Brownian Motion and Ito s Lemma

Transcription

1 Brownian Motion and Ito s Lemma 1 The Sharpe Ratio 2 The Risk-Neutral Process

2 Brownian Motion and Ito s Lemma 1 The Sharpe Ratio 2 The Risk-Neutral Process

3 The Sharpe Ratio Consider a portfolio of assets indexed by i. If asset i has expected return α i, the risk premium is defined as where r denotes the risk-free rate. The Sharpe ratio is defined as RiskPremium i = α i r SharpeRatio i = RiskPremium i σ i = α i r σ i, where σ i stands for the volatility of the asset i We can use the Sharpe ratio to compare two perfectly correlated claims, such as a derivative and its underlying asset Two assets that are perfectly correlated must have the same Sharpe ratio, or else there will be an arbitrage opportunity

4 The Sharpe Ratio Consider a portfolio of assets indexed by i. If asset i has expected return α i, the risk premium is defined as where r denotes the risk-free rate. The Sharpe ratio is defined as RiskPremium i = α i r SharpeRatio i = RiskPremium i σ i = α i r σ i, where σ i stands for the volatility of the asset i We can use the Sharpe ratio to compare two perfectly correlated claims, such as a derivative and its underlying asset Two assets that are perfectly correlated must have the same Sharpe ratio, or else there will be an arbitrage opportunity

5 The Sharpe Ratio Consider a portfolio of assets indexed by i. If asset i has expected return α i, the risk premium is defined as where r denotes the risk-free rate. The Sharpe ratio is defined as RiskPremium i = α i r SharpeRatio i = RiskPremium i σ i = α i r σ i, where σ i stands for the volatility of the asset i We can use the Sharpe ratio to compare two perfectly correlated claims, such as a derivative and its underlying asset Two assets that are perfectly correlated must have the same Sharpe ratio, or else there will be an arbitrage opportunity

6 The Sharpe Ratio Consider a portfolio of assets indexed by i. If asset i has expected return α i, the risk premium is defined as where r denotes the risk-free rate. The Sharpe ratio is defined as RiskPremium i = α i r SharpeRatio i = RiskPremium i σ i = α i r σ i, where σ i stands for the volatility of the asset i We can use the Sharpe ratio to compare two perfectly correlated claims, such as a derivative and its underlying asset Two assets that are perfectly correlated must have the same Sharpe ratio, or else there will be an arbitrage opportunity

7 The Sharpe Ratio: Two stocks with the same source of uncertainty Consider two nondividend-paying stocks modeled as ds t = α 1 S t dt + S dz t d S t = α 2 St dt + σ 2 St dz t where Z is a standard Brownian motion The stock price processes S and S are perfectly correlated since they have the same driving Brownian motion Let us suppose that they have different Sharpe ratios and demostrate that there is arbitrage opportunity in the market

8 The Sharpe Ratio: Two stocks with the same source of uncertainty Consider two nondividend-paying stocks modeled as ds t = α 1 S t dt + S dz t d S t = α 2 St dt + σ 2 St dz t where Z is a standard Brownian motion The stock price processes S and S are perfectly correlated since they have the same driving Brownian motion Let us suppose that they have different Sharpe ratios and demostrate that there is arbitrage opportunity in the market

9 The Sharpe Ratio: Two stocks with the same source of uncertainty Consider two nondividend-paying stocks modeled as ds t = α 1 S t dt + S dz t d S t = α 2 St dt + σ 2 St dz t where Z is a standard Brownian motion The stock price processes S and S are perfectly correlated since they have the same driving Brownian motion Let us suppose that they have different Sharpe ratios and demostrate that there is arbitrage opportunity in the market

10 The Sharpe Ratio: Two stocks with the same source of uncertainty (cont d) Without loss of generality assume that α 1 r > α 2 r σ 2 Buy 1/S shares of asset S Short-sell 1/ Sσ 2 shares of asset S Invest/borrow the risk-free bond in the amount of the cost difference 1 1 σ 2 The return of the above strategy is 1 S ds 1 σ ds 2 + ( 1 1 ) dt = 2 S σ 2 ( α1 r α ) 2 r dt > 0 σ 2

11 The Sharpe Ratio: Two stocks with the same source of uncertainty (cont d) Without loss of generality assume that α 1 r > α 2 r σ 2 Buy 1/S shares of asset S Short-sell 1/ Sσ 2 shares of asset S Invest/borrow the risk-free bond in the amount of the cost difference 1 1 σ 2 The return of the above strategy is 1 S ds 1 σ ds 2 + ( 1 1 ) dt = 2 S σ 2 ( α1 r α ) 2 r dt > 0 σ 2

12 The Sharpe Ratio: Two stocks with the same source of uncertainty (cont d) Without loss of generality assume that α 1 r > α 2 r σ 2 Buy 1/S shares of asset S Short-sell 1/ Sσ 2 shares of asset S Invest/borrow the risk-free bond in the amount of the cost difference 1 1 σ 2 The return of the above strategy is 1 S ds 1 σ ds 2 + ( 1 1 ) dt = 2 S σ 2 ( α1 r α ) 2 r dt > 0 σ 2

13 The Sharpe Ratio: Two stocks with the same source of uncertainty (cont d) Without loss of generality assume that α 1 r > α 2 r σ 2 Buy 1/S shares of asset S Short-sell 1/ Sσ 2 shares of asset S Invest/borrow the risk-free bond in the amount of the cost difference 1 1 σ 2 The return of the above strategy is 1 S ds 1 σ ds 2 + ( 1 1 ) dt = 2 S σ 2 ( α1 r α ) 2 r dt > 0 σ 2

14 The Sharpe Ratio: Two stocks with the same source of uncertainty (cont d) Without loss of generality assume that α 1 r > α 2 r σ 2 Buy 1/S shares of asset S Short-sell 1/ Sσ 2 shares of asset S Invest/borrow the risk-free bond in the amount of the cost difference 1 1 σ 2 The return of the above strategy is 1 S ds 1 σ ds 2 + ( 1 1 ) dt = 2 S σ 2 ( α1 r α ) 2 r dt > 0 σ 2

15 Brownian Motion and Ito s Lemma 1 The Sharpe Ratio 2 The Risk-Neutral Process

16 The True Price Process The model ds t = S t [(α δ) dt + σ dz t ] where δ denotes the dividend yield on S The drift contains the average appreciation of the stock The uncertainty is driven by the stochastic process Z To facilitate calculations (recall the binomial model!) we look at the process S under a new probability measure which renders the price process to be a martingale. This is different from the physical measure - whatever that may be...

17 The True Price Process The model ds t = S t [(α δ) dt + σ dz t ] where δ denotes the dividend yield on S The drift contains the average appreciation of the stock The uncertainty is driven by the stochastic process Z To facilitate calculations (recall the binomial model!) we look at the process S under a new probability measure which renders the price process to be a martingale. This is different from the physical measure - whatever that may be...

18 The True Price Process The model ds t = S t [(α δ) dt + σ dz t ] where δ denotes the dividend yield on S The drift contains the average appreciation of the stock The uncertainty is driven by the stochastic process Z To facilitate calculations (recall the binomial model!) we look at the process S under a new probability measure which renders the price process to be a martingale. This is different from the physical measure - whatever that may be...

19 The True Price Process The model ds t = S t [(α δ) dt + σ dz t ] where δ denotes the dividend yield on S The drift contains the average appreciation of the stock The uncertainty is driven by the stochastic process Z To facilitate calculations (recall the binomial model!) we look at the process S under a new probability measure which renders the price process to be a martingale. This is different from the physical measure - whatever that may be...

20 The True Price Process The model ds t = S t [(α δ) dt + σ dz t ] where δ denotes the dividend yield on S The drift contains the average appreciation of the stock The uncertainty is driven by the stochastic process Z To facilitate calculations (recall the binomial model!) we look at the process S under a new probability measure which renders the price process to be a martingale. This is different from the physical measure - whatever that may be...

21 The Risk-Neutral Measure Under the risk-neutral measure P the SDE for the stock-price reads as ds t = S t [(r δ) dt + σ d Z t ] with Z a standard Brownian motion under P Note that the volatility is not altered

22 The Risk-Neutral Measure Under the risk-neutral measure P the SDE for the stock-price reads as ds t = S t [(r δ) dt + σ d Z t ] with Z a standard Brownian motion under P Note that the volatility is not altered