Part 2: Basics of Derivative Pricing and Valuation Flashcards
Risk-neutral pricing
The determination of the no-arbitrage derivative price, the same as no-arbitrage pricing or the price under a no arbitrage condition.
e.g. replication, combining a risky bon with a credit protection derivative to replicate risk free bond: risky bond + credit protection = bond valued at risk-free rate.
Replication
We replicated the payoffs on one asset or portfolio with those of a different asset or portfolio.
Example of risk-neutral pricing & replication:
Investor who buys a share of stock, sells a call on stock at 40, and buys put on stock at 40 with same expiration date as the call.
The investor will receive 40 at option expiration regardless of stock price, because:
- If stock price is 40 at expiration, the put and call are both worthless at expiration.
- If stock price > 40 at expiration, the call will be exercised, the stock will be delivered for 40, and put will expire worthless.
- the stock price is < 40 at expiration, the put will be exercised, the stock will be delivered for 40, and the call will expire worthless.
Forward and futures intuition:
- The value of futures and forward contracts is zero at initiation, when forward price is no arbitrage value.
- As price of underlying asset changes during the life of the contract, the value of a futures or forward contract position may increase or decrease.
e. g. long position in forward contract to by the underlying asset in the future at $50 which is the forward contract price. - The initiation of contract, the value is zero, but contact price is $50, where if spot price of underlying asset increases, the value of long contract position ill increases, ad value of short position will decrease.
- The contract price at which the long forward will purchase the asset in the future does not change over the life of the contract, but value of forward contract almost surely will.
Convenience yield
This is the non monetary benefit of holding an asset, a difficulty to measure and only significant for some assets, primarily commodities.
If an asset is difficult to sell short in the market, owning it may convey benefit in circumstances where selling asset is advantageous.
e.g. shortage of asset may drive prices up, making sale of asset in ST profitable, while ability to look at painting or sculpture provides nonmonetary benefits to its owners, this is unlikely with corn and other commodities.
Net cost of carry
= PV(benefits of holding the asset) - PV(costs of holding the asset)
This value is positive if benefits (cash flow yield and convenience yield) > costs (storage and insurance) of holding the asset.
Forward rate agreement (FRA)
A derivative contract that has a future interest rate, rather than an asset as its underlying.
The point of entering into an FRA s to lock in a certain IR for borrowing and lending at some future date.
- on party will pay the other party the difference (based on an agreed-upon notional contract value) between the fixed interest rate specified in FRA and market interest rate at contract settlement.
e. g. LIBOR often used as the underlying rate US dollar LIBOR referred to the rates on Eurodollar time deposits, interbank US dollar loans in London.
Uses of FRA:
- these are used by firms to hedge risk of (remove uncertainty about) borrowing and lending they intend to do in the future.
- company that intends to borrow funds in 30 days could take long position in FRA, receiving payment if future 90-day LIBOR (and its borrowing cost) increase, and making payment if future 90 day LIBOR (and its borrowing cost) decreases, over 30 day life of FRA.
perfect hedge = firms borrowing costs will not be higher if rates increases, but firms borrowing costs will not be lower if IR decreases.
- for funds to lend (invest) in the future, a short position in FRA can hedge its interest rate risk.
- decline in rates would decrease return on funds loaned at future date, but positive payoff on FRA would augment returns so the return from both short FRA and loaning funds is no-arbitrage rate that is the price of FRA at initiation.
Banks use of FRA:
- synthetic FRA = a bank can create the same payment structure with 2 LIBOR loans, borrowing money for 120 days and lend amount for 30 days.
- end of 30 days, the bank receives funds from repayment of 30 day loan it made, and use of these funds for the next 90 days at effective rate determined by original transactions.
- effective rate of interest on 90 day loan depends on 30 day and 120 day LIBOR at the time the money is borrowed and loaned to third party.
- this rate is the contract rate on the 30 day FRA on 90 day LIBOR.
Difference in forward and futures prices
- same function in gaining exposure to or hedging specific risks, but differ in degree of standardisation, liquidity and counterparty risk.
- future gains and losses are settled each day, and the margin balance is adjusted accordingly.
- gains put margin balance above initial margin level, any funds in excess of that level can be withdrawn.
- if losses put margin value below min. margin level, funds must be deposited to restore account margin to its initial (required) level.
- forwards typically do not require/provide funds in response to fluctuations in value during their lives.
- if IR are constant, or uncorrelated with futures prices, the prices of futures and forwards are the same.
- a positive correlation between IR and future prices means for long position daily settlement provides funds (excess margin) when rates are high, and earn more interest, but requires funds when rates are low and OC of deposited funds is less.
- this causes futures prices to be higher than forward when IR and futures prices are positively correlated, and vice versa.
Simple interest rate swap
When one party pays a floating rate and the other pays a fixed rate on the notion of principle amount.
A series of forward contracts, specifically FRA, each with forward contract rate = swap fixed rate.
Since forward contract rate all equal in FRAs that are equivalent to swap, these would not be zero value forward contracts at initiation of swap.
An increase in ST future rates will produce a positive value for fixed rate payer in IR swap.
A decrease in ST future rates produces a negative value as promised FRP have more value than expected floating rate payments over life of swap.
Off market forward
- A forward contract created with a contract rate that gives it a non-zero value at initiation.
i. e. forward contracts equivalent to the series of swap payments. - since swap itself has zero value to both parties at initiation, it must consist of some off market forwards with positive present values and negative present values so sum of PV = 0.
Moneyness
This refers to whether an option in in or out of the money.
If immediate exercise of option generates a positive payoff, it is in the money, and vice versa.
When current asset price = exercise price, the exercise will generate neither gain or loss, and the option is at the money.
Exercise value (of option)
- The maximum of zero (payoff at expiration) and the amount that the option is in the money, that is exercise value is amount of option in the money if it is in money or zero if option is at or out of the money.
- The value of option if exercised immediately.
Time value (of an option)
The amount by which the option premium (price) exceeds the exercise value sometimes known as the speculative value of the option.
option premium = exercise value + time value
- at any life of an option, its value will be typically > than exercise value, as there is some probability that underlying asset price will change in amount that gives option a positive payoff at expiration > current exercise value.
