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Tutorial 5

Tutorial 5. Yufeng Yao. The Black-Scholes differential equation. Assumptions: The stock price follows a Geometric Brownian Motion. The short selling of securities is permitted. There are no transactions costs or taxes. All securities are perfectly divisible (1/3 stocks).

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Tutorial 5

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  1. Tutorial 5 Yufeng Yao

  2. The Black-Scholes differential equation • Assumptions: • The stock price follows a Geometric Brownian Motion. • The short selling of securities is permitted. • There are no transactions costs or taxes. • All securities are perfectly divisible (1/3 stocks). • There are no dividends during the life of the option. • There are no arbitrage opportunities. • Security trading is continuous. • The risk-free rate of interest is constant and the same for all maturities. Assumptions 2, 3, 4 is the assumption for arbitrage trading. When the arbitrage trading is allowed, then there are no arbitrage opportunities (Assumption 6). The assumption 7 is condition of assumption 1, because Geometric Brownian Motion is stochastic process. The stochastic process assume stock price change continuously.

  3. The Black-Scholes differential equation

  4. N( • 1-N( • The implied volatility is not observable. • If implied volatility is not observable, how can we calculate the option price? • We estimate the implied volatility. • We plug market price into the formula to calculate the implied volatility. • Black Scholes formula assumes the volatility is constant.

  5. BS formula VS Binominal Tree model • BS formula is best pricing model for European option on non-dividend paying stock. • In binominal tree model, if the time interval is infinitely small and the number of time interval is infinitely big, thebinominaltreemodelis equivalent tothe BSformula • Youcanusebinominaltreemodelpricinganyoption,aslongasyouknow the payoff function of derivatives.

  6. Black-Scholes formula • In Black-Scholes formula, we assume stock price movement is follow Geometric Brownian Motion, which means the stock price movement is smooth, there is not big jump. • Can you use Black-Scholes formula to calculate the option price during Global Financial Crisis (GFC)? • Interest rate is not constant • Volatility is not constant

  7. European Options on a dividend-paying stock • You can use Black-Scholes formula to calculate the price of European options on a dividend-paying stock, only the you known the amount and timing of the dividends during the life of a European option. • =S-

  8. Calculate the price of a 3 months European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum, and the volatility is 30% per annum. What is Call option price with same maturity and same strike price? c +

  9. Calculate the price of a 3-month European put option on a dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum, and the volatility is 30% per annum, a dividend of $1.5 is expected in 2 months. • =S-

  10. Calculate the price of a 3-month European put option on a dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum, and the volatility is 30% per annum, the dividend yield is 1.2%.

  11. What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is three months?

  12. The American call options on non-dividend paying stock are never optimal to exercise early. • The American call options on non-dividend paying stock could be exercise the option immediately prior to an ex-dividend date if

  13. Consider an American call option when the stock price is $18, the exercise price is $20, the time to maturity is 6 months, the volatility is 30% per annum, and the risk-free interest rate is 10% per annum. Two equal dividends are expected during the life of the option with ex-dividend dates at the end of 2 months and 5 months. Assume the dividends are 40 cents. How high can the dividends be without the American option being worth more than the corresponding European option?

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