7 Derivatives

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Derivatives

• Derivative Markets andInstruments
  • Definition and introduction
    • Definition: A derivative is a financial instrument whose value is transformed from the performance of an underlying asset
    • 特点: Traded in future,while spot market is traded now
    • Derivative act as insurance: the underlying is the source of risk; derivative allow transfer of risk
  • Four types of derivatives
    • 类型
      • Forward
        • 定义: one party agrees to buy and the counterparty to sell a physical or financial asset at a specific price on a specific date in the future
        • Terms
          • The physical or financial asset is the underlying
          • A specific price is forward price
          • A specific date is maturity date
        • The long and short position
          • Long position: the party will buy the underlying at time T for a price of K
          • Short position: the party will deliver the underlying at time T for a price of K
        • 特点
          • OTC traded
          • Customized contracts
          • Low liquidity
          • Settlement at maturity
          • No margin deposit required
          • Deliver asset or settle by cash
          • Default risk is present
          • Largely unregulated
          • Trade with counterparty
        • Payoff
          • K =Delivery price ST =Price of the underlying at maturity
        • Purpose of forward contracts
          • Hedge: forward contracts lock buying or selling price of an underlying asset in the future
          • Speculation: assume the risk laid off by hedgers and make bets on the direction of the underlying asset
        • Settlement
          • Deliverable forwards: settled by physical delivery, the long pays K and receives an asset worth ST
          • Non-deliverable forwards (cash-settled forwards): settled by cash, the short simply pay cash to the long in the amount of (ST-K) at expiration
        • Forward rate agreement (FRA)
          • 定义: a derivative contract that has a future interest rate, rather than an asset, as its underlying, and the purpose to enter a FRAis to lock the borrowing rate in the future
          • Long/short
            • Long position: Borrower- earn a profit when market price> forward price
            • Short position: Lender- earn a profit when market price< forward price
          • 表示
            • e.g.,2*3FRA
          • Pricing
            • e.g.,60-LIBOR=3%, 90-LIBOR-3.9%
              • (1+3%*60/360)(1+FRA*30/360)=(1+3.9%*90/360)
          • Synthetic FRA
      • Futures
        • 定义: a standardized forward contract that one party agrees to buy and the counterparty to sell a physical or financial asset (the underlying) at a specific price (futures price) on a specific date in the future
        • 特点
          • Mark to market & Daily settlement
            • The most important distinctive characteristic of future contracts is the daily settlement of gains and losses
            • Mark to market/daily settlement: at the end of trading days, clearinghouse will determine the settlement price, and all contracts are to be marked to the settlement price
          • Price limits: Exchange prohibits members from executing trades at prices outside (settlement price of previous day±price limits)
          • Margin
            • Initial margin (初始保证金): when two parties enter into a futures contract, both parties are required to deposit a minimum sum of money, which is referred to as the initial margin
            • Maintenance margin (维持保证金):during the holding period, each party must maintain the required amount of money, which is referred to as the maintenance margin. Maintenance margin is always lower than initial margin
            • Margin call (追加保证金通知): when the amount of money in the margin account is below the maintenance margin requirement, then the party will get a margin call to bring the balance up to the initial margin
        • 实例
          • Margin: assume initial margin is 15%, maintenance margin is 8%
        • Settlement methods
          • At maturity
            • Asset delivered: settled by physical delivery, the long pays K and receives an asset worth ST, the contract is worth ("#-K) to the long at expiration
            • Cash settled: settled by cash, the short simply pay cash to the long in the amount of (ST-K).
          • Prior to maturity
            • Offsetting: making an exact opposite transaction to current position, the clearinghouse will net positions out
      • Neither party makes a payment at the initiation,V0=0
        • 比较
      • Swaps
        • 定义: two parties agree to exchange a series of payments on periodic settlement dates
          • At each settlement date, the two payments are "netted", only one party with the greater liability makes a payment
        • Similarity to forwards
          • At initiation, require no payment by either party
          • Custom instruments
          • Are not traded in any organized secondary market
          • Are largely unregulated
          • Default risk is present
          • Most participants are large institutions
          • Individuals are rarely swaps market participants
        • 实例: Simplest type of swap - plain vanilla interest rate swap
          • Also called fixed-for-floating interest rate swap, which is based on a notional principal (not exchanged)
          • Fixed payments payer: the party makes fixed-rate interest payments, in return receiving floating-rate payments
          • Floating payments payer: the party makes floating-rate interest payments, in return receiving fixed-rate payments
      • Options
        • 定义: the buyer pays a sum of money (option price; option premium) to the seller, and receive the right to either buy or sell an underlying asset at a fixed price (exercise price or strike price) either on or before a specific expiration date
        • Classification of options
          • 根据未来标的物资产价格涨跌的判断
            • 基本关系
            • 图示
              • Long call
                • 当ST<X ,看涨期权的多头方不会行权；当ST>X ,看涨期权的多头方会选 择行权，且随着ST价格越高，其收益和净利润越高
              • Short call
                • 一份看涨期权的多头方和空头方的收益和净利润的图形是对称的，也就是说看涨期权的多头方的收益相对于就是看涨期权的空头方的亏损。 当ST<X ,看涨期权的多头方不会行权；当ST>X,看涨期权的多头方会选择行权，且随着ST价格越高，看涨期权的多头方收益和净利润越高，相对应的看涨期权的空头方亏损越大
              • Long put
                • 当ST<X ,看跌期权的多头方会选择行权，且随着ST价格越低，看跌期权的 多头方收益越高；当ST>X,看跌期权的多头方不会行权
              • Short put
                • 看跌期权的多头方和空头方的收益和净利润图形也是关于横坐标对称的，也就是说多头方的收益相对应就是空头方的亏损，当 ST < X ,看跌期权的多头方会选择行权，看跌期权的空头方只有义务以行权价格买入标的物资产；当ST≥X ,看跌期权的多头方不会行权，看跌期权空头方收获期权费用
          • 根据期权是否可以提前交割
            • European Options: can be exercised only on the expiration date
            • American options: can be exercised on or before the expiration date
          • 根据多头方的到期日是否获利分为一一实值期权、平价期权和虚值期权
        • Value of options
          • Intrinsic Value(or exercise value)
            • call option: cT= max[0, ST — X]
            • Intrinsic value of put option: pT= max[0,X — ST]
          • Time Value
            • 定义: The difference between the price(or value) of an option and its intrinsic value
            • 计算: option value=intrinsic value+time value
            • 分类
              • Prior to expiration: option value>intrinsic value
              • At expiration: option value=intrinsic value
      • Credit derivatives
        • Definition: a contract that the credit protection seller (lender or bondholder) provides protection to the credit protection buyer against a downgrade or a default by the borrower
        • Types
          • Total return swap
            • Underlying: typically a bond or loan
            • Consists of all interest and principal paid by the borrower plus any changes in the bond's market value. In return, the credit protection seller typically pays the credit protection buyer either a fixed or a floating rate of interest
            • If the bond incurs a loss, as it surely will if it defaults, the credit protection seller effectively pays the credit protection buyer
          • Credit-link­ed note (CLN)
            • The credit protection buyer holds a bond or loan that is subject to default risk (the underlying reference security) and issues its own security (the credit-linked note)
            • The buyer of the credit-­linked note effectively insures the credit risk of the underlying reference securi
          • Credit default swap(CDS)--most common type
            • CDS is essentially an insurance contract against default. A bondholder pays a series of cash flows to a credit protection seller and receives a payment if the bond issuer defaults
          • Credit spread option
            • credit spread option is typically a call option that is based on a bond's yield spread relative to a benchmark. If the bond,s credit quality decreases, its yield spread will increase and the credit protection buyer will get a payoff on the option
    • 分类方式
      • 按照买卖双方的权利义务划分
        • Forward commitments: force the two parties to perform some actioni n the future
          • Forwards
          • Futures
          • Swaps
        • Contingent claims: provide the right to buy or sell the underlying that depends on a particular event
          • Option
          • Credit derivative
          • ABS
      • 按照交易场所划分
        • Over-the-Counter: no central location, largely unregulated markets
          • Forwards
          • Swaps
          • Options
        • Exchange-Traded: with central location, backedby clearinghouse, regulated markets
          • Furures
          • Options
        • OTC v.s. Exchange-traded
  • The benefits and criticism of derivatives markets
    • Benefits
      • Risk allocation, transfer and management
      • Provide price information
      • Operational advantages: low transaction costs/greater liquidity/low capital requirements/go short
      • Improve market efficiency
    • Criticisms
      • Speculative devices: too risky
      • Destabilization and systemic risk
• Basics of Derivative Pricing and Valuation
  • The risk aversion of the investor
    • Risk averse: risk-averse investors require a positive premium on risky asset
    • Risk neutral: risk-neutral investors require no risk premium on risky asset
    • Risk seeking: risk-seeker prefer risk over certainty, even if the investment implying negative risk premium, the risk-seeker is willing to engage
  • Pricing of the underlying
    • No benefits or cost during holding and the asset is risk-free asset
      • The asset at time 0, spot price is
    • No benefits or cost during holding and the asset is risky asset
      • Assume investors are risk-averse, so they require positive premium for risky asset
      • The asset at time 0, spot price is
    • Incur benefits or cost during holding and the asset is risky asset
      • Assume investors are risk-averse, so they require positive premium for risky asset
      • Assume during holding period, generate benefits and incur costs
      • The asset at time 0, spot price is
      • Benefits and costs
        • Storage costs: incurred in owning commodities, such as gold, oil, or wheat would incur some costs in storing them Convenience yieId: refers to nonmonetary benefits from holding an asset. Convenience exist when the asset is difficult to sell short or unusually tight
    • The holding period from time 0 to time; The investor forecasts the expected price of the asset at time T is E(ST)
  • The basis of Derivative pricing and valuation: No-Arbitrage Principle
    • No-arbitrage principle
      • Arbitrage
        • In a well-functioning markets with low transaction costs and a free flow of information, arbitrage opportunities arise when the same assets are mispriced
        • Law of one price: The same assets (the identical assets or two assets or portfolios with identical future cash flows) should be sold at identical price
      • Limits to arbitrage
        • Transaction costs
        • Require capital
        • Some assets can be difficult to short
      • Arbitrage opportunity
    • Replication
      • 定义: replicate the payoffs on one asset or portfolio with those of a different asset or portfolio
      • 方法: An asset and a derivative on the asset can be combined to produce a risk-free asset
      • 计算
        • Asset + Derivative = Risk -free asset
        • Equation transformation
          • Asset - Risk-free asset = -Derivative
          • Derivative - Risk-free asset = -Asset
  • Pricing and valuation for four types of derivatives
    • Forwards
      • At time 0
        • No benefits and costs during holding period
          • Price
          • Value
        • Incur benefits and costs during holding period
          • Price
          • Value
      • At time t (t<T)
        • No benefits and costs during holding period
          • Value to long position
          • Value to short position
        • Incur benefits and costs during holding period
          • Value to long position
          • Value to short position
      • At time T
        • Value to long position
        • Value to short position
    • Futures
      • The value of futures at time 0, V0=0
      • Futures marked to market on a daily basis: futures price is reset to the settlement price and value goes to zero
      • Differences in futures price and forward price
        • ρ is the correlation coefficient between the spot price of the underlying and risk-free rate
    • Swaps
      • Swaps are similar to a series of forwards
        • Time periods of forwards are different, the forwards prices are not equal:
        • For swaps, all the fixed payments are equal,
        • Replicate swaps with a series of forwards, the forwards prices should be all the same. Use off-market forward--a forward transaction that starts with a nonzero value. A series of off-market forwards can be equivalent to the swaps
      • General idea of pricing and valuation
        • Pricing of swaps is to find the fixed rate (the contract rate)
        • Valuation
          • An increase in floating rate —benefit fixed-rate payer—increase contract value to fixed-rate payer
          • A decrease in floating rate —benefit floating-rate payer —increase contract value to floating-rate payer
    • Options
      • 公式
        • Put-call parity
          • 基本关系
          • Synthetic equivalents
          • Put-call-forward parity
            • The put-call forward parity relationship is derived by substituting the synthetic asset for the underlying asset in the put-call parity relationship
        • Binomial valuation of options
          • Binomial model: is based on the idea that, over the next period, some value will change to one of two possible values (binomial)
          • Risk-neutral probability an up move is Πu; a down move is Πd
          • Pricing and valuation of options
          • Hedge ratio
        • Minimum value of options
          • 表示方法
            • c0—European call option premium; C0—American call option premium
            • p0—European call option premium; P0—American call option premium
          • 关系
      • Values of European and American options differ
        • The only difference between European and American options is that the holder of an American option has the right to exercise prior to expiration. If the American option exercise at the expiration, then both options are equivalent
        • Circumstances that makes values of European and American options are different
          • American call options
            • If the underlying asset has no cash flow during the life of the option, there is no advantage to early exercise
            • If the underlying asset pays cash flow during the life of the option, early exercise can be valuable
          • American put options
            • Cash distributions rather than cash flows make American put options more valuable
            • If the American put options are deep in the money, early exercise can be valuable
      • Factors that determine option prices