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102 Cards in this Set
- Front
- Back
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Forward contract
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an agreement between two parties in which one party, the buyer (“long”), agrees to buy from the other party, the seller (“short”), an underlying asset or other derivative, at a future date at a price established at the start of the contract
It locks in a price No money is exchanged at the start Can require delivery of the underlying asset, or cash settlement – where the net cash value is exchanged on the delivery date To terminate a forward contract – create a new forward contract expiring at the same time as the original forward contract, taking the position of the seller, instead of the buyer – in this situation you would be long and short the asset, with the sole exposure being the price differential between when both contracts were created |
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Traditional “Value” definition
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is what you can sell something for or what you must pay to acquire something; valuation is the process of determining the value of an asset or service
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Derivatives definition of Pricing:
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to determine the forward price or forward rate
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Derivatives definition of Valuation:
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to determine the amount of money that one would need to pay or would expect to receive to engage in the transaction; or if you already had a position – to determine the amount of money one would either have to pay or expect to receive in order to get out of the position
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Off market FRAs "start"
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start the contract with an initial value (so money does change hands) – if the negotiated value is positive, the long pays that amount up front to the short; if negative, the short pays that amount up front to the long
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Equity forward contracts
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Adjust for dividends that occur over the life of the forward contract
For discrete dividend payments will need to find the PV of all dividends that are paid during the contract time period For continuous dividend payments – need the constant dividend rate (Dc), and to convert the RFR into a continuously compounding rate Rc = ln(1+r) |
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Fixed-income security forward contract
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identical to equity forward contract valuation but coupons instead of dividends need to be considered
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forward rate agreement (FRA)
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Contracts where the underlying is simply an interest payment made in dollars, Euribor, or any other currency at a rate appropriate for that currency
Cannot simply “divide by 365” as FRAs have specific day counts and therefore an expiration prior to the maturity date of the forward contract Forward price (rate) = FRA(0,h,m) where “h” is the FRA expiration date and “m” is the time between “h” and “T”, where “T” is the maturity date of the forward contract and also = h+m; “L” stands for LIBOR rate used (similar to price at any given time within the contract) FRA(0,h,m) = {(1 + (Lo at h+m*T/360))/(1 + (Lo at h*(h/360))) – 1} * (360/m) FRA value at “g” (random time during the contract) |
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Currency forward contract
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used to “lock” in forward currency rates
Interest rate parity suggests that the forward rate will exceed the spot rate if the domestic interest rate exceeds the foreign interest rate (or selling at a premium – solely based on rates) |
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Credit risk in forward contracts
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Credit risk is if one party is unable to complete its obligation forcing the receiver back to the market at current rates, rather than the rates “locked in” with forward contracts
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How market value is a measure of exposure to a party in a forward contract
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Market value is a measure of exposure, as the difference between forward contract value and market value is the “simple profit or loss” potential within any given contract and time period
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Why the future price must converge to the spot price at expiration?
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Unlike forward contract prices, futures prices fluctuate in an open and competitive market
In order to avoid arbitrage opportunities, the futures price must converge to the spot price at expiration: fT(T) = ST If this did not occur and fT(T)<ST, a trader could buy the futures contract, let it immediately expire, pay fT(T) to take delivery of the underlying, and receive an asset worth ST |
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Value of a future contract throughout its life
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Since no money changes hand, the value of a futures contract at the initiation date is zero
During the life of a futures contract, the value is = ft(T) – ft-1(T) or the gain or loss accumulated since the account was last marked to market Very simplified compared to forward contracts, because its actively traded and marked to market daily |
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Why forward and futures prices differ?
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Futures contracts settle daily and are essentially free of default risk
When futures prices are negatively correlated with interest rates, traders will prefer not to mark to market, so forward contracts will carry higher prices and vice versa When there is zero correlation, there would be a very small difference between forward and futures prices, therefore: VT(T) = ST – fo(T)=ft(T) – fo(T), or just fo(T) = ST Futures price is = fo(T) = So * (1+r)^T to prevent arbitrage |
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Monetary and non-monetary benefits and costs associated with holding the underlying asset, and explain how they affect the futures price
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Storage costs/carrying costs: physical characteristics of the underlying asset that may make storage or the cost to physically hold onto the underlying asset, expense or difficult
FV(SC,0,T) represents those storage costs, just like the dividends or interest pmts in forwards, but this time costs that could be compensated fo(T) = So*(1+r)^T + FV(SC, 0, T) When the underlying asset produces cash flow – similar to our dividends or interest payments, this is a detractor form futures price fo(T) = So*(1+r)^T – FV(CF, 0,T) Non-monetary benefits – a home may fall into this category, because it provides an alternate use (not just monetary benefits) as a place to live - but pricing this type of convenience yield (to reduce futures price) is difficult FV(CB,0,T) = costs of storage – non-monetary benefits (or the convenience yield fo(T) = So(1+r)^T + FV(CB, 0, T) called the cost-of-carry model |
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Contango
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futures price exceeds spot price, because it is more common for the costs + the interest to exceed the benefits
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Normal contango
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the futures price exceeds the expected spot price
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Backwardation
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futures price are less than spot price, benefits exceed costs
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Normal backwardation
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the futures price is lower than the expected spot price
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The relation between future prices and expected spot prices
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Some consider futures price to be an unbiased predictor of the future spot price, but this theory does not factor in the risk premium, and why a trader would want to take on the risk of the asset (or the transferal of risk, without being compensated)
Risk premium – represents a discount off of the expected value that is embedded in the current price – this is done in order to compensate the trader (because of the transfer of risk) – this risk premium transfers from the holder of the asset to the buyer of the futures contract If the risk premium is not consider - you tend to overshoot the futures price and undershoot the future spot price |
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Difficulties in pricing Eurodollar futures
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No risk-free combination of the Eurodollar time deposit and a Eurodollar futures contract can be constructed – a mismatch in design of the two markets, but can still be used as a hedging tool – it’s just an imperfect hedge
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Adjusting for mispricing of futures compared to stocks
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if the future contract is selling for more than the calculated futures price, you can buy the stocks and sell the futures; meanwhile collect and reinvest the dividends and at expiration receive a gain greater than the RFR
If the futures contract is selling for less than the calculated futures price, you can sell short the stocks and buy the futures; meanwhile pay the dividends while holding the stocks, and then buying the stocks back at a price that implies he borrowed money and paid it right back at a rate less than the RFR |
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Role of derivatives markets
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Derivative markets provide price discovery and risk management, make the markets for the underlying assets more efficient, and permit trading at low transaction costs
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Options
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give you the right, but not the obligation, to buy or sell an underlying asset. They, however, require payment (“option price” sometimes called the “option premium” or “premium”) upfront from the option buyer (“long” or “option holder”) to the option seller (“short” or “option writer”). The terms are open and customizable between two parties – therefore they can be privately created OTC, and are subject to credit risk. There are options traded publically, in which case, they are standardized and the clearinghouse takes on the credit risk. The underlying of an option could be a forward or futures contract. All options have a predetermined exercise or strike price, as well as, an expiration date
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Call
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an option that grants the right to buy the underlying; the minimum value is 0, the maximum value is the underlying price (same for American and European calls)
Call options have a lower premium the higher the exercise price Both call and put options are cheaper the shorter the time to expiration – all along the same vein that the “opportunity” for upside has diminished causing the option to be less valuable. |
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Put
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an option that grants the right to sell the underlying; the minimum value is 0, the maximum value is the exercise price for an American put, or the PV of the exercise price = X/(1+r)^T for a European put
Put options have a lower premium the lower the exercise price Both call and put options are cheaper the shorter the time to expiration – all along the same vein that the “opportunity” for upside has diminished causing the option to be less valuable. |
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European option
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the option can be exercised only on its expiration day
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American option
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an option that can be exercised on any day through the expiration day – should therefore be worth more than a European option
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Moneyness
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the relationship between the price of the underlying and the exercise price with in-the-money options being those in which exercising the option would produce a cash inflow that exceeds the cash outflow, and out-of-the-money producing the opposite
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Payoff - options
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at expiration, an option is worth either zero or the difference between the underlying price and the exercise price, whichever is greater (a call takes the position of an increasing underlying, St – X is > 0 while a put takes the position of a decreasing underlying, X – St >0)
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Fiduciary call
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buy a call and buy a bond
consists of a European call and a risk-free bond that matures on the option expiration day and has a face value equal to the exercise price of the call. This allows protection against downside losses and is thus faithful to the notion of preserving capital. |
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Protective put
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buy a put and buy an underlying asset – a European put and the underlying asset. Must be equal to a fiduciary call to avoid arbitrage opportunities
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Put-call parity
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a fiduciary call = a protective put OR co + X/(1 + r)^T = po + So
by rearranging this formula can create synthetic options and determine whether to go long or short – i.e. if we solve for co and it’s a negative, short the co, if its position, go long the co Only European options are used in this simplified example Investors create these instruments to exploit arbitrage opportunities when the put-call parity does not balance. Selling the overpriced “side” the protective put or the fiduciary call, and buying the other “side” to make the spread |
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Interest rate impact on option prices
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when interest rates are higher, call option prices are higher (you save more money by not paying for the underlying until a later date – collecting interest until you exercise your option) and put option prices are lower (because investors lose more interest locked up in an underlying, while waiting to sell the underlying when using puts, the opportunity cost is greater)
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Volatility impact on option prices
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higher volatility increases call and put option prices because it increases possible upside values and increases possible downside values of the underlying. Does not have a negative impact on either the call or the put, because of the minimum, “0” value, creates a floor regardless of how “out of the money” it is. Volatility cannot be measured, only estimated
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Delta
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the sensitivity of the option price to a change in the price of the underlying (duration)
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Gamma
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a measure of how well the delta sensitivity measure will approximate the option price’s response to a change in the price of the underlying (convexity)
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Rho
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the sensitivity of the option price to the RFR (the continuous RFR within Black-Scholes)
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Theta
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the rate at which the time value decays as the option approaches expiration, or to 0
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Vega
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the sensitivity of the option price to volatility
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The final price of the cap
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= the one-period caplet cost + the found cap price (using the binomial tree)
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Zero cost collar
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long cap and a short floor with exercise rates such that the premium on the cap is equal to the premium on the floor
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Assumptions underlying the Black-Scholes-Merton model
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Assumes a continuous-time world
Assumes the underlying price follows a lognormal probability distribution as it evolves through time; lognormal = one in which the log return is normally distributed; log return is ln (1 + r) Assumes the RFR is constant and known Assumes the volatility of the underlying asset (specified in the form of standard deviation of the log return) is known and constant at all times Assumes there are no taxes or transaction costs Assumes the options are European (use the binomial model for American options) |
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Five inputs in the Black-Scholes-Merton model
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the underlying price
the exercise price the RFR the time to expiration the volatility |
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Factors impacting Call option price in Black-Scholes-Merton model
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Call options should move higher, the higher the underlying, the longer the time to expiration, the higher the volatility, the higher the RFR, and the lower the exercise price.
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Factors impacting Put option price in Black-Scholes-Merton model
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Put option should move higher, the higher the exercise price, the higher the volatility, the lower the underlying and the lower the RFR.
Can be either higher or lower the longer the time to expiration under European, whereas under American, put options are always higher the longer the time to expiration (but American is not applicable under Black-Scholes) |
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Call Delta
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change in the option price/change in the underlying price – will converge towards “1”
Change in the option price = delta (put or call) * change in the underlying price Delta defines the sensitivity of the option price to the change in the price of the underlying |
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Put Delta
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call delta – 1
will converge towards “-1” if in-the-money or “0” if out-of-the-money Change in the option price = delta (put or call) * change in the underlying price Delta defines the sensitivity of the option price to the change in the price of the underlying |
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Dynamic hedging =
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delta hedging: adjusting the price of the call or the put based on movements in the price of the underlying
The larger the move in the underlying the less effective, or the worse accuracy |
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The gamma effect on an option’s delta and how gamma can affect a delta hedge
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If you consider duration to be similar to delta, gamma is similar to convexity, or 2nd order curve
Officially, gamma is a numerical measure of how sensitive the delta is to the change in the underlying – or how much the delta changes |
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Effect of the underlying asset’s cash flows on the price of an option
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Adjusting for cash flows is done in the same manner with others, subtracting the PV of cash flows over the life of the option from the underlying to start, and then going through the same process
It effectively lowers the price of the option |
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Historical volatility
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taking past volatility and applying it the future; estimate the standard deviation of the continuously compounded return of the underlying (if its publicly traded) – the farther “back you go” the less current, and potentially the less accurate
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Implied volatility
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tapping the market for comparable options and extracting their volatility variable from the current market price of that option – but this is not an easy variable to solve for; often found via trial and error (getting volatility within a small range and then honing in
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Put-call parity for options on forwards (or futures)
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Needs the assumption of constant interest rates, futures prices and forward prices are the same
For forwards: co + (X – F(0,T))/(1+r)^T = po * remember the initial value of the underlying, So = 0 This is the same idea, a call + forward bond = put +So (0 in the case of forwards) Can be rearranged to this: F(0,T) = (co – po)*(1 + r)^T + X |
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American vs. European options on forwards and futures
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Early exercise is never justified for American calls on underlying assets that make no cash payments or forwards that do not pay out until expiration, but can be justified for American call options on futures, and therefore for calls on futures American pricing should be> European
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Appropriate pricing model for European options
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European pricing model: the Black model: fo(T) = So*(e^rcT) and either solve for So, and apply Black-Scholes-Merton model
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Swap
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is an agreement between two parties to exchange a series of future cash flows with most often one party making payments determined by random outcomes, such as an interest rate, a currency rate, an equity return, or a commodity price.
Parties are often defined as floating or fixed-rate payer Most swaps involve multiple payments, or a series of payments Swaps are essentially a series of forward contracts There is no value at the start of the contract/initiation with the exception of currency swaps where money is exchanged in different currencies, but is an equivalent value for each party Swap market is almost exclusively OTC, so customized by party, but therefore default is possible – credit risk exists To terminate the swap – one party can pay out in cash (upon approval of the other party), or can enter into a separate and offsetting swap, or can sell the swap to another counterparty (not common), or use a swaption (the option to enter into a swap at terms that are established in advance) |
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Swaption
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the option to enter into a swap at terms that are established in advance
have expiration dates like options, can be American or European 2x5 swaption = means that the underlying is a five-year swap at the time the swaption is initiated and will be a three-year swap when the swaption expires |
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Plain vanilla swap
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is simply an interest rate swap in which one party pays a fixed rate and the other pays a floating rate, with both sets of payments in the same currency
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FRAs
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forward contracts on interest rates, with a single fixed rate and floating rate payment made between parties
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Pricing vs valuation of swaps
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Pricing of swaps: to determine the terms of the swap prior to its initiation
Valuation of swaps: no market value at the start, but this will change over the life of the swap, from zero to either an asset or a liability for each party’s perspective (depending on which side of the swap they’re on) |
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Equivalence of interest rate swaps to a series of off-market forward rate agreements (FRAs)
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a swap is essentially the combination of a series of (equal fixed payments – means this would have to be done off-market, otherwise pricing would vary) forward contracts into a single transaction; here also, the next payment that each party makes is known
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Equivalence of a plain vanilla swap to a combination of an interest rate call and an interest rate put
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equates the payment required to buy or short an option to a swap payment with the same payment dates
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pricing an interest rate swap
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means finding the fixed rate that equates the PV of the fixed payments to the PV of the floating payments, a process that also sets the market value of the swap to zero at the start
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Four type of currency swaps
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pay one currency fixed, receive the other fixed
pay one currency fixed, receive the other floating pay one currency floating, receive the other fixed pay one currency floating, receive the other floating (no fixed rate in this scenario) |
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Currency swap properties
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All currency swaps have two notional principals, one in each currency
The fixed rate for a currency swap = one for each “side” that is fixed, calculated the same way The notional principal on a currency swap = one for each side, converted to the appropriate currency exchange rate with provided current exchange rates (the inverse, typically) The estimated market value during the life of the swap = same methodology, but be sure to adjust for the inverse of the original exchange rate * the new exchange rate in the final step when the PV of fixed and floating remaining payments have already been calculated - essentially convert foreign value into domestic and subtract the difference to compare the value of all four currency swaps |
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Three types of equity swaps
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A swap to pay a fixed rate and receive the return on the equity
value = (St/So) – the discount term at expiration from “the new set” – (fixed rate) * (sum of the PV discount factors from “the new set”) Where St and So represent the equity index value at “t” and “o” A swap to pay a floating rate and receive the return on the equity value = (St/So) – (1 + first floating rate *(T/306))*(first PV discount factor of “the new set”) A swap to pay the return on one equity and receive the return on another Value = (St/So) – (Ft/Fo) where F’s are a different equity index value at time “t” and “o” |
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Payer swaption
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allows the holder to enter into a swap as the fixed-rate payer and the floating-rate receiver
a put option |
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Receiver swaption
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allows the holder to enter into a swap at the floating-rate payer and the fixed-rate receiver: a call option
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Who uses swaptions?
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Swaptions are used by parties who anticipate the need for a swap at a later date but would like to establish the fixed rate today, while providing the flexibility to not engage in the swap later or engage in the swap at a more favorable rate in the market; OR may be used by parties entering into a swap to give them the flexibility to terminate the swap; OR used to speculate about interest rates
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Ways to exercise a swaption
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When a swaption is exercised it creates a stream of annuity payments that are equal to the different between the exercise rate and the market rate on the underlying swap when the swaption is exercised
Ways to exercise a swaption: i. Exercise the payer swaption, thereby entering into a swap to pay the predetermined swaption fixed rate and receive-floating swap ii. Exercise the payer swaption, thereby entering into a swap to pay the predetermined swaption fixed rate, receive-floating swap and then enter into a receive-fixed swap, pay-floating swap at the market rate (floating “cancels” and you receive the fixed spread between market and swaption rates) iii. Exercise of swaption with offsetting swap netted |
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swap credit risk for each party and during the life of the swap
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The party holding the positive-value swap assumes the credit risk, as the counterparty could declare bankruptcy; netting is therefore used to reduce the total liability to just the difference
Credit risk varies through the life of a swap – interest rate or equity has no final principal payments, so there is not much risk near the end, and at the beginning you could assume you would never enter a swap if there was a credit risk issue – greatest credit risk in the middle! For currency swaps, the greatest credit risk is the middle to the end, given the notional principal swap at the end of its life |
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Current credit risk
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when payment is immediately due and cannot be made by one party
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Potential credit risk
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the ever-present possibility that, although a counterparty may currently be able to make payments, it may be unable to make future payments
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Marking to market a swap
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finding the value of the swap after a payment and then repricing the fixed rate before the next payment (instead of using the same one the whole time) – essentially terminating the swap and reestablishing it on a predetermined schedule (to reduce credit risk)
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Swap spread
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the spread of the swap rate over the default-free rate; IS NOT A MEASURE OF CREDIT RISK FOR THE INDIVIDUAL, BUT RATHER THE GLOBAL ECONOMY
When the swap spread widens, recession or credit concerns are abound When the swap spread tightens = economy doing well, good credit quality |
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Interest rate cap
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when one party agrees to the pay the other when the reference rate exceeds a predetermined level (or strike rate) – pays the difference between market and cap rate
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Interest rate floor
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when one party agrees to the pay the other when the reference rate falls below a predetermined level (or strike rate) – pays the difference between market and floor rate
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Equivalence of buying a cap (long cap)
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is the equivalent to buying a package of puts on a fixed-income instrument
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Equivalence of buying a put (long floor)
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is the equivalent to buying a package of calls on a fixed-income instrument, as caps and floors can be seen as packages of options on interest rates
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Cap equivalence
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is equivalent to a package of call options on an interest rate, as the call option’s value increases as interest rates increase and vice versa
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Caplet
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each of the interest rate options comprising a cap
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Floor equivalence
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is equivalent to a package of put options on an interest rate, as the put option’s value decreases as interest rates increase and vice versa
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Floorlet
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each of the interest rate options comprising a floor
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Cap payoff
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the difference between the market rate and the cap * notional principal
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Floor payoff
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the difference between the floor and the market rate * notional principal
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Interest rate collar
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a cap and a floor – the net premium that the borrower who wants a collar must pay, is the difference between the premium paid to purchase the cap and the premium received to sell the floor
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Characteristics of a Credit default swap (CDS)
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the building blocks of credit derivatives (1/2 of volume) – a bilateral contract between the protection buyer (short the credit) and a protection seller (long the credit) that exchanges the credit risk of a specific issuer/entity
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Credit events within CDS
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events include: bankruptcy, a material default, and debt restructuring for a specified reference asset – if a credit event occurs, the protection buyer delivers the reference obligation and the protection seller returns the notional amount; some are also settled in cash
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Cost variance of CDS
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Yield spreads on CDS (what it costs, bps) widen in high credit risk environments and tighten in good times
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CDS vs. Corporate bonds
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CDS differs from corporate bonds, because it only reflects credit risk vs. bonds that encompass the RFR, funding risk, and credit risk
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Advantages of using credit derivatives over other credit instruments
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Portability of pure risk
An efficient way to short credit vs. shorting corporate bonds which can be difficult to construct and may be very expensive depending on demand Liquidity is growing in credit derivatives market – meaning most names are more readily traded in the CDS market; more options and can be done confidentially |
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Use of credit derivatives by the various market participants
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Banks – 35-40% and decline – used CDS to hedge their risk (loan portfolios) and to obtain regulatory capital relief
Securities houses (or investment banks) – are the market makers in CDS – the biggest group, used to hedge their underlying corporate bond portfolios and provide access and liquidity to the rest of the participants in the market Hedge funds – about 15% of the market and largest growing segment – as credit hedge funds emerge and opportunistically trade a number of different strategies – more trading orientated, creating essential liquidity for the rest of the market participants Insurance companies – mainly act as sellers of protection (going long credit), or use CDS to enhance yield and provide diversification in their portfolios |
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Basis trades - CDS trading strategy
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cash default basis – the difference between the CDS premium and the asset swap spread on the bond; typically look for 10-20bps of upside for these trades, and tough because it can quickly be arbitraged away
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Curve trades - CDS trading strategy
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enables an investor to express views on how a company’s credit profile will change over time; can either match the notional of the long and short trades to be default neutral, but not duration neutral, OR to be duration neutral and the investor weights both sides of the trade by adjusting for duration
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Flattener - curve trade
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buying short protection and selling long protection to establish forward exposure – believes that the company has overall good credit, but may have short-term uncertainty
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Steepener - curve trade
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sells the short protection and buys the long protection – has a bearish view of the long-term, but believes the company has good short-term liquidity
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Index trades - CDS trading strategy
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an efficient and inexpensive way to take a macro view on credit as an asset class; macro capital structure strategy = investor goes long a credit index and short an equity index
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Receiver Option trade - CDS trading strategy
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the right to sell protection at a specified level at some date in the future (assumes a bullish view on credit and will make money when spreads tighten)
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Payer Option trade - CDS trading strategy
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the right to buy protection at a specified level at some date in the future (assumes a bearish view on credit and will make money when spreads widen)
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Capital structure trades - CDS trading strategy
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Taking a stance between a holding company and its operating company
Senior vs. subordinate debt spread Credit vs. equity trades |
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Correlation trades - CDS trading strategy
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the likelihood that any two credits in a portfolio will default together
First to default swap basket – if one of the group goes, the contract ceases and a payout ensues |
