Derivatives

Novation

Process where a central clearinghouse effectively takes the opposite position to each side of an exchange-traded derivative trade

Forward commitment

Legally binding promise to perform some action in the futures

Contingent claim

Payoff depends on a future events

Forward contract

One part commits to buy and another party commits to sell an underlying at a specific price on a specific date in the future

Futures contract

One part commits to buy and another party commits to sell an underlying at a specific price on a specific date in the future, but standardized and exchange traded.

Marking to market

Adjusts the margin accounts of both buyers and sellers for gains and losses. Daily basis for future contracts

Initial margin

Amount of cash or collateral that a party must deposit before entering a futures position

Maintenance margin

Minimum amount the party must have in the margin account to keep a futures position

Swaps

Exchange a series of payments on multiple settlement dates over a specific period. Settles periodically at the end of each period. The sum of all FRA values = 0 (even though individual FRAs might have positive or negative values)

Long Swap:

PV(To be received Expected Floating Payments) - PV(To be paid Fixed Payments)

ex) If rates increase, swaps increase in value

Swap rate

The fixed interest rate paid or received in an interest rate swap, which is mathematically determined at inception to equate the present value of all expected future floating-rate payments with the present value of all future fixed-rate payments, ensuring the contract has an initial value of exactly $0.

Credit default swap (CDS)

One party makes fixed periodic payments in exchange for a payment to be made if there is a credit event

If a credit event occurs, the protection seller must pay an amount that offsets the loss in value of the reference security.

American options

May be exercised at any time up to and including the contract’s expiration date

European options

Exercised only on the contract’s expiration date

Hedge accounting

Permits companies to recognize gains / losses on some derivative hedge at the same time as changes in the values of assets or liabilities being hedged

Cash flow hedge

A hedge designed to protect against the volatility of future cash flows associated with a recognized asset/liability or a highly probable future transaction (e.g., hedging the future purchase price of jet fuel)

Fair value hedge

A hedge designed to protect against changes in the fair value of a recognized asset, liability, or firm commitment that is already on the books

Net Investment Hedge

A hedge designed to protect against foreign currency risk associated with a company's net investment in a foreign subsidiary or operation

Convenience yield

The non‑monetary benefit from physically holding a commodity rather than holding a futures contract

Forward Price

F₀(T) = [ S₀ + PV₀ (Costs) - PV₀ (Benefits) ] (1 + R_f )^T

Forward Value at Initiation

V_t​(T)= [S_t​+PV_t​(costs)−PV_t​(benefits)] − [F₀​(T) / (1+rf​)^(T−t)]

Forward Value at Settlement

V_T​(T)= [S_T - F₀​(T)]

Forward Rate Agreement (FRA)

The long counterparty will pay a fixed rate on a national amount of principal while the short counterparty will pay the MMR at the date on the same amount of notional principal. Parties will only exchange the net amount owed. Settles at the beginning of the interest period

Futures vs. Forward Prices w/ Interest Rate Correlations

• Positive Correlation (Asset price & interest rates): Futures > Forwards. (Futures are more desirable because long positions generate cash to reinvest when rates are high).

• Negative Correlation: Forwards > Futures. (Forwards are more desirable).

• Zero Correlation (or constant rates): Futures = Forwards.

Interest rate futures

100 - (100 * MRR_{A, B-A})

BPV of an Interest rate futures

notional principal * period * 0.01%

Convexity Bias: Interest Rate Forwards vs. Futures What is the bias, and when does it actually cause their prices to differ?

The Bias: Forwards have a natural mathematical curve (convexity). A rate drop increases a forward's value more than an equal rate hike decreases it. (Futures lack this because they are strictly linear).

When does it impact pricing?

• Long-term rates: Convexity effect is large → Significant price differences between forwards and futures.

• Short-term rates: Convexity effect is tiny → Virtually identical prices in practice.

Put-Call Parity for European Options

S₀ + p = c + X / (1+r_f)^T

Put-Call Forward Parity for European Options

F₀(T) / (1+R_f)^T + p = c + X / (1+r_f)^T

Risk neutral probability

π = ((1+ r) - d) / (u - d)

Binomial option model

c₀ = (π c⁺ + (1 - π ) c^-) / (1+r)

Hedge ratio

n = (c⁺ - c^-) / (s⁺ - s^-)