This question deals with the dynamics of the implied volatility surfaces, and in particular with the...

 

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This question deals with the dynamics of the implied volatility surfaces, and in particular with the properties of a particular dynamic long volatility strategy when faced by a sticky strike regime. We assume that from time 0 to time T the sticky strike regime is in place. We construct the strategy not with a call (as in lecture note on Volatility Trading in Black Scholes) or short two log contracts (as in the lecture note on Volatility Derivatives) but with a sort of risk-reversal that is long a call with strike K and short Tas (St,1,K,01v (K2,So)) calls T(St,t,K1,0v (K1,So)) with strike K < K2. Notice that one further difference from those two cases is that here the portfolio of derivatives is not held constant (such as one call or minus two log contracts) Tas (St.t,K2,0v (K2,S0)) BS but is adjusted continuously over time since the holdings of K calls, is TBS (St,t,K1,0v (K1,So))' continuously changing. We assume that interest rates are zero, r = 0, so the MMA is priced at B = 1 all t. The fact that the portfolio is changing (in the previous two cases, only the amount of stocks was changing, here both the stocks and the calls K held are changing) means we need to make sure that our volatility portfolio with value (V) is self-financing. A portfolio comprised of -a, stocks, 1 call K (with price C), -a, calls K (with a unitary price of C) and be units of the money-market account (with unitary price B = 1, so we have b, Bt = bt in the MMA) is self-financing if + f bedl VT = bo + bed Bt - ao So - fort and and + C - Chad which is intuitive as it says that the value of the portfolio at T equals its starting value plus all the gains between 0 and T. a. [30 marks] Construct this portfolio, with the MMA used to make the portfolio self- financing as well as requiring no initial investment, with portfolio strategy a, as specified above and for a chosen to make V delta-neutral. Notice that as you buy and sell calls with strike K over time, those calls are transacted at time t market prices. Following the methodology in the lecture note Volatility Derivatives, derive Vr and show that b. [20 marks] Interpret your findings. T P&LT = [(0w (K1, So)) (0w (K2, So))] [* Ts(t, K2, ow (K2, So))dt X This question deals with the dynamics of the implied volatility surfaces, and in particular with the properties of a particular dynamic long volatility strategy when faced by a sticky strike regime. We assume that from time 0 to time T the sticky strike regime is in place. We construct the strategy not with a call (as in lecture note on Volatility Trading in Black Scholes) or short two log contracts (as in the lecture note on Volatility Derivatives) but with a sort of risk-reversal that is long a call with strike K and short Tas (St,1,K,01v (K2,So)) calls T(St,t,K1,0v (K1,So)) with strike K < K2. Notice that one further difference from those two cases is that here the portfolio of derivatives is not held constant (such as one call or minus two log contracts) Tas (St.t,K2,0v (K2,S0)) BS but is adjusted continuously over time since the holdings of K calls, is TBS (St,t,K1,0v (K1,So))' continuously changing. We assume that interest rates are zero, r = 0, so the MMA is priced at B = 1 all t. The fact that the portfolio is changing (in the previous two cases, only the amount of stocks was changing, here both the stocks and the calls K held are changing) means we need to make sure that our volatility portfolio with value (V) is self-financing. A portfolio comprised of -a, stocks, 1 call K (with price C), -a, calls K (with a unitary price of C) and be units of the money-market account (with unitary price B = 1, so we have b, Bt = bt in the MMA) is self-financing if + f bedl VT = bo + bed Bt - ao So - fort and and + C - Chad which is intuitive as it says that the value of the portfolio at T equals its starting value plus all the gains between 0 and T. a. [30 marks] Construct this portfolio, with the MMA used to make the portfolio self- financing as well as requiring no initial investment, with portfolio strategy a, as specified above and for a chosen to make V delta-neutral. Notice that as you buy and sell calls with strike K over time, those calls are transacted at time t market prices. Following the methodology in the lecture note Volatility Derivatives, derive Vr and show that b. [20 marks] Interpret your findings. T P&LT = [(0w (K1, So)) (0w (K2, So))] [* Ts(t, K2, ow (K2, So))dt X

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Analysis of Dynamic Long Volatility Strategy with Sticky Strike Regime a Portfolio Construction and Derivation of Vr 1 DeltaNeutral Portfolio To make ... View the full answer