7 Derivatives

2023年CFA二级科目，难度较大，核心的三块知识分别为:远期合约的定价和估值，尤其是在合约未到期之前某一时点的情形;互换合约的估值，包括利率互换的估值和货币互换的估值;期权的定价和希腊字母，特别需要掌握利用二叉树模型对期权定价的方法。

Derivatives

• Pricing and Valuation of Forward Commitments
  • Introduction
    • Arbitrage(law of one price)：In a well-functioning markets with low transaction costs and a free flow of information, arbitrage opportunities arise when the same assets are mispriced
    • Assumptions
      • Replicating instruments are identifiable and investable
      • Market frictions are nil
      • Short selling is allowedwith full use of proceeds
      • Borrowing and lending are available at a known risk-free rate
    • The main difference between forward and futures contract
  • Carry Arbitrage Model
    • • If we assume continuous compounding (rc), then FV(S0) =0
      • If we assume annual compounding (r), then FV(S0) = S0(1 + r)^T
    • Carry Arbitrage Model When Underlying Has Cash Flows
      • F0 = Future value of the underlying adjusted for carry cash flows= FV[S0 + Carry costs – Carry benefits]
      • If there are no explicit carry costs (CC0 = 0) as with many financial assets, For a stock paying a dividend (D)
  • Pricing Forwards and Futures
    • Valuation principle
      • Value of Long Forward Contract Prior to Maturity (Time t) =Vt (long) = Present value of the difference in forward prices
    • Day count and compounding conventions
    • Different types
      • Forwards
        • T-bill Forward(no Cash flow during holding period)
          • The price of T-bill at initiation is S0 and forward price is FP
          • Pricing : FP = S0(1 + rf)^T
          • Valuation to long position
            • At initiation: V0 = 0
            • During the holding period
            • At expiration
        • Equity Forward Contracts with discrete dividends
          • 特点: During holding period, the underlying will receive discrete dividends
          • Pricing
          • Valuation(long)
            • At initiation: V0 = 0
            • During the holding period
        • Equity Forward Contracts With Continuous Dividends
          • 特点: During holding period, the underlying will receive continuous dividends and the risk-free rate is also continuously compounded
          • Pricing
          • Valuation(long)
            • At initiation: V0 = 0
            • During the holding period
        • Fixed income securities Forward Contracts With Coupons
          • Similar to Equity Forward Contracts with discrete dividends, simply substituting the present value of expected coupon payments (PVC) for present value of expected discrete dividends (PVD)
          • Pricing
          • Valuation(long)
            • At initiation: V0 = 0
            • During the holding period
        • Forward rate agreement(FRA)
          • Long/short
            • Long position → Borrower → earn a profit when market price ＞forward price
            • Short position → Lender → earn a profit when market price ＜forward price
          • Valuation
            • 时间关系
            • At maturity
            • Before maturity
      • Futures
        • Definition: is a standardized forward contract that 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
        • Bond futures
          • Each bond is given a conversion factor
          • The futures price
            • full price = clean price + accrued interest = clean price + AI0
            • AI=t/T*coupon amount
          • The quoted futures price(QFP)
          • Cheapest-to-deliver bond: the short will deliver the bond that is least expensive
          • The value to the long at time t: Vt = PV [Ft – F0]
  • Pricing and Valuing Swap Contracts
    • General idea of pricing: the pricing of swaps is to find the fixed rate (the contract rate)
    • Interest rate swap
      • Pricing
        • 时间关系
        • Floating payer
        • Fixed payer (C=fixed rate)
      • Valuation
        • PV(floating PMT)=NP
        • PV(Fixed PMT)=[ C × PV1′+ C × PV2′+(1+C) × PV3′]×NP
        • Value to fixed-rate payer=PV(floating PMT)-PV(Fixed PMT)=NP- [ C × PV1′+ C × PV2′+(1+C) × PV3′]×NP=ΣPV'(C new-C old)*np
    • Currency swap
      • Definition: Currency swap is a contract that two parities agree to exchange future interest payments in different currencies
      • Interest payments: can be based on fixed rate or floating rate
      • 类型(以US dollar 和 UK pond为例, 从UK pond payer角度分析)
        • Pay GBP fixed and receive USD fixed
        • Pay GBP fixed and receive USD floating
        • Pay GBP floating and receive USD fixed
        • Pay GBP floating and receive USD floating (no pricing problem)
      • Steps
        • Pricing: When GBP payment based on fixed rate, first we solve for the GBP fixed rate using the method introduced earlier to price an interest rate swap, that is the fixed rate will make the present value of the fixed-rate GBP payments equal to £1.00 in Great Britain
        • Valuation: If we value the swap in USD, then we have to convert GBP payments to USD
          • Convert GBP notional principle to the equivalent of $1 notional principle at exchange rate of 0
            • Use GBP notional principle to calculate the present value of GBP payments
              • Convert PV(GBP PMTs) to dollars at the current exchange rate
                • Calculate PV(receive) - PV(pay) = PV(USD PMTs) - PV(GBP PMTs)
    • Equity swap
      • Definition: equity swap is a contract that two parties agree to exchange a series of cash flows whereby one party pays a variable series that is determined by an equity return and the other party pays: (1) fixed series; (2) variable series determined by a floating rate; (3) variable series determined by another equity
      • Three types of equity swap:
        • receive-equity return, pay-fixed
        • receive-equity return, pay-floating
        • receive-equity return, pay-another equity return
      • Pricing
      • Valuation: PV(receive)-PV(pay)
• Valuation of Contingent Claims
  • Valuation models
    • Binomial Option Valuation Model
      • Introduction
        • A binomial model is based on the idea that, over the next period, the value of an asset will change to one of two possible values (binomial)
        • Conditions to construct a binomial model
          • The beginning asset value S0
          • The size of the two possible changes: u denotes the up factor, d denotes the down factor; S + denotes the outcome when the underlying goes up, S - denotes the outcome when the underlying goes down
          • The probability of each of these changes occurring: πu denotes the probability of an up move, πd denotes the probability of a down move, and πu + π d = 1
      • One-Period Binomial Model
        • 形状
        • 计算步骤
          • Calculating the payoff of the option at maturity in both the up-move and down-move states
          • Calculating the expected value of the option in one year as the probability-weighted average of the payoffs in each state
          • Discounting the expected value back to today at the risk-free rate
        • 其他部分计算
          • Risk-neutral probability
          • Pricing and valuation of options
            • Call
            • Put
          • Hedge ratio: The fractional share of stock needed in the arbitrage trade (commonly referred to as the hedge ratio or delta)
      • Two-Period Binomial Model
        • 形状
        • 计算步骤
          • Calculate the stock values at the end of two periods:S++, S+-and S--
          • Calculate the three possible option payoffs at the end of two periods: C++, C+-and C--
          • Calculate the expected option payoff at the end of two periods (t = 2) using the up- and down-move probabilities
          • Discount the expected option payoff (t = 2) back one period at the risk-free rate to find the option values at the end of the first period (t = 1)
            • European option
            • American option
          • Calculate the expected option value at the end of one period (t = 1) using up-and down-move probabilities
          • Discount the expected option value at the end of one period (t = 1) back one period at the risk-free rate to find the option value today (t=0)
        • Using hedge ratio calculate option value
      • Interest Rate Options
        • 图形
        • 条件
          • The interest at each node is one-period forward rate
          • The interest rates are selected so that πu= πd = 0.5, the probabilities are risk-neutral probabilities
    • Black–Scholes–Merton（BSM) Option Valuation Model
      • Assumptions
        • The underlying asset price → geometric Brownian motion process
        • The price the underlying asset → lognormal distribution
        • The return on the underlying asset → normal distribution
        • The (continuously compounded) r f , is constant and known
        • The σ return is constant and known
        • Markets are "frictionless"—No taxes, no transactions costs, and can short sales
        • If the underlying instrument pays a yield, it is expressed as a continuous known and constant yield at an annualized rate
        • The options are European options
      • Options on stocks without dividend
        • 公式
          • N(*) = cumulative standard normal probability ; N(-x) = 1 -N(x)
        • 说明
          • The BSM can be interpreted as a portfolio of the stock and zero-coupon bond. And the portfolio will replicate the payoff of options. For both call and put, initial cost of the portfolio is ns×S + nB×B
      • Options on stocks with dividend
        • 公式
        • δ=continuously compounded dividend yield
      • Options on currencies (quoted as P per B or Domestic currency per Foreign currency)
        • 公式
        • 说明
          • r(P) = continuously compounded price currency interest rate
          • r(S) = continuously compounded base currency interest rate
    • Black Option Valuation Model
      • European Options on Futures
        • 公式
        • The value of a call option on futures is equal to the value of a portfolio with a long futures position (the PV of the futures price multiplied by N(d1）and a short bond position (the PV of the exercise price multiplied by N(d2)
      • Interest Rate Options
        • Denotation
          • AP is accrued period
          • FRA(0, tj-i, tm) denote the fixed rate on a FRA at Time 0 that expires at Time tj-i, where the underlying matures at time tj
        • 公式
      • Swaptions(Options on swap)
        • 关系
          • Payer Swaption
            • Given the right as fixed rate payer
              • The holder expect interest rate will increase
          • Receiver Swaption
            • Given the right as fixed rate receiver
              • The holder expect interest rate will decrease
        • Valuation
          • Payer: PAY=AP*PVA[SFR*N(d1)-XN(d2)]*NP
          • Receiver: REC=AP*PVA[XN(-d2)-SFR*N(-d1-]*NP
  • Put-call Parity
    • 公式
    • Put-call-forward parity
  • Option Greeks and Implied Volatility
    • Delta
      • 公式
      • 不同情况
        • Deltas for call and put options on stock without dividend
        • Deltas for call and put options on dividend-paying stock
      • 取值范围
        • Call:[0,1]
        • Put:[-1,0]
      • 图形
        • Delta
        • Delta与时间关系
      • Dynamic Hedging
        • Goal: Delta-neutral portfolio(setting the portfolio delta to zero)
        • Key consideration: the delta-neutral portfolio is only risk free for a very small change in the underlying
        • Rebalance: The delta-neutral portfolio must be continually rebalanced to maintain the hedge ratio
        • 计算
          • Call option
          • Put option
    • Gamma(Γ)
      • 计算
      • 评价
        • Long positions in calls and puts: Γ>0
        • Gamma is highest for at-the-money options and in-the-money or deep out-of-the-money have low gamma
      • 作用: Improve the precision with which change in option value for large changes in stock price
      • Gamma与到期时间关系
      • Gamma risk: the risk that stock price might abruptly "jump", leaving an otherwise delta-hedged portfolio unhedged
    • Vega
      • 作用: measures the sensitivity of the option price to changes in the volatility of returns on the underlying asset
      • All else equal, the higher the volatility, both call and put options are more valuable
      • Vega is positive for both calls and puts
      • Vega gets larger as the option gets closer to being at-the-money
      • Implied volatility is the standard deviation of continuously compounded asset returns that is "implied" by the market price of the option
      • 其中横轴为volatility,纵轴为call/put option price
    • Rho(ρ): measures the sensitivity of the option price to changes in the risk-free rate. The ρ of a call option is positive and is negative in put option
    • Theta(θ)
      • 作用: measures the sensitivity of the option price to the passage of time and is less than zero
      • As time passes and a call option approaches maturity, its speculative value declines, option theta is the rate at which the option time value declines
      • The relationship between option value and time to maturity is positive and the passage of time is negative
      • 图形