Forward Contracts

Equity forward contract

FP = (S₀ – PVD) × (1 + R_f)^T

 to long position

Equity index forward contract

Forward contract on fixed income security

FP = (S₀ − 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.
• mark-to-market 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 mark-to-market price

• higher reinvestment rate for gains & lower borrowing costs to fund losses -> preference for mark-to-market futures
• if investors prefer mark-to-market feature, futures price > forward price, interest rates // asset values

• Storage/holding costs -> increase no-arbitrage futures price
• Monetary benefits (e.g. dividend, coupon bonds) -> decrease no-arbitrage futures price
• Convenience yield: return from non-monetary 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 LIBOR-based deposits priced as add-on yield -> Deposit value not perfectly hedged -> Eurodollar futures cannot be priced using standard no-arbitrage framework

Futures no-arbitrage price

• Treasury bond

• Stock

• Equity index

• Currency

Options

Synthetic position created by combining the other three terms from put-call 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 pure-discount riskless bond
• Synthetic European put option = C + X - S
• Synthetic stock position = C + X - P
• Synthetic pure-discount 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 put-call parity doesn’t hold, an arbitrage profit is available.

Option price calculation on an equity using a two-period binomial model:

1. Calculate the stock values at the end of two periods
2. Calculate option payoffs at the end of two periods
3. Calculate expected values using the up- and down-move probabilities
4. Discount these back at Rf

Option valuation on a fixed-income instrument using a binomial interest rate tree:

1. Price the bond at each node using the projected interest rates
2. Establish the intrinsic value of the option at each node at the maturity of the option.
3. Bring the terminal option values determined in Step 2 back to today.
4. Assume probability of an up-and-down move in the interest rate tree is always 50%

Interest rate caplet: European-style call option on interest rates

Interst rate floorlet: European-style put option on interest rates

Black-Scholes-Merton model

Assumptions

• No-arbitrage condition
• Underlying asset price follows a lognormal distribution(eleminates probability of negative prices)
• Risk-free 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.

Delta-neutral 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 one-unit change in underlying security price

1. 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 Black-Scholes-Merton 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 off-market 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 n-period 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 (n-m) year swap after m years
• primary usage
  1. lock in a fixed rate
  2. speculate on interest rates
  3. terminate swap termination.
• Payer swaption permits the holder to enter a swap as fixed-rate payer; valuable when swap rates increase
• Receiver swaption permits the holder to enter a swap as fixed-rate 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 risk-free rate for

1. Interest rate(funding) risk of the bond
2. 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 10-year 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 4-year maturity, a customized contract could be devised using 3- and 5-year maturity swaps.
• Confidentiality: Credit derivatives are confidential, over-the-counter 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 short-term prospects for a bond issuer. In the flattener, the investor believes that the issuer has some short-term instability, but that its long-term prospects are sound. In the steepener, the investor believes that the issuer has the ability to subsist in the short term, but that its long-term 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 market-wide 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.