asian option between the underlying asset value, and its average prior to the exercise date An average which gives more importance to the latest observations is typically higher than an equally weighted average à The payoff of the option would be lower Part 5.6: real options Real Option: is essentially an opportunity, but not the obligation, to take a deferred action/decision A call option found in most projects where one has Deferral option the possibility to delay the start of the project Abandon a project for a certain price is essentially a Option to abandon put option Expand by paying to scale up the operations is a call Option to expand option Extend the life of a project by paying a supplement is Option to extend also a call option The possibility to stop (or start) operations with the Switching option further possibility to re-start (or re-stop)

UE 2 - Capital Markets and Financial Instruments PDF

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