This question explores the links between the implied volatility skew and the shape of the risk-neutral...

   

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This question explores the links between the implied volatility skew and the shape of the risk-neutral density. Consider calls and puts as well as digitals. Current time is t. Fix a maturity T>t. The implied volatility curve for that maturity is giv(K,T; S, t). Assume that r is a constant and that there are no dividends paid between now and T included. In this question, we are going to compare the Black-Scholes world to a world with a skew, keeping F(T) constant. a. [4 marks] Imposing No Arbitrage (NA), show via the FTAP that the forward price on the stock at t for delivery of one unit of the stock at T must satisfy F: (T) = e(T-) St b. [4 marks] Argue by contradiction that if the IV curve has a skew, then the Black- Scholes-Merton (BSM) model cannot be true. c. [4 marks] Show that the risk-neutral density of log excess returns R = In (ST) is sidi (R,r) <0, provided that. excess returns covary negatively with the realised variance term (RT-ERT])2. d. [4 marks] Argue that the price of a digital call at strike K satisfies negatively skewed in the statistical sense of E(-ER) ] < 0, dC D = dk e. [8 marks] Consider the prices of a digital call D (you do not need to derive this expres- sion): OCBS (S, K, aw (K,S,)..) CBS (S, K, ow(K,S). ) dow(K,S,) ak K D = - Assume that day(K) <0 has the usual equities shape. Contrast the price of the digital at t in the BSM case to the skew case. What does your result imply about the mass of the risk-neutral density around any given strike K when comparing the BS and the skew case (given F)? Show also that QB (ST K; or (K, S)) > Q(ST K) Interpret especially for small K F. g. [8 marks] Compare your findings of the last two questions and describe the apparent paradox. h. [10 marks] Resolve the puzzle. What deformation to the BSM stock annualised excess log-return density does such a negative skew imply? Illustrate the density deformation from BSM to skew graphically. This question explores the links between the implied volatility skew and the shape of the risk-neutral density. Consider calls and puts as well as digitals. Current time is t. Fix a maturity T>t. The implied volatility curve for that maturity is giv(K,T; S, t). Assume that r is a constant and that there are no dividends paid between now and T included. In this question, we are going to compare the Black-Scholes world to a world with a skew, keeping F(T) constant. a. [4 marks] Imposing No Arbitrage (NA), show via the FTAP that the forward price on the stock at t for delivery of one unit of the stock at T must satisfy F: (T) = e(T-) St b. [4 marks] Argue by contradiction that if the IV curve has a skew, then the Black- Scholes-Merton (BSM) model cannot be true. c. [4 marks] Show that the risk-neutral density of log excess returns R = In (ST) is sidi (R,r) <0, provided that. excess returns covary negatively with the realised variance term (RT-ERT])2. d. [4 marks] Argue that the price of a digital call at strike K satisfies negatively skewed in the statistical sense of E(-ER) ] < 0, dC D = dk e. [8 marks] Consider the prices of a digital call D (you do not need to derive this expres- sion): OCBS (S, K, aw (K,S,)..) CBS (S, K, ow(K,S). ) dow(K,S,) ak K D = - Assume that day(K) <0 has the usual equities shape. Contrast the price of the digital at t in the BSM case to the skew case. What does your result imply about the mass of the risk-neutral density around any given strike K when comparing the BS and the skew case (given F)? Show also that QB (ST K; or (K, S)) > Q(ST K) Interpret especially for small K F. g. [8 marks] Compare your findings of the last two questions and describe the apparent paradox. h. [10 marks] Resolve the puzzle. What deformation to the BSM stock annualised excess log-return density does such a negative skew imply? Illustrate the density deformation from BSM to skew graphically.

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a Under no arbitrage the forward price must satisfy FtT EQSTFt erTt St b If the IV curve has a skew ... View the full answer