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code๐ Derivatives โโโ ๐ Chapter 1: Pricing and Valuation of Forward Commitments โ โโโ ๐น Principles of Arbitrage-Free Pricing and Valuation โ โโโ ๐น Pricing and Valuing Generic Forward and Futures Contracts โ โโโ ๐น Carry Arbitrage Model and its Variations โ โโโ ๐น Pricing Equity Forwards and Futures โ โโโ ๐น Pricing Interest Rate Forward and Futures Contracts โ โโโ ๐น Pricing Fixed-Income Forward and Futures Contracts โ โโโ ๐น Pricing and Valuing Currency Swap Contracts โโโ ๐ Chapter 2: Pricing and Valuing Equity Swap Contracts โ โโโ ๐น Equity Swap Mechanics and Valuation โโโ ๐ Chapter 3: Valuation of Contingent Claims โ โโโ ๐น Principles of No-Arbitrage Approach to Valuation โ โโโ ๐น One-Period Binomial Model โ โโโ ๐น Two-Period Binomial Model: Call Options โ โโโ ๐น Two-Period Binomial Model: Role of Dividends โ โโโ ๐น Black-Scholes-Merton (BSM) Option Valuation Model โ โโโ ๐น BSM Model: Carry Benefits and Applications โ โโโ ๐น Black Option Valuation Model and European Options on Futures โ โโโ ๐น Interest Rate Options โ โโโ ๐น Swaptions โ โโโ ๐น Option Greeks and Implied Volatility: Delta โ โโโ ๐น Gamma โ โโโ ๐น Theta, Vega and Rho โ โโโ ๐น Implied Volatility
What this chapter covers: This chapter covers the core principles of pricing and valuing forward commitments, including forwards, futures, and swaps. It emphasizes arbitrage-free pricing, the carry arbitrage model, and the application of these concepts to equity, interest rate, and fixed-income instruments. The chapter also covers swap contracts, providing a foundation for understanding complex derivative instruments.
| Concept/Formula | Definition/Equation | When to Use | Quick Check |
|---|---|---|---|
| Arbitrage-Free Pricing | Prices adjust to prevent risk-free profit. | Pricing derivatives | No arbitrage opportunity exists. |
| Carry Arbitrage Model | F<sub>0</sub> = S<sub>0</sub>(1 + r)<sup>T</sup> - Benefits + Costs | Pricing forwards | Replicates forward payoff. |
| Forward Rate Agreement (FRA) | FRA Rate = [(1 + r<sub>2</sub>T<sub>2</sub>) / (1 + r<sub>1</sub>T<sub>1</sub>) - 1] / (T<sub>2</sub> - T<sub>1</sub>) | Determining FRA rate | Compare to market rates. |
Type A: Calculating Forward Price (No Dividends) Setup: "Given spot price S<sub>0</sub>, risk-free rate r, and time T" Method: F<sub>0</sub> = S<sub>0</sub>(1 + r)<sup>T</sup> Example: S<sub>0</sub> = 100, r = 5%, T = 1 year. F<sub>0</sub> = 100(1.05) = 105
Type B: Calculating Forward Price (With Dividends) Setup: "Given spot price S<sub>0</sub>, risk-free rate r, time T, and dividend D" Method: F<sub>0</sub> = S<sub>0</sub>(1 + r)<sup>T</sup> - D(1 + r)<sup>T-t</sup> Example: S<sub>0</sub> = 100, r = 5%, T = 1 year, D = 2, t = 0.5 years. F<sub>0</sub> = 100(1.05) - 2(1.05)<sup>0.5</sup> = 102.95
Problem: A stock is trading at $50. The risk-free rate is 6%. What is the theoretical price of a 6-month forward contract?
Given: S<sub>0</sub> = $50, r = 6%, T = 0.5 years
"โSolution: F<sub>0</sub> = S<sub>0</sub>(1 + r)<sup>T</sup> = 50(1 + 0.06)<sup>0.5</sup> = 50(1.02956) = $51.48
"โAnswer: $51.48
โ Mistake 1: Forgetting to annualize the interest rate. โ How to avoid: Always ensure the interest rate and time period are in the same units (e.g., annual rate with years).
โ Mistake 2: Incorrectly accounting for dividends. โ How to avoid: Subtract the future value of the dividends from the future value of the spot price.
Remember the carry arbitrage model: Forward Price = Spot Price + Cost of Carry - Benefit of Carry. This helps visualize the components affecting forward pricing.
What this chapter covers: This chapter focuses on equity swaps, where two parties agree to exchange cash flows based on an equity return and a fixed or floating rate. It explains how to price and value equity swaps, taking into account the cash flows generated by the underlying equity.
| Concept/Formula | Definition/Equation | When to Use | Quick Check |
|---|---|---|---|
| Equity Swap | Exchange of cash flows based on equity return and fixed/floating rate. | Hedging equity exposure | Cash flows match agreement. |
| Receive-Equity, Pay-Fixed | Receive equity return, pay fixed rate on notional principal. | Gaining equity exposure | Compare to direct investment. |
| Swap Valuation | PV(Expected Cash Flows) | Determining swap value | Discount cash flows properly. |
Type A: Calculating Swap Cash Flows (Receive Equity, Pay Fixed) Setup: "Given notional principal, equity return, and fixed rate" Method: Cash Flow = Notional Principal * (Equity Return - Fixed Rate) Example: Notional = 1M * (0.10 - 0.05) = $50,000
Type B: Valuing an Existing Equity Swap Setup: "Given current market conditions, remaining cash flows, and discount rate" Method: Discount each expected cash flow to present value and sum them up. Example: Cash flow of 50,000 / 1.04 = $48,076.92
Problem: A company enters into an equity swap to receive the return on an equity index and pay a fixed rate of 4% on a notional principal of $10 million. If the equity index return is 8%, what is the net cash flow?
Given: Notional Principal = $10 million, Equity Return = 8%, Fixed Rate = 4%
"โSolution: Net Cash Flow = 400,000
"โAnswer: $400,000
โ Mistake 1: Incorrectly calculating the equity return. โ How to avoid: Ensure the equity return is calculated correctly based on the index or stock performance.
โ Mistake 2: Forgetting to discount future cash flows when valuing the swap. โ How to avoid: Always discount future cash flows to their present value using the appropriate discount rate.
Visualize equity swaps as a combination of buying the equity index and shorting a bond. This helps understand the cash flow dynamics.
What this chapter covers: This chapter introduces option valuation, focusing on the binomial option valuation model and the Black-Scholes-Merton (BSM) model. It covers the principles of no-arbitrage valuation, the assumptions of the models, and the calculation of option values and hedge ratios.
| Concept/Formula | Definition/Equation | When to Use | Quick Check |
|---|---|---|---|
| Binomial Option Pricing | C = [pC<sub>u</sub> + (1-p)C<sub>d</sub>] / (1+r) | Valuing options with discrete time steps | Option value is non-negative. |
| Black-Scholes-Merton (BSM) | C = S<sub>0</sub>N(d<sub>1</sub>) - Xe<sup>-rT</sup>N(d<sub>2</sub>) | Valuing European options | Ensure inputs are annualized. |
| Delta (ฮ) | Change in option price per change in underlying asset price. | Hedging option positions | Delta ranges from 0 to 1 for calls. |
Type A: One-Period Binomial Model Setup: "Given stock price, strike price, up factor, down factor, and risk-free rate" Method: Calculate option values at each node and discount back to the present. Example: Stock = 50, Up = 1.2, Down = 0.8, r = 5%. Calculate C<sub>u</sub> and C<sub>d</sub>, then discount.
Type B: Black-Scholes-Merton Model Setup: "Given stock price, strike price, time to expiration, risk-free rate, and volatility" Method: Use the BSM formula to calculate the call or put option value. Example: S<sub>0</sub> = 50, T = 0.5, r = 5%, ฯ = 20%. Plug into BSM formula.
Problem: Using the BSM model, calculate the price of a European call option with the following parameters: S<sub>0</sub> = 105, T = 0.5 years, r = 5%, ฯ = 0.2.
Given: S<sub>0</sub> = 105, T = 0.5, r = 5%, ฯ = 0.2
"โSolution: Calculate d<sub>1</sub> and d<sub>2</sub>, then use the BSM formula. d<sub>1</sub> = [ln(S<sub>0</sub>/X) + (r + ฯ<sup>2</sup>/2)T] / (ฯโT) = [ln(100/105) + (0.05 + 0.02)0.5] / (0.2โ0.5) = -0.076 d<sub>2</sub> = d<sub>1</sub> - ฯโT = -0.076 - 0.2โ0.5 = -0.217 C = S<sub>0</sub>N(d<sub>1</sub>) - Xe<sup>-rT</sup>N(d<sub>2</sub>) = 100N(-0.076) - 105e<sup>-0.05*0.5</sup>N(-0.217) = 100(0.4697) - 105(0.9753)(0.4139) = 46.97 - 42.24 = $4.73
"โAnswer: $4.73
โ Mistake 1: Incorrectly calculating d1 and d2 in the BSM model. โ How to avoid: Double-check the formulas and ensure all inputs are correctly annualized.
โ Mistake 2: Using the wrong N(d) values from the standard normal distribution table. โ How to avoid: Carefully look up the correct values and remember to adjust for negative d values if necessary (N(-d) = 1 - N(d)).
Remember that the BSM model is a continuous-time model, so ensure all inputs are annualized. Also, practice calculating d1 and d2 to avoid errors.
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