For call options, a larger gamma means _______

Number of calls needed to hedge (formula)

For call options larger gamma means that as the asset price increases, the delta of option A increases more than the delta of option B.

The number of calls to hedge is:

(– 1/delta)x(number of shares)

Normal backwardation means that expected futures spot prices are greater than futures prices. It suggests that when hedgers are net short futures contracts, they must sell them at a discount to the expected future spot prices to get investors to buy them. The futures price rises as the contract matures to converge with spot prices.

Normal contango occurs when the futures price is greater than the expected asset price at contract expiration. The statement that high demand to buy the contract could increase the contract price is also correct. Note the contrast with contango, which means the futures price is above the asset's spot price. (LOS 49.f)

Put-call parity for options on forward contracts is:

c₀ + (X – FT) / (1+R)^T = p₀

c = call option

FT = forward price at time t

A payer swaption gives one the right to enter a swap in the future as the fixed-rate payer / (pay a fixed rate below market if rates rise)

When interest rate changes are negatively correlated with the price changes of the asset underlying a futures/forward contract:

(Futures/forwards) contracts are higher

 Exercising an in-the-money swaption effectively generates

an annuity over the term of the underlying swap 

A positive payoff to a receiver swaption each quarter is the:

Portfolio variance = σ²_p =

(1 / n) σ ²₁ + [(n − 1) / n]cov

To initiate an arbitrage trade if the futures contract is underpriced, the trader should:

(long/short) the asset

(invest/lend) at risk free rate

(buy/sell) futures

short the asset, invest at the risk-free rate, and buy the futures.

Delta of an instantaneously riskless hedged portfolio:

A riskless portfolio is delta neutral; the delta is zero.

A large OAS indicates (2 things):

A large OAS indicates a wider risk-adjusted spread and lower relative price

Option cost measures:

prepayment risk

In general, the (highest/lowest) OAS and (highest/lowest) option cost is most attractive

highest OAS; lowest option cost

The spread (k) that must be added to all of the spot rates along each interest rate path that will force equality between the average present value of the path’s cash flows and the market price (plus accrued interest) for the MBS being evaluated

An investor who anticipates the need to exit a pay-fixed interest rate swap prior to expiration might:

(buy/sell) a (payer/receiver) swaption

buy a receiver swaption

The swap spread will increase with:

The swap spread is the spread between the fixed-rate on a market-rate swap and the Treasury rate on a similar maturity note/bond. Since the fixed rate is calculated from the reference rate yield curve, it is increased as the credit spread embedded in the reference rate yield curve increases.

Gamma measures rate of change in delta as underlying stock price changes

Gamma, the curvature of the option-price/asset-price function, is greatest when the asset is at the money.

The price of a forward contract:

is the settlement price for the underlying asset. 

The price of a forward contract is the price of the underlying asset that the long will pay to the short at settlement (for a deliverable contract). The value of a forward contract comes from the difference between the forward contract price and the market price for the underlying asset. This difference between price and value is a key concept to understand. A forward contract has only one price, which applies to both the long and to the short.

An arbitrage transaction involving asset swap spread relative to CDS spread can be problematic when:

asset swap spread is less than CDS spread

When asset swap spreads are less than CDS spreads on an equivalent credit exposure, the indicated arbitrage is problematic because it requires a short position in the asset swap. A short position in an asset swap requires that the subject bond must be shorted; however, many bonds cannot be shorted because they have no liquid market.

Regarding deep in-the-money options on forwards, it is _______ worthwhile to exercise calls and/or puts early

NEVER

Unlike futures, forwards do not generate any cash at exercise even when they are deep in-the-money so there is no advantage to early exercise.

CMBS are non-recourse. Residential mortgages are recourse, meaning that the lender can go back to the homeowner for payment if the collateral is insufficient.

What is the difference between spot and futures prices at expiration?

The difference between the spot and the futures price must be zero at expiration to avoid arbitrage.

Value of currency forward contract to long (formula):

Forward price (formula):

V_t= [S_t / (1 + R_f)^(t/365)] − [FP/ (1 + R_d)^(t/365)]  

FP(currency) = S₀ x [(1+R_D)^t / (1+R_F)^t]

Figure out payment in a plain-vanilla swap

A U.S. firm (U.S.) and a foreign firm (F) engage in a four year plain-vanilla annual pay currency swap. The U.S. firm pays fixed in the FC and receives floating in dollars. The fixed rate at initiation and at the end of the swap was 5%. The variable rate at the end of year 1 was 4%, at the end of year 2 was 6%, and at the end of year 3 was 7%. At the beginning of the swap, $2 million was exchanged at an exchange rate of 2 foreign units per $1. At the end of the swap period the exchange rate was 1.75 foreign units per $1. 

At the end of year 3, firm F will pay firm U.S.: 

A plain-vanilla currency swap pays floating on dollars and fixed on foreign. The floating rate cash flows on the settlement date are based on the previous period's ending floating interest rate 0.06 x $2,000,000 = $120,000. 

Prepayment tranching is also referred to as:

time tranching

Prepayment tranching refers to when an asset or mortgage backed security is subdivided so some components are exposed to more prepayment risk than others. 

The delta of an option is (formula for call and put + def):

Delta_call = ($change in option price) / ($change in asset price)

Delta_put = Delta_call - 1

the dollar change in option price per $1 change in the price of the underlying asset.

Price does not change over life.

The price of a swap, quoted as the fixed rate in the swap, is determined at contract initiation and remains fixed for the life of the swap.

What is the situation called when a futures price continuously increases over its life because most hedging strategies are short hedges?

Normal backwardation means that expected futures spot prices are greater than futures prices. It suggests that when hedgers are net short futures contracts, they must sell them at a discount to the expected future spot prices to get investors to buy them. The futures price rises as the contract matures to converge with spot prices.

Never

Unlike futures, forwards do not generate any cash at exercise even when they are deep in-the-money so there is no advantage to early exercise.

Have a fixed principal repayment schedule that must be satisfied as long as support tranches exist

PAC tranche has significant protection against prepayment risk at the expense of support/companion tranches

When interest rates and asset values are positively correlated, the futures price tends to be _____ than forward price? When IR and asset price negatively correlated?

Positively correlated - the futures price tends to be higher 

Negatively correlated - the futures price tends to be lower

If r_f < dividend yield (on futures contract)

Compared to purchasing bonds directly, an investor may participate in the CDS market as a:

Market value CDO vs Cash Flow CDO

The manager of a market value CDO will actively manage the portfolio to generate sufficient cash flows.

This is in contrast to a cash flow CDO, where the portfolio is structured at inception in such a way that its principal and interest payments can pay the tranches and trading profits will not be needed to support the cash flows of the CDO.

Early maturing tranches offer relatively greater protection against _______ risk.

longer-term tranches offer relatively greater protection against _______ risk.

(1) Extension

(2) Contraction

Put-call parity for options on forward contracts at the initiation of the option where the forward price at that time (time=0) is FT, can best be expressed as: (in terms of put)

Put call parity for stocks (with discrete time discounting) is: c₀ + X / (1 + R)^T − S₀ = p₀.

Noting that for the forward contract on an asset with no underlying cash flows, S₀ = FT / (1 + R)^T, and substituting, we get c₀ + (X − FT) / (1 + R)^T = p₀. 

Prepayment = (SMM)(Principal Bal. Outstand. - Monthly Principal Pmt.)

=Gross weighted avg coupon - Servicing Fee 

=Spread Available to pay Tranches

- Net Weighted Average Coupon

= Excess Servicing Spread

The writer of a receiver swaption has:

an obligation to enter a swap in the future as the fixed-rate payer

Excellent Swap value question (hard) (formula)

Question ID#:88315 

Consider a fixed-rate semiannual-pay equity swap where the equity payments are the total return on a $1 million portfolio and the following information:

180-day LIBOR is 5.2% 

360-day LIBOR is 5.5% 

Dividend yield on the portfolio = 1.2%

What is the fixed rate on the swap?

5.4234%. 

Question ID#: 127305

Calculate the price (expressed as an annualized rate) of a 1x4 forward rate agreement (FRA) if the current 30-day rate is 5% and the 120-day rate is 7%.

A 1x4 FRA is a 90-day loan, 30 days from today.

FR (x,y) = [(1+ R(x+y))/(1+ R(x))] – 1

R(x) = Current x-day rate * (x/360)

R(y) = Current y-day rate * (y/360)

Annualized rate is FR(x,y) * (360/t)

t = t-day loan (90 in example), do y-x months

The actual rate on the 30-day loan is: R30 = 0.05 x 30/360 = 0.004167
The actual rate on the 120-day loan is: R120 = 0.07 x 120/360 = 0.02333

The annualized 90-day rate = 0.0190871 x 360/90 = .07634 = 7.63% 

FR (30,90) = [(1+ R120)/(1+ R30)] – 1 = (1.023333/1.004167) – 1 = 0.0190871

The measure of prepayments associated with securities backed by auto loans is called:

absolute prepayment speed

SBA loan payments are based on:

the reference rate at the beginning of each period

Number of call options needed to create a delta neutral hedge when LONG a stock:

 (# shares of stock long) / (Delta of a call option)

=X options or (X/100) contracts.

If long the stock, short the calls.

Futures Arbitrage:

If futures market price < no-arbitrage price

*If futures market price > no-arbitrage price, use C-a-C

(futures contract ovepriced)

Use cash and carry (futures contract underpriced)

Initiation:

(1) Borrow $ for term of contract (RFR)

(2) Buy underlying asset at spot price

(3) Sell a futures contract at current F price

At expiration:

(1) Deliver the asset and receive the futures contract price

(2) Repay the loan plus interest

(2) Backwardation

(3) Contango

(4) Normal backwardation

(5) Normal contango

(2) Futurs price < spot price

(3) Futures price > spot price

(4) Futures price < expected spot price

(5) Futures price > expected spot price

Futures Arbitrage:

If futures market price < no-arbitrage price

Reverse cash and carry (futures contract underpriced)

Initiation:

(1) Sell asset short

(2) Lend short sale proceeds at RFR

(3) Buy futures contract at market price

Expiration:

(1) Collect loan proceeds

(2) Take delivery of asset for futures price and cover short sale commitment

Put-call parity for European Options

(a) left side of equation called

(b) right side called

C₀ + X/(1+R_f)^t = P₀ + S₀

(a) fiduciary call

(b) protective put

Size of downward movement (D) factor is:

D = 1/U

Risk-neutral probabability of an upward movement:

Pr(Up) = (1+R_f-D) / (U-D)

PV(Call) = Weight Avg. of Payoffs / (1+r_f)^t

Expiration (Caplet) = Max[0,(1-yr rate - cap rate)*Notional Principal] / (1+1-yr rate)

Expiration (floorlet) = Max[0, (Floor rate-1-yr rate)*NP] / (1+1-yr rate)

Value of cap (floor) = sum of caplets (floorlets)

(1) Price of underlying asset follows lognormal distribution

(2) Continuous RFR is constant and known

(3) Volatility of underlying asset is constant and known

(4) Markets are frictionless

(5) Underlying asset generates no cash flows

(6) European options only

(1) Valuing IR options and options on bond prices

(2) When assumption of constant/known volatility of underlying asset is violated

(3) Significant taxes and transaction costs

(4) American options