Lecture no43 option pricing

Option Pricing Lecture No 43 Chapter 13 Contemporary Engineering Economics Copyright © 2016 Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Elements of Financial Option Pricing Model Stock price Strike price Intrinsic value of The option Time until expiration Volatility Probability of a profitable move Dividends Adjustment To share price Interest rates Cost of money Option Pricing Model Theoretical Option Value Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Option Valuation Approaches Two approaches to value options o Discrete-time o Binomial Lattices o Continuous-time o Black-Shores Model Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Discrete-Time Approach Assumptions o The underlying asset follows a discrete, binomial, multiplicative stochastic process throughout time o Arbitrage-free pricing o The law of one price, which states that if two portfolios are equal in value at time T, then they must have equivalent values today Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Option Pricing: Mathematical Symbols • • • • • • • • • Δ = Number of shares to purchase b = the amount cash borrowed R = (1 + r), where r = risk-free rate S0 = value of the underlying asset today • Cd = downward movement in the value of the call option uS = upward movement in the value of S dS = downward movement in the value of S K = strike (exercise) price of the option C = value of the call option Cu = upward movement in the value of the call option Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved How Would You Price a One-Day Call Option? Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Three Different Approaches to Valuing a Financial Option o Replicating-Portfolio  All three approaches o Risk-Free Financing lead to the same valuation, but the risko Risk-Neutral Probability neutral probability approach is most commonly adopted Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Approach 1: Replicating-Portfolio Concept with a Call Option o Create an arbitrage portfolio that contains two risky assets: the share of stock and the call option on the stock o An arbitrage (replicating) portfolio is a portfolio that earns a sure return Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Creating an Arbitrage Portfolio • Arbitrage portfolio o Objective: Select the value of Δ such that the total value of the portfolio is the same regardless of the value of the share of stock at option expiration o Mathematical expression 315  15  285   0.5 o What it means o Long: 0.5 shares o Short: call option o The value of portfolio at day Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Pricing Option Value at Day  Option values between day and day Establishing equivalence between two options values by using discounting factor (r), a risk-free rate:  Option value calculation at r = 6% per year or 0.016% per day Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Approach 2: Risk-Free Financing Approach  Create a replicating portfolio that consists of a stock with a risk-free bond Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Creating a Hedge Portfolio ΔS = $150, b = −$142.48 A portfolio needs to be formed with $150 worth of stock financed in part by $142.48 at the risk-free rate of 6%  Option value on day C = ΔS + b = $150 − $142.48 = $7.52 Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Approach 3: Risk-Neutral Probability Approach    Value the option in a riskfree world by calculating a risk-neutral probability The objective probability (p) never enters into the option value calculation In other words, the probability of a stock price moving a typical direction will not affect the option value This risk-neutral property permits us to use a riskfree interest rate in valuing an option Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Risk-Neutral Probability Concept Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Example 13.5: A Put Option Valuation with Continuous Discounting  Given: u = 1.2, d = 0.8, T = 1, r = 6%, S0 = $50, and K = $55 Find: C Formula: Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Two-Period Binomial Lattice Option Valuation o Step 1: Calculate q o Step 2: Determine the call option value at day o Step 3: Determine the call option value at day Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Multi-period Binomial Lattice Option Valuation At issue: As we increase the number of steps in a year, what would happen to the resulting price distribution? Observation: As the timeincrement approaches zero, then the underlying asset’s probability distribution approaches the lognormal distribution Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Important Relationship A smaller Δt for the binomial lattice will provide option values closer to its continuous-time counterpart (the Black-Scholes equation) Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Example 13.6: Construction of a Binomial Lattice Model  Given: μ = 25%, σ = 50%, and Δt = 1/52  Find: p, u, d, and binomial lattice Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Black-Scholes Option Model: Continuous Model Black, Scholes, and Merton were the first to derive the option value using the replicating portfolio concept Some of the key assumptions in deriving their model are: o Constant interest rate o A continuously operating market, where asset values' returns are normal, which implies that the distribution of terminal asset values is lognormal; this process is known as a geometric Brownian motion o No arbitrage opportunities exist, which implies a risk-neutral world Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved The B-S Call and Put Equations The Black-Scholes equation for European calls and puts are: C call  S0N(d1 )  Ke  r T N(d2 ) Cput  Ke  r T N(d2 )  S0N(d1 ) where ln(S0 K )  (r   )T d1   T ln(S0 K )  (r   )T d2   d1   T  T S0 = Underlying asset price today K = Exercise price T = Time to expiration r = Risk-free rate N(.) = the cumulative standard normal distribution Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Example 13.4: Option Valuation under a Continuous-Time Process  Given: S0 = $40, K = $44, r = 6%, T = years, and σ = 40% Find: Option premiums for both call and put options  Comments: The call option value is greater than the put option value, indicating the upside potential is higher than the downside risk Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Excel Worksheet to Evaluate the B-S Formulas Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved American Options and B-S Model o Generally speaking, the Black-Scholes formula cannot value American call or put options o For our purpose, we will use the binomial lattice approach to value American options Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Key Point • As long as a portfolio consisting of 0.25 shares of stock plus a short position in one call option is set up, the value of this portfolio at expiration will equal $10 in both the up-state and the down-state • In essence, this portfolio mitigates all risk associated with the underlying asset’s price movement • Because all risk has been ‘hedged’ away, the appropriate discount rate to account for the time value of money is the risk-free rate Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved ... Rights Reserved Pricing Option Value at Day  Option values between day and day Establishing equivalence between two options values by using discounting factor (r), a risk-free rate:  Option value... rates Cost of money Option Pricing Model Theoretical Option Value Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Option Valuation... 13.4: Option Valuation under a Continuous-Time Process  Given: S0 = $40, K = $44, r = 6%, T = years, and σ = 40% Find: Option premiums for both call and put options  Comments: The call option