Binomial Pricing and BSM Flashcards
What are factors affecting price of an option
- Current price of the underlying asset
- Time of expiration
- Strike price
- Rate of interest
- Random characteristics of the price of the underlying asset
What are the stochastic assumptions of binomial and black Scholes option
- Price is distributed lognormally where the logarithm of the future price follows the normal distribution
- Stock price follows a discrete time distribution for binomial model
- BSM follows a continuous time distribution union
- BSM and binomial option pricing are capable of derving an exact option price on arbitrage alone
What is valuation by replication
Constructing a portfolio containing the stock and diskless asset who’s payoffs are equal to the derivative at every time period and balance needs to continuously achieve this
What is the derivative of the expected payoff discounted with in a risk neutral world
Risk free rate
Why is stock expected return irrelevant
Valuing an option in the real world, the probability of up and down movements in the real world are irrelevant and this is already incorporated in the stock price
What are the assumptions of the binomial model
- Assume that there’s no arbitrage opportunities
- Portfolio has no risk
- There’s two securities: Stock and stock option
- there’s 2 possible outcomes
- Portfolio is diskless when delta is chosen so that the final value of the portfolio is the same for both alternatives
What are the assumptions of the risk neutral world
All individuals are indifferent of risk and expected return on all securities is the risk free rate
What are implications of American options
American options will be greater than European option payoff therefore it should be exercised early
What are the implications of option price
Always equal to expected payoff in a risk neutral world discounted at the risk free rate
What is delta
Ratio of change in price of the stock option to the change in the price of the underlying stock
What are the signs for delta call
Positive
What are the signs of delta put
Negative
What is Girsanov’s theorem
Moves from a world with one set of risk and expected growth rates change but volatilities stay the same
What are the assumptions on options on stock indices
- Assume that stocks of underlying assets provide a dividend yield of rate q
- Valuation of an option on stock index is similar to valuation of option on stock paying a known dividend yield
What are the assumptions for foreign currencies
Foreign currencies can be regarded as an asset providing a yield at a risk free rate of interest
What are the options on futures
In a risk neutral world future price should be in a growth rate expected at zero
Explain the no arbitrage approaches and risk neutral valuation approaches to value a European option
- Set up a risk less portfolio consisting a position in the option and position of the stock
- Set portfolio equal to risk free rate to value the option
- Using risk neutral valuation, we choose probabilities for the branches so that the return will equal the risk free rate
- Calculate the expected payoff and discounting this expected payoff at the risk free interest rate
Why is it not possible to set up a position in a stock and the option that remains riskless for the whole of life of the option
Because delta changes during the life of the option and the riskless portfolio must change
What is the distribution of the rate of return for BSM
Lognormal property of stock prices showcases the probability distribution of the continously compounded rate of return earned on a stock between 0 and 1
What are the expected returns like in BSM
- High Risks = high return
- Returns depend on interest rate
- Higher interest rates = higher return
What does volatility measure
Measures uncertainty returns provided by the stock
What are volatility of typical stocks
15-60%
How are stocks usually observed
At fixed intervals of time
How many trading days in a year
252
What are the assumption of the BSM
- Set up a riskless portfolio consisting a position in the derivative and a position of the stock
- Return of portfolio must be the risk free interest rate bc the stock price and derivative are affected by uncertainty
- Price of the derivative is perfectly correlated with price of the underlying stock
- Stock price follows with mean and variance are constant
- Short selling of securities with full use of proceeds is permitted
- No transaction costs/taxes since all securities are perfectly divisible
- No dividends during the life of the derivative
- no riskless arbitrage
- Security trading is continuous
- Risk free interest is constant and the same for all maturity
What is the difference between BSM and Binomial
- BSM is only riskless for a short while and must be rebalanced frequently
What are the key boundaries for EU call
F = MAX(S-K,0) when t = T
What are the key boundaries of EU put
f= MAX(K-S,0) t=T
How do you keep EU call and put riskless
Change the relative proportions of the derivative and stock in the portfolio
What does risk neutral valuation in BSM mean
Simplifies the analysis by:
1. Assuming expected mean = risk free rate
2. Calculate expected payoff of derivative
3. Discount the expected payoff at risk free interest rate
What happens when we move from risk neutral to risk averse
Expected growth rate in stock price changes
Discount rate must be used for any payoffs from derivative changes
How do we analyze eu options with dividends with BSM
Assume that stock prices is the sum of:
1. Riskless component that corresponds to the known dividends during the life of the option
2. Risky component
What are the riskless components for EU options
- PV of all dividends during the life of the option discounted from ex dividend dates to the present at the risk free rate
- As option matures, dividends have paid, riskless component no longer exists
What is implied volatility
The volatility that makes the BS price of an option equal to the market price calculated using an iterative procedure
