L14 - Black-Scholes-Merton Model

Question 1

How does the CCR model move towards a continuous model?

Answer 1

• If we extend the model (tree steps) to a minimum time step we create a solution where the three steps can be considered as the bid-ask prices
  
  • The higher frequency the more it tends to the BSM model
• Another implication with be analysed with the BSM model

Question 2

Why is it called implied volatility?

Answer 2

• because it cant be observed so we estimate it
  
  • Market makers take this risk so they estimate the BS implied volatility and use this to estimate the price
  • we take the price and use that in our own calculation of what the option price should be

Question 3

What did Merton say about loans?

Answer 3

• Lending is the same as a short put –> based on put-call parity

Question 4

What do we want to use the BSM model to learn?

Answer 4

• Just What an option is worth?
• In truth, this is a very difficult question to answer.
• • At expiration, an option is worth its intrinsic value.
• • Before expiration, put-call parity allows us to price options. But,
  
  • – To calculate the price of a call, we need to know the put price.
  • – To calculate the price of a put, we need to know the call price.
• • So, we want to know the value of a call option:
• – Before expiration, and
• – Without knowing the price of the put

Question 5

History of the BSM model?

Answer 5

• The Black-Scholes option-pricing model allows us to calculate the price of a call option before maturity (and, no put price is needed).
  
  • – Dates from the early 1970s
  • – Created by Fischer Black and Myron Scholes
  • – Made option pricing much easier—The CBOE was launched soon after the Black-Scholes model appeared.
• • Today, many finance professionals refer to an extended version of the model as the Black-Scholes-Merton option pricing model.
  
  • – To recognize the important theoretical contributions by Robert Merton.

Question 6

What are the six main factors that are used to value a stock option?

Answer 6

• can value an option at time zero and throughout its life
• S/K ratio –> called moneyness
  
  • if equal to 1 –> option is At the money
  • if it differs from this then its ITM or OTM
• use the risk free rate because we need to consider the cost of borrowing
  
  • assume you are cashless so must borrow

Question 7

How to you price a call/put option on a singe share of common stock?

Answer 7

Question 8

Assumptions of the BSM model?

Answer 8

• No taxes
• No transactions costs
• Unrestricted short-selling of stock, with full use of short sale proceeds
• Shares are infinitely divisible
• The constant riskless interest rate for borrowing/lending
• European options (or American calls on non-dividend paying stocks)
• Continuous trading – The stock price evolves via a specific ‘process’ through time

Question 9

What is the BSM model formula?

Answer 9

• Don’t need to estimate the NPV of the stock because S is the price of the stock today
  
  • This is because we estimate that the asset price is based on the dividend yield
  • When you invest I a call option you invest in the risk-free rate so lose out on the dividend hence y is negative –> replacer with (r-y) when dealing with dividend

IN the BSM formula there are three common functions used to price call and put options:

• exp(-rt) is the natural exponent of the value of -rt *in common terms it is the discount factor)
• Ln(S/K) is the natural log of the moneyness term, S/K
• N(d1) and N(d2) denotes the standard normal probability for the values of d1 and d2
  
  • If you look these up in a Normal Distribution stats table you are looking for the Z value -> then looking at the corresponding probability
• In addition the formula makes use of the fact that N(-d1) = 1 - N(d1)
  *

Question 10

Why under BSM model is a OTM option not valued at zero?

Answer 10

• Due to time value of money
  
  • It could still move in the money so it is still work something even if it a minute amount

Question 11

How should you check if your BSM model answer it correction?

Answer 11

• Using the put-call parity
  
  • The formula below is a variation when using a dividend

Question 12

What is implied Volatility?

Answer 12

• Of the six input factors for the Black-Scholes-Merton stock option pricing model, only the stock price volatility is not directly observable.
• A stock price volatility estimated from an option price is called an implied standard deviation (ISD) or implied volatility (IVOL).
• Calculating an implied volatility requires:
  
  • – All other input factors, and – Either a call or put option price
  • All other input factors and
  • Either a call or put option price

Question 13

What is the Brenners-Subramanyam formula (1988) for Implied volatility?

Answer 13

Question 14

What is Corrado-Miller general model for implied Volatility?

Answer 14

Question 15

What is Corrado-Miller fitted model for implied Volatility?

Answer 15

• Based on if the call option is ITM or OTM

Question 16

How can we check on implied volatilities for stock indexes?

Answer 16

• The CBOE publishes data for two implied volatility indexes:
  
  • – S&P 100 Index Option Volatility, ticker symbol VIX
  • – Nasdaq 100 Index Option Volatility, ticker symbol VXN
• • Each of these volatility indexes are calculating using ISDs from eight options: –> take the average of the following
  
  • – 4 calls with two maturity dates:
    
    • • 2 slightly out of the money
    • • 2 slightly in the money
  • – 4 puts with two maturity dates
    
    • : • 2 slightly out of the money
    • • 2 slightly in the money

Question 17

What the problem with implied volatility?

Answer 17

• Volatility skews around the strike price creating a smile
  
  • high implied standard deviation(implied volatility) the further ITM and OTM you get
    
    • Only buy a high volatility OTM option because there is a chance it will move into the money
    • Only sell a high volatility ITM option because there a chance it could move out of the money
  • the volatility of the stock price cannot depend on the strike price

Volatility skews(or volatility smiles) describe the relationship between implied volatilities and strike prices for options.

• – Recall that implied volatility is often used to estimate a stock’s price volatility over the period remaining until option expiration.

Question 18

What is stochastic volatility?

Answer 18

• Logically, there can be only one stock price volatility, because price volatility is a property of the underlying stock.
• However, volatility skews do exist. There is widespread agreement that the major cause is stochastic volatility.
• Stochastic volatility is the phenomenon of stock price volatility changing randomly over time.
• Recall that the Black-Scholes-Merton option pricing model assumes that stock price volatility is constant over the life of the option. •
• Nevertheless, the Black-Scholes-Merton option pricing model yields accurate option prices for options with strike prices close to the current stock price.