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Bus 316 > Binomial Pricing and BSM > Flashcards

Binomial Pricing and BSM Flashcards

1
Q

What are factors affecting price of an option

A
  • Current price of the underlying asset
  • Time of expiration
  • Strike price
  • Rate of interest
  • Random characteristics of the price of the underlying asset
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2
Q

What are the stochastic assumptions of binomial and black Scholes option

A
  1. Price is distributed lognormally where the logarithm of the future price follows the normal distribution
  2. Stock price follows a discrete time distribution for binomial model
  3. BSM follows a continuous time distribution union
  4. BSM and binomial option pricing are capable of derving an exact option price on arbitrage alone
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3
Q

What is valuation by replication

A

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

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4
Q

What is the derivative of the expected payoff discounted with in a risk neutral world

A

Risk free rate

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5
Q

Why is stock expected return irrelevant

A

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

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6
Q

What are the assumptions of the binomial model

A
  1. Assume that there’s no arbitrage opportunities
  2. Portfolio has no risk
  3. There’s two securities: Stock and stock option
  4. there’s 2 possible outcomes
  5. Portfolio is diskless when delta is chosen so that the final value of the portfolio is the same for both alternatives
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7
Q

What are the assumptions of the risk neutral world

A

All individuals are indifferent of risk and expected return on all securities is the risk free rate

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8
Q

What are implications of American options

A

American options will be greater than European option payoff therefore it should be exercised early

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9
Q

What are the implications of option price

A

Always equal to expected payoff in a risk neutral world discounted at the risk free rate

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10
Q

What is delta

A

Ratio of change in price of the stock option to the change in the price of the underlying stock

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11
Q

What are the signs for delta call

A

Positive

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12
Q

What are the signs of delta put

A

Negative

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13
Q

What is Girsanov’s theorem

A

Moves from a world with one set of risk and expected growth rates change but volatilities stay the same

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14
Q

What are the assumptions on options on stock indices

A
  • 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
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15
Q

What are the assumptions for foreign currencies

A

Foreign currencies can be regarded as an asset providing a yield at a risk free rate of interest

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16
Q

What are the options on futures

A

In a risk neutral world future price should be in a growth rate expected at zero

17
Q

Explain the no arbitrage approaches and risk neutral valuation approaches to value a European option

A
  1. Set up a risk less portfolio consisting a position in the option and position of the stock
  2. Set portfolio equal to risk free rate to value the option
  3. Using risk neutral valuation, we choose probabilities for the branches so that the return will equal the risk free rate
  4. Calculate the expected payoff and discounting this expected payoff at the risk free interest rate
18
Q

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

A

Because delta changes during the life of the option and the riskless portfolio must change

19
Q

What is the distribution of the rate of return for BSM

A

Lognormal property of stock prices showcases the probability distribution of the continously compounded rate of return earned on a stock between 0 and 1

20
Q

What are the expected returns like in BSM

A
  1. High Risks = high return
  2. Returns depend on interest rate
  3. Higher interest rates = higher return
21
Q

What does volatility measure

A

Measures uncertainty returns provided by the stock

22
Q

What are volatility of typical stocks

A

15-60%

23
Q

How are stocks usually observed

A

At fixed intervals of time

24
Q

How many trading days in a year

A

252

25
Q

What are the assumption of the BSM

A
  1. Set up a riskless portfolio consisting a position in the derivative and a position of the stock
  2. Return of portfolio must be the risk free interest rate bc the stock price and derivative are affected by uncertainty
  3. Price of the derivative is perfectly correlated with price of the underlying stock
  4. Stock price follows with mean and variance are constant
  5. Short selling of securities with full use of proceeds is permitted
  6. No transaction costs/taxes since all securities are perfectly divisible
  7. No dividends during the life of the derivative
  8. no riskless arbitrage
  9. Security trading is continuous
  10. Risk free interest is constant and the same for all maturity
26
Q

What is the difference between BSM and Binomial

A
  1. BSM is only riskless for a short while and must be rebalanced frequently
27
Q

What are the key boundaries for EU call

A

F = MAX(S-K,0) when t = T

28
Q

What are the key boundaries of EU put

A

f= MAX(K-S,0) t=T

29
Q

How do you keep EU call and put riskless

A

Change the relative proportions of the derivative and stock in the portfolio

30
Q

What does risk neutral valuation in BSM mean

A

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

31
Q

What happens when we move from risk neutral to risk averse

A

Expected growth rate in stock price changes
Discount rate must be used for any payoffs from derivative changes

32
Q

How do we analyze eu options with dividends with BSM

A

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

33
Q

What are the riskless components for EU options

A
  1. PV of all dividends during the life of the option discounted from ex dividend dates to the present at the risk free rate
  2. As option matures, dividends have paid, riskless component no longer exists
34
Q

What is implied volatility

A

The volatility that makes the BS price of an option equal to the market price calculated using an iterative procedure

Bus 316 (2 decks)

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