R46 BASICS OF DERIVATIVE
PRICING AND VALUATION
-General-
Term
Types of investors:
risk-averse
- averse: strongly dislike sth
- So the discount rate for higher risk asset will need a positive premium (higher return) plus risk-free rate
risk-neutral
- would require no risk premium
- discount rate = risk-free rate
Arbitrage
Definition:
- refers to a transaction in which an investor purchases one asset or portfolio of assets at one price and simultaneously sells an asset or portfolio of assets that has the same future payoffs, regardless of future events, at a higher price, realizing a risk-free gain on the transaction
Apply No-arbitrage condition in valuation of derivatives securities
- arbitrage opportunities may be rare
- the same future payoffs as the derivative, regardless of future events
- small differences in price may persist because the arbitrage gain is less than the transactions costs of exploiting it
Derivatives:
Definition: A derivative is a financial instrument that derives its performance from theperformance of an underlying asset
Types:
The two principal types of derivatives are forward commitments and contingent claims
Forward commitment: A forward commitment is an obligation to engage in a transaction in the spot market at a future date at terms agreed upon today. There are three types of forward commitments: forward contracts, futures contracts, and swap
contracts
A forward contract is an over-the-counter derivative contract in which two parties agree that one party, the buyer, will purchase an underlying asset from the other party, the seller, at a later date at a fixed price they agree upon when the contract is signed
A futures contract is a standardized derivative contract created and traded on a futures exchange in which two parties agree that one party, the buyer,will purchase an underlying asset from the other party, the seller, at a later
date at a price agreed upon by the two parties when the contract is initiated and in which there is a daily settling of gains and losses and a credit guarantee by the futures exchange through its clearinghouse
A swap contract is an over-the-counter derivative contract in which two parties agree to exchange a series of cash flows whereby one party pays a variable series that will be determined by an underlying asset or rate and
the other party pays either 1) a variable series determined by a different underlying asset or rate or 2) a fixed series
Contingent claims:
- A contingent claim is a derivative in which the outcome or payoff is determined by the outcome or payoff of an underlying asset, conditional on some event occurring Contingent claims include options, credit derivatives, and asset-backed securities.
Pricing the underlying
- The value of the financial asset is the expected future price plus any interim payments such as dividends or coupon interest discounted at a rate appropriate for the risk assumed.
The Formation of Expectations
- the underlying does not pay interest or dividends, nor does it have any other cash flows attributable to holding the asset
The Required Rate of Return on the Underlying Asset
- We use the symbol k to denote this currently unknown discount rate, which is often referred to as the required rate of return and sometimes the expected rate of return or just the expected return. At a minimum, that rate will include the risk-free rate of interest, which we denote as r.
The Risk Aversion of the Investor
- three potential types of investors by how they feel about risk: risk averse, risk neutral, or risk seeking
Risk neutral thì return là risk-free rate, risk averse thì yêu cầu a risk premium, còn risk seeking chấp nhận negative risk premium
The Pricing of Risky Assets
- So: current price, E(St) expected cash flow with no interim cash flow, r: risk free rate and anpha: risk premium
Other Benefits and Costs of Holding an Asset
- We use the symbol θ (theta) to denote the present value of the costs and γ (gamma) as the present value of any benefits
- The net of the costs and benefits is often referred to by the term carry, or sometimes cost of carry
- There is substantial evidence that some commodities generate a benefit that is somewhat opaque and difficult to measure. This benefit is called the convenience yield.
The principle of arbitrage
The (in)frequency of arbitrage opportunities
Arbitrage and derivatives
Arbitrage and replication
Risk Aversion, Risk Neutrality, and Arbitrage-Free Pricing
Limits to Arbitrage