In the Black-Scholes model, although the stock price process (St)+20 makes fluctuations infinitely often, the drift...

 

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In the Black-Scholes model, although the stock price process (St)+20 makes fluctuations infinitely often, the drift is usually positive and hence the prices tend to increase on average. This is better seen in the dynamics dS₁ S₁ = (1 + 7/20²) Sidt + SidW... On the other hand, many stochastic processes in finance (and in physics!) are mean- reverting in the sense that they tend to fluctuate around a long-term mean level m > 0. A typical model for a mean-reverting process (X+)+20 is described by the so-called Ornstein-Uhlenbeck (OU) process whose dynamics are given by the following stochas- tic differential equation: dx₁ = (m-X₂)adt + odW₁. Here, a > 0 is the rate of mean-reversion. Whenever the process is above the long-term mean at some time t, the drift term pulls it down to m as m-X₁ <0. Symmetrically, the drift term pulls the process up if it is below m. Mean-reversion is used in modeling stochastic interest rate and stochastic volatility in finance, and the stochastic velocity of a particle in physics. a. (8) Define the standard OU process (V+)tzo by V₁ = X₁-m-(Xo-m)e-at, t≥0, where XO E R is deterministic. Write down the dynamics of (V+)+20 as a stochastic differential equation. b. (9) Solve the stochastic differential equation for (V+)2o. Then, find a representation of (X+)+20 involving a stochastic integral. C. (5) Find the expected value of X₁. d. (8) Find the variance of Xt. In the Black-Scholes model, although the stock price process (St)+20 makes fluctuations infinitely often, the drift is usually positive and hence the prices tend to increase on average. This is better seen in the dynamics dS₁ S₁ = (1 + 7/20²) Sidt + SidW... On the other hand, many stochastic processes in finance (and in physics!) are mean- reverting in the sense that they tend to fluctuate around a long-term mean level m > 0. A typical model for a mean-reverting process (X+)+20 is described by the so-called Ornstein-Uhlenbeck (OU) process whose dynamics are given by the following stochas- tic differential equation: dx₁ = (m-X₂)adt + odW₁. Here, a > 0 is the rate of mean-reversion. Whenever the process is above the long-term mean at some time t, the drift term pulls it down to m as m-X₁ <0. Symmetrically, the drift term pulls the process up if it is below m. Mean-reversion is used in modeling stochastic interest rate and stochastic volatility in finance, and the stochastic velocity of a particle in physics. a. (8) Define the standard OU process (V+)tzo by V₁ = X₁-m-(Xo-m)e-at, t≥0, where XO E R is deterministic. Write down the dynamics of (V+)+20 as a stochastic differential equation. b. (9) Solve the stochastic differential equation for (V+)2o. Then, find a representation of (X+)+20 involving a stochastic integral. C. (5) Find the expected value of X₁. d. (8) Find the variance of Xt.

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a The dynamics of the standard OrnsteinUhlenbeck OU process denoted by V can be written as a stochas... View the full answer