Forward Contracts
Equity forward contract
 FP = (S_{0} – PVD) × (1 + R_{f})^{T}
 to long position
Equity index forward contract
Forward contract on fixed income security
 FP = (S_{0} − PVC) × (1 + R_{f})^{T}
 The value of an FRA at maturity is the interest savings to be realized at maturity of the underlying "loan" discounted back to the date of the expiration of the FRA at the current LIBOR rate. The value of an FRA prior to maturity is the interest savings estimated by the implied forward rate discounted back to the valuation date at the current LIBOR rate.
Currency Forwards
Credit risk in a forward contract
 The party with the position that has positive value has credit risk in this amount because the other party would owe them that amount if the contract were terminated.
 The contract value and, therefore, the credit risk, may increase, decrease, or even change sign over the remaining term of the contract.
 marktomarket part way through in order to reduce credit risk.
Futures
 Spot (cash) price: price of a commodity or financial asset for immediate delivery.
 Futures price: price today for delivery at some future point in time (the maturity date)
 Basis = spot price – futures price
 As the maturity date nears, the basis converges toward 0. Spot price = futures price at expiration
Contrast with forwards
 accumulate value changes over contract term
 marked to market daily > margin deposit adjusted for daily gains/losses
 futures contract value strays from zero only during the trading periods
 Value of futures contract = current futures price − previous marktomarket price
 higher reinvestment rate for gains & lower borrowing costs to fund losses > preference for marktomarket futures
 if investors prefer marktomarket feature, futures price > forward price, interest rates // asset values
 Storage/holding costs > increase noarbitrage futures price
 Monetary benefits (e.g. dividend, coupon bonds) > decrease noarbitrage futures price
 Convenience yield: return from nonmonetary benefits, e.g. having readily supply to avoid temporary shortage of inputs
 Backwardation: FV < S0
 Contango: FV > S0
 Normal backwardation: FV < E(Sf) > most likely situation in which futures prices are biased predictors of spot rates
 Normal Contango: FV > E(Sf)
 Eurodollar futures priced as a discount yield while LIBORbased deposits priced as addon yield > Deposit value not perfectly hedged > Eurodollar futures cannot be priced using standard noarbitrage framework
Futures noarbitrage price
Options
Synthetic position created by combining the other three terms from putcall parity
 Synthetic European call option = P + S  X
 Buy a European put option with the same exercise price X
 Buy the stock
 Short the present value of X worth of purediscount riskless bond
 Synthetic European put option = C + X  S
 Synthetic stock position = C + X  P
 Synthetic purediscount riskless bond = P + S  C
 Raison:
 To price options by using combinations of other instruments with known prices.
 To earn arbitrage profits by exploiting relative mispricing among the four securities. If putcall parity doesn’t hold, an arbitrage profit is available.
Option price calculation on an equity using a twoperiod binomial model:
 Calculate the stock values at the end of two periods
 Calculate option payoffs at the end of two periods
 Calculate expected values using the up and downmove probabilities
 Discount these back at Rf
Option valuation on a fixedincome instrument using a binomial interest rate tree:
 Price the bond at each node using the projected interest rates
 Establish the intrinsic value of the option at each node at the maturity of the option.
 Bring the terminal option values determined in Step 2 back to today.
 Assume probability of an upanddown move in the interest rate tree is always 50%
Interest rate caplet: Europeanstyle call option on interest rates
Interst rate floorlet: Europeanstyle put option on interest rates
BlackScholesMerton model
 Assumptions
 Noarbitrage condition
 Underlying asset price follows a lognormal distribution(eleminates probability of negative prices)
 Riskfree rate is constant and known.
 Volatility of the underlying asset is constant and known.
 Markets are “frictionless”; no taxes, no transactions costs, and no restrictions on short sales
 No cash flow present, such as dividends or coupon payments
 Options valued are European options, which can only be exercised at maturity.
Greek 
Input 
Calls 
Puts 
Delta 
Asset Price (S) 
+// 
// 
Vega 
Volatility (s) 
+// 
+// 
Rho 
Riskfree rate (r) 
+// 
// 
Theta 
Time to Expiration (T) 
>0 
>0 

Exercise price (X) 
// 
+// 
Deltaneutral hedge: combine a long stock position with a short call position, so portfolio value does not change when the value of the stock changes
 Delta = change in option price for a oneunit change in underlying security price

 calls to sell = # shares hedged / delta of call option
 only holds for very small changes in underlying stock value
 must be continuously rebalanced to maintain the hedge (dynamic hedge)
Gamma measures the rate of change in delta as the underlying stock price changes
 at its maximum(1) when option is at the money and close to expiration
 measure of adjustment extent of dynamic hedge when asset price changes
Existence of cash flows on the underlying asset
 decrease the value of a call option.
 increase the value of a put option.
 decreases spot price
Future volatility estimation methods
 Historical volatility
 Convert a time series of N prices to returns.
 Convert the returns to continuously compounded returns:
 Ric = ln(1 + Ri)
 Calculate the variance and standard deviation of the continuously compounded returns.
 Implied volatility:
 This implied volatility is the value for standard deviation of continuously compounded rates of return that is “implied” by the market price of the option.
 Set the BlackScholesMerton model price equal to market price of the option. This "infers" the volatility.
 No mark to market on forwards > no gains on early exercise > American & European forward option values are same
Swaps
Interst rate swaps = series of Forward Rate Agreements(FRA)
 sum of offmarket FRAs => 0
Plain vanilla swap = combination of interest rate call & put(long & short)
 floating rate payments for fixed rate payments
Fixed periodic rate on an nperiod swap at initiation:
Currency swap = foreign currency receivables  domestic currency earnings payment
Equity swap = equity/index investment  payment for loan
Swaption: an option which gives the holder the right to enter into an interest rate swap
 m x n swaption  right to enter into (nm) year swap after m years
 primary usage
 lock in a fixed rate
 speculate on interest rates
 terminate swap termination.
 Payer swaption permits the holder to enter a swap as fixedrate payer; valuable when swap rates increase
 Receiver swaption permits the holder to enter a swap as fixedrate receiver; valuable when swap rates decrease
Swap credit risk: probability that a counterparty will default on required payments
 credit risk highest at the middle term
 risk lowered through the netting process and marking to market
Swap spread = swap rate  comparable maturity treasury notes
 default premium in LIBOR reflected in the swap rate
Et Cetera
Interest rate cap: periodic payments when the benchmark interest rate > cap rate(strike rate specified)
 usually based on portfolios of caplets(call options on LIBOR)
Interest rate floor: periodic payments when the benchmark interest rate < floor rate
 usually based on portfolios of floorlets(put options on LIBOR)
Collar?
Credit Derivatives Credit default swap(CDS): insurance contract for payments (buying protection)
 reference obligation: fixed income security on which the swap is written
 if defaults occurs on the reference obligation, swap holder receives payment from the seller
 Corporate bond yield spread reflects compensation over the riskfree rate for
 Interest rate(funding) risk of the bond
 Credit risk of the issuer
Advantages over other credit instruments:
 Risk management: Credit derivatives allow credit risk to be managed separately from interest rate risk.
 Short positions: A short position can be taken in fixed income securities by buying a credit derivative. In contrast, shorting the underlying credit obligation can be challenging and cost prohibitive, especially if the obligation is in high demand in the repo market.
 Liquidity: The credit derivative market is more liquid than the underlying cash market. Over time, trading in CDSs has increased such that 3, 5, 7, and 10year maturity swaps are fairly liquid.
 Flexibility: Credit derivatives facilitate credit, maturity, and currency positions not otherwise available in the underlying cash market. For example, if an investor wanted a position with a 4year maturity, a customized contract could be devised using 3 and 5year maturity swaps.
 Confidentiality: Credit derivatives are confidential, overthecounter contracts. In contrast, in a loan, the issuer has knowledge of the contract.
Usage
 Commercial banks use credit derivatives to hedge their exposures arising from their loan portfolios and to satisfy regulators by buying credit protection. They are the largest participant in the market.
 Investment banks act as dealers in the credit derivatives market, providing liquidity to the rest of the market. They also use credit derivatives to hedge their corporate bonds, and they have trading desks that seek to exploit mispricing.
 Hedge funds now specialize in the trading of credit risk in addition to traditional convertible arbitrage and distressed debt opportunities. In their pursuit of relative value opportunities, they have become quite active and are important providers of liquidity to the market. Hedge funds represent the fastest growing segment of the credit derivatives market.
 Life insurance, property and casualty insurance, reinsurers, and monoline companies take long positions in credit by selling protection.
Strategies
 Basis trade: The CDS premium is compared to the asset swap spread of the underlying bond. The asset swap spread should reflect the credit risk of the bond. If it is higher than the CDS premium, the basis is negative and, to exploit the arbitrage opportunity, the investor should buy the bond and buy the CDS.
 Curve trade: The investor has different opinions than the market about the long term versus shortterm prospects for a bond issuer. In the flattener, the investor believes that the issuer has some shortterm instability, but that its longterm prospects are sound. In the steepener, the investor believes that the issuer has the ability to subsist in the short term, but that its longterm prospects are poor.
 Index Trade: Credit indices represent opportunities to use CDS to exploit perceived mispricing. Credit index strategies include: 1) A short index position to hedge a portfolio or to exploit an expected increase in marketwide credit risk, 2) If an investor is bullish on a market or sector but bearish on particular issues, he could go long an index and short specific issues, and 3) If an investor is bearish on bonds and bullish on stocks, she could go short the credit index and long an equity index.
 Options Trade: There are European receiver options and payer options available in the market. These options will change in value as the value of the underlying changes. They can be used to provide leverage, hedge, take a position in volatility, or to create straddles and other option strategies.
 Capital Structure Trade: The investor uses CDSs to exploit different views on a firm’s various securities, as in the following examples: 1) The investor believes that a subsidiary has less credit risk than the parent, so he sells a CDS on the subsidiary and buys a CDS on the parent. 2) The investor uses his opinion on a firm’s recovery rates to sell a CDS on the firm’s subordinated debt and buy a cheaper CDS on the senior debt, earning the difference in the CDS premiums. 3) The investor has differing views on the firm’s debt and equity.
 Correlation Trade: Instead of selling protection on several individual CDSs, an investor could sell protection on a basket of CDSs. The higher the number of CDSs in the basket, the higher the basket’s premium. Higher spreads on the individual CDSs result in higher basket premiums. Higher default correlations result in lower premiums.