A derivatives trader is modelling the volatility of an equity index using the following discrete-time model:...

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A derivatives trader is modelling the volatility of an equity index using the following discrete-time model: Model 1: 0, 0.12+0.401 +0.05e,, 1=1,2,3,... where o, is the volatility at time I years and ₁,2.... are a sequence of independent and identically-distributed random variables from a standard normal distribution. The initial volatility of equals 0.15. (1) Determine the long-term distribution of o,. [3] The trader is developing a related continuous-time model for use in derivative pricing. The model is defined by the following stochastic differential equation (SDE): do, = -a(0, -μ)dt + Baw, where o, is the volatility at time years. W, is standard Brownian motion and the parameters a, ß and all take positive values. Model 2: (ii) (a) Show that for this model: (b) -0 0₁ = 0 +(-e) + Beaw, Hence determine the numerical value of and a relationship between the parameters a and B if it is required that o, has the same long-term mean and variance under each model. State another consistency property between the models that could be used to determine precise numerical values for a and B. [7] The derivative pricing formula used by the trader involves the squared volatility V₁ = 7, which represents the variance of the returns on the index. (iii) Determine the SDE for V, in terms of the parameters a, p and μ. [2] A derivatives trader is modelling the volatility of an equity index using the following discrete-time model: Model 1: 0, 0.12+0.401 +0.05e,, 1=1,2,3,... where o, is the volatility at time I years and ₁,2.... are a sequence of independent and identically-distributed random variables from a standard normal distribution. The initial volatility of equals 0.15. (1) Determine the long-term distribution of o,. [3] The trader is developing a related continuous-time model for use in derivative pricing. The model is defined by the following stochastic differential equation (SDE): do, = -a(0, -μ)dt + Baw, where o, is the volatility at time years. W, is standard Brownian motion and the parameters a, ß and all take positive values. Model 2: (ii) (a) Show that for this model: (b) -0 0₁ = 0 +(-e) + Beaw, Hence determine the numerical value of and a relationship between the parameters a and B if it is required that o, has the same long-term mean and variance under each model. State another consistency property between the models that could be used to determine precise numerical values for a and B. [7] The derivative pricing formula used by the trader involves the squared volatility V₁ = 7, which represents the variance of the returns on the index. (iii) Determine the SDE for V, in terms of the parameters a, p and μ. [2]

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Here are the stepbystep workings 1 Longterm distribution of t in Model 1 t is defined recursively as ... View the full answer