Flashcards
Ch 15: The BSM Model
Cards in this deck (17)
- 1BSM model for option pricing- Predict the distributions of future share price movements to get expected payoff
- Option price = PV(expected option payoff) - 2Geometric Brownian MotionStock price evolves according to a random process called GBM:
=> Stock prices evolve smoothly (no jumps)
=> Follow a log-normal distribution with mean μ and constant volatility σ - 3Log-normal assumptionStock returns expressed in continuously-compounded terms are normally distributed
- 4Volatility assumption- Standard deviation of annual log-returns is constant
- Can be estimated from:
1. Historical data (historical volatility)
2. Option prices (implied volatility) - 5BSM model assumptions1. GBM
2. Short selling permitted
3. No taxes or transaction costs
4. No dividends during life of the derivative
5. No arbitrage opportunities
6. Risk-free interest rate is constant
7. Options are European style - 6BSM propertiesAs stock price increases and volatility approaches zero:
1. Call option more likely to be exercised
=> price of call option increases
=> d1 and d2 become larger; N(d1) and N(d2) become close to 1
2. Put option less likely to be exercised
=> price of put option decreases
=> -d1 and -d2 become smaller and N(-d1) and N(-d2) become close to 0 - 7Pricing calls by replication
- 8BSM call formula: interpretation
- 9BSM put formula: interpretation
- 10Historical volatility- Estimated from data on past stock returns
- Volatility = std dev x √252
- 11Implied volatility- Estimated from prices of traded options
- Forward-looking => represents "market" perception of volatility over option life
- Volatility level that would make observed option prices consistent with the BSM formula, given values for the other parameters - 12Introducing dividends
- 13Early exercise of calls: non-dividend paying shares- Never pays to exercise early
- Exercise if K - Ke^-rT > time value of call
- This is never true as a call option always has time value prior to maturity
- 14Early exercise of calls: dividend paying shares- May pay to exercise early, (just before the ex-div day) to capture the large dividend
- Exercise if D > K - Ke^-rT - 15Early exercise of puts: non-dividend paying shares- May pay to exercise early, if interest earned on the shares sold exceeds the time value of the put
- E.g. deep ITM put
- Exercise if K - Ke^-rT > time value - 16Early exercise of puts: dividend paying shares- May pay to exercise early, if interest earned on the shares sold exceeds the dividend
- Exercise if K - Ke^-rT > D - 17Warrants and employee stock options- Exercise of a regular call option does not affect no. company shares outstanding
- Exercise of warrants and employee stock options leads to the company issuing more shares and selling them at K
- As K < market price, this dilutes the interest of the existing shareholders
- However, EMH => stock price already reflects potential dilution from outstanding warrants and employee stock options
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- Business/Economics (BFC2751: Derivatives 1) Flashcards on Ch 15: The BSM Model, created by Melina Mackey on 06/06/2019.
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- Melina Mackey
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- Created by Melina Mackey · Mar 29, 2019
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