Lecture 6 - Option Pricing Methods

How do we replicate a future and a protective put (stock and put option)?

• To replicate a future, we can borrow money to buy the spot and store them

• To replicate a protective put (stock and put option), we can buy a bond and a call option (put-call parity)

What is non-arbitrage pricing?

• If we can achieve some goal with two different ways, the cost of each way should be the same

• To price an option, if we can find some portfolio that can replicate the payoff of the option, the cost to build this portfolio must be the option price

• This is called non-arbitrage pricing

Option Valuation:

• A two stage single period model, assume:

  • A European call option expires in one period and has an exercise price of $50

  • The stock price today is equal to $50 and the stock pays no dividends

  • In one period, the stock price will either rise or fall by $10

  • The one period risk-free rate is 6%

• The payoffs can be summarised in a binomial tree, draw this tree

Given a call option with an exercise price of $50, a current share price of $50 and possible future share prices of $60 or $40, how do you set up the replicating portfolio?

Using the simultaneous equations, calculate Δ and B. What does the resulting replicating portfolio consist of?

How do you calculate the price of the call option today using the Law of One Price?

What assumptions and equations are used in the binomial pricing formula?

What are the general formulas for calculating the replicating portfolio and the value of the option today in the binomial model?

Using the binomial option pricing model, how do you set up and solve for the replicating portfolio parameters (Delta and B)

Based on the replicating portfolio parameters (Delta = -0.3333, B = $23.30) for the European put option (S = $60), what is the composition of the replicating portfolio and what is the current price of the put option (P)?

What is Option Delta (or the hedge ratio), how is it calculated in a two-state binomial model, and what does a Delta of 0.5 mean for the relationship between the stock price and the option price?

What is a "delta hedged" portfolio, how is its overall risk eliminated in a binomial model, and what formula proves that its price change relative to stock price change equals 0?

If you hold 3 units of stock (where stock Delta = 1), and a call option has a Delta of 0.75, how many units of the call option must you LONG or SHORT to achieve an overall portfolio Delta of 0?

What is the core definition of risk-neutral probability, and how does the concept of pretending all investors are risk-neutral allow us to set up an alternative method for option valuation? 

• Alternative way to value the option is to pretend that all investors are indifferent about risk

  • The expected return on the stock must equal the risk-free interest rate

• We don’t know the probabilities of each state

• In a risk neutral world, if ρ is the probability that the stock price will increase, then (1 - ρ) is the probability that it will go down

• ρ and (1 - ρ) are risk neutral probabilities

Draw the 1-period binomial tree diagram showing the stock prices (S, Su, Sd), option payoffs (Cu, Cd), and probabilities (p, 1-p). In a risk-neutral world, what assumption do we make about investor risk preferences, and what cost of capital do all financial assets share? 

Draw the 1-period numerical binomial tree diagram for the stock and bond (S=50, Su=60, Sd=40, rf=6%). How do you set up the stock price present value equation to solve for p, and what is the resulting value of p?

Using the solved risk-neutral probability (p = 0.65), set up the equation to calculate the present value of the expected payouts for the call option (K = 50). What is the resulting option price, and how does it compare to the value from the replicating portfolio method?

What are the 2 sequential steps listed for applying risk-neutral methods to value an option across time in a binomial tree?

• At each step and each date, computer the probabilities such that the expected return on the stock equals the risk-free return

• Work backwards in time from the last date to value the option

  • Using the payoffs or next period prices on the option, find the risk-free PV of the option using the created probabilities

  • At each date and state, the price of the option will equal this PV

What are the general formulas for:

1. Solving for the risk-neutral probability p, including its simplified form p = (a-d)/(u-d) and the definitions of a, d, and u? 

2. Calculating the option value Ct at any date using p, Cu, Cd, and rf?

Calculate the possible prices of Buffelhead stock

Calculate the associated call option values

Calculate the probability of the price doubling and halving in the future (risk neutral probabilities)

Calculate the call option value at each branch backward (Pt 1)

Calculate the call option value at each branch backward (Pt 2)

Calculate the call option value at each branch backward (Pt 3)

Describe multi-period binomial models

What are the general binomial method formulas?

What is the Black-Scholes option pricing model?

• Binomial tree model is based on discrete time

• Taking the limit of the binomial model as t ⇒ 0 gives “continuous time pricing”

• The continuous-time pricing formula for a call option is called the Black-Scholes formula

• European put option price can be found from the Black-Scholes call option price using put-call parity

What is the Black-Scholes formula for call option price?

What 5 inputs are needed for the Black-Scholes formula?

• Stock price

• Strike price

• Exercise date

• Risk-free rate

• Volatility of the stock

How is a call option replicated using the Black-Scholes model?

Calculate d1 and d2

What is the price of the call option and how can you replicate it?

What is the Black-Scholes formula for a put option price

Describe the real option to delay

Describe the real option to expand

Describe the real option to abandon

What is the NPV for the investment of a new machine with an immediate expenditure of £1m, future cash flows of £250k per year for 5 years, at 10% discount rate?

What are the options for the company in year 0 and end of year 5?

Calculate d1 and d2

What is the option price and NPV of the project if expansion occurs?

What is the choice of models (Black-Scholes and Binomial)?

• Black-Scholes

  • Option value is function of 6 inputs: Current price of the underlying asset, strike price of the option, life of the option, variance in price of the underlying asset, the risk-free rate, and dividends

  • Restrictive assumptions (volatility fixed over option life, no early exercise, no price jumps)

• Binomial Model

  • Less restrictive: Early exercise, change in variance

  • But have to estimate prices at each branch

  • A lot of information required

What are some key insights from real options?

• Out-of-the-money real options have value

• In-the-money real options need not be exercised immediately

• Writing is valuable

• Delay investment expenses as much as possible

• Tackle the hardest problem first

• Create value by exploiting real options