Question 1 (15 marks) Consider a financial market with two instruments: a money market account yielding the risk-free rate r > 0, and a risky stock with price process (St)eo. You can lend and borrow freely at the riskless rate. a. Write down an example of continuous-time dynamics for S satisfying the following condi- tions: (i) the price process has continous trajectories; (ii) the log-price is an affine process; (iii) the instantaneous volatility is a deterministic function of time. (5 marks) b. Consider a contingent claim Hr that needs to be priced within the setup outlined in the previous point. Write down a risk-neutral valuation formula yielding the price H of H, at some time t [0, T). Is the price unique? Discuss. (5 marks) c. Assume that point a. (iii) is now replaced by (iii) the instantaneous volatility is a stochastic process. How would your answer to question b. change? (5 marks) Question 1 (15 marks) Consider a financial market with two instruments: a money market account yielding the risk-free rate r > 0, and a risky stock with price process (St)eo. You can lend and borrow freely at the riskless rate. a. Write down an example of continuous-time dynamics for S satisfying the following condi- tions: (i) the price process has continous trajectories; (ii) the log-price is an affine process; (iii) the instantaneous volatility is a deterministic function of time. (5 marks) b. Consider a contingent claim Hr that needs to be priced within the setup outlined in the previous point. Write down a risk-neutral valuation formula yielding the price H of H, at some time t [0, T). Is the price unique? Discuss. (5 marks) c. Assume that point a. (iii) is now replaced by (iii) the instantaneous volatility is a stochastic process. How would your answer to question b. change