Investment Guarantees Chapter 7 Investment Guarantees Chapter 7: Option Pricing Theory Mary Hardy (2003) Video By: J. Eddie Smith, IV, FSA, MAAA Investment Guarantees Chapter 7 1 / 15 Key Exam Topics in This Lesson Replication The Binomial Model One-Period Binomial Model Two-Period Binomial Model and Dynamic Hedging Black-Scholes-Merton BSM Assumptions European Puts and Calls BSM Option Valuation Investment Guarantees Chapter 7 2 / 15
Replication Law of One Price Option prices can be replicated with portfolios containing combinations of risk-free and risky assets Option Value = Replicating Portfolio Value = Risk-Free Asset Amount + Risk Asset Amount = ae r + bs 0 Investment Guarantees Chapter 7 Replication 3 / 15 Investment Guarantees Chapter 7 Replication The Binomial Model One-Period Binomial Model Two-Period Binomial Model and Dynamic Hedging Black-Scholes-Merton Investment Guarantees Chapter 7 The Binomial Model 4 / 15
One-Period Binomial Model Put Option Set = Stock Price S 0 S u S d Put Payoff P 0 (K S u ) + (K S d ) + Replicating Portfolio a + bs u ae r + bs 0 a + bs d Set = You can solve directly for the units of risk-free and risky assets This gives you the replicating portfolio value at time 0 Investment Guarantees Chapter 7 The Binomial Model 5 / 15 One-Period Binomial Q-Measure We can also value the put option using a risk-neutral probability distribution P 0 = ( C u (1 p ) + C d p ) e r p = S u S 0 e r S u S d Requirements for Q-measure = Risk-neutral probability that S goes down 1. Must be equivalent to P-measure: p u + p d = 1 2. E Q [S 1 ] = p us u + p d S d = S 0 e r Investment Guarantees Chapter 7 The Binomial Model 6 / 15
One-Period Binomial Quiz Your company just issued a single-premium variable annuity based on a stock index that will rise by 10% or fall by 5% over a 3-year period. The VA also guarantees minimum cumulative account growth of 5%. The risk-free rate is 2% all years. Determine the value of the embedded option in this product as a percentage of premium at issue. Investment Guarantees Chapter 7 The Binomial Model 7 / 15 Pause Video No peeking! Investment Guarantees Chapter 7 The Binomial Model 8 / 15
Solution 1 Calculate Cost of a Put Option Directly Index Level 100 110 95 Put Payoff (K = 105) P 0 (105 110) + = 0 (105 95) + = 10 First, solve for the risk-neutral probability that the index falls: p = S u S 0 e 3r S u S d = 0.2544 The put price is: P 0 = ( C u (1 p ) + C d p ) e 3r = = ( 0 + 10(0.2544) ) e 0.02(3) 110 100e3(0.02) 110 95 = 2.4 2.4% of premium needed to hedge guarantee Investment Guarantees Chapter 7 The Binomial Model 9 / 15 Solution 2 Construct a Replicating Portfolio The company can hedge the guarantee by holding a portfolio of risk-free and risky assets: ae 3r + bs 0 To get a and b, set the replicating portfolio equal to the ultimate put payoffs a + bs u = 0 a + 110b = 0 a + bs d = K S d a + 95b = 10 This gives us a = 73.33 and b = 2/3, which means holding a portfolio with A long position of 73.33e 3(0.02) = 69.05 in risk-free assets A short position of 2 3 (100) = 66.67 in the risky index RP Value = 69.05 66.67 = 2.4 Investment Guarantees Chapter 7 The Binomial Model 10 / 15
Two-Period Binomial Model and Dynamic Hedging S 0 S u S uu S ud Break into 3 one-period models: 1. Solve for the value of P u and P d at time 1: S d S dd P u = ( C uu (1 pu) + C ud pu ) e r P d = ( C ud (1 pd ) + C ddpd ) e r P 0 P u C uu C ud 2. Solve for the put price at time 0 P 0 = ( P u (1 p0) + P d p0 ) e r P d C dd Process is self-financing Investment Guarantees Chapter 7 The Binomial Model 11 / 15 Investment Guarantees Chapter 7 Replication The Binomial Model Black-Scholes-Merton BSM Assumptions European Puts and Calls BSM Option Valuation Investment Guarantees Chapter 7 Black-Scholes-Merton 12 / 15
Black-Scholes-Merton Assumptions 1. S t follows Geometric Brownian motion (GBM) with constant variance σ 2 Lognormal, IID returns 2. Frictionless markets (no transaction costs, taxes) 3. Short selling allowed 4. Continuous trading 5. Interest rates are constant Investment Guarantees Chapter 7 Black-Scholes-Merton 13 / 15 European Puts and Calls Put Payoff = (K S T ) + Call Payoff = (S T K) + $ K $ Call Payoff Put Payoff 0 K S T 0 K S T Investment Guarantees Chapter 7 Black-Scholes-Merton 14 / 15
BSM Value of European Puts and Calls Put-Call Parity Ke r(t t) + BSC t = S t + BSP t Standard Normal Curve Φ( d 1 ) BSP t = Ke r(t t) Φ( d 2 ) S t e d(t t) Φ( d 1 ) BSC t = S t e d(t t) Φ(d 1 ) Ke r(t t) Φ(d 2 ) d 1 = ln S t K + (T t)(r d + σ2 /2) σ T t d 2 = d 1 σ T t d 1 0 d 1 Φ( z) = 1 Φ(z) Investment Guarantees Chapter 7 Black-Scholes-Merton 15 / 15