Hedging beyond BlackScholes The focus is on the theory of hedging in models beyond the BlackScholes model for stock prices. Black and Scholes developed a theory of pricing options on a stock based on the delta-hedging trading strategy. Subsequent authors have considerably generalized their theory, and hedging is still an active area of research. The compulsory parts of the project require you to: Part 1: Write a literature review highlighting aspects of the development of the theory of hedging that interest you; Part 2: Develop a numerical simulation of hedging in Vasiceks interest rate model. You should then develop these ideas further in a direction of your choosing for Part 3 of your project. Part 1: Literature review Give a literature review of aspects of the theory of hedging that you consider to be of particular interest. The theory is too large for it to be realistic to review the entire theory, so it is important to focus on those aspects of the theory that will be important in Part 2 and Part 3 of your project. You should be sure to provide enough background for the reader to be able to understand the numerical work in Part 2. You may therefore want to illustrate the more general theory with the example of hedging interest rate products. As an entry point to the literature you may wish to study [2] which discusses interest rate modelling in general, and in particular devotes a chapter to Arbitrage Theory including the theory of hedging and pricing. Classical references include Black and Scholes original paper [1], Vasiceks paper [5] and the papers of Harrison and Kreps [3] and Harrison and Pliska [4]. 1 Parameter Value a 0.5 b 0.05 s 0.05 r0 0.02 ? -1 S 1 T 0.5 U 2 N 100 Table 1: Parameter values for the numerical simulation You should be proactive in researching the literature, which involves published journal papers and books. Working papers should be used mostly for orientation, given that their content has not been peer reviewed. It is particularly important that you synthesize the information gathered from these sources and presents it as a flowing story that is consistent both in terms of notation and mathematical and financial content. Part 2: Numerical analysis In this part of the project you will illustrate the theory of hedging in the Vasicek model numerically. Suppose that the short-term interest rate follows the process drt = a(b - rt)dt + sdWQ t in the pricing measure, Q. This statement is intended to imply that bonds of all maturities T are available in the market at the risk-neutral prices given by Vasicek model. You should use this to compute the prices of bonds in the calculations below. You should discuss which P measure models are consistent with this Qmeasure process for rt. In the calculations below you should use the specific P-measure model given by drt = (a(b - rt) - ?s)dt + sdWP t where ? is a constant. The parameter values you should use are summarized in Table 1 In this model, a trader can replicate the payoff of a T-bond (i.e. zero coupon a bond with maturity T) by a continuous time trading strategy in: (i) The money market account, which has instantaneous interest rate of rt; (ii) An S-bond, with maturity S > T. 2 You should give the price for a bond with maturity T in this model and compute how many S-bonds should be held at time t ? [0, T] to implement this replication strategy. We will call this quantity Qt. In practice, continuous time trading is not possible. Suppose that a trader has initial wealth 0 and receives the T bond price at time 0 and in exchange, promises to pay 1 unit of currency at time T. They then hedge their risk by trading at time intervals of dt = T /N for some integer N starting from time 0. At each discrete time point they ensure that they are holding Qt units of the S-bond and use the rest of their wealth to purchase zero coupon bonds with maturity t + dt. Derive equations for the traders wealth at each time point. Simulate the process rt. Use this to numerically compute the following statistics for the strategy in the given P-measure: (i) The 1 day 95% Value at Risk of the hedging portfolio at time 0. This hedging portfolio consists of -1 unit of the T bond together with the traders holding in the bonds with maturity dt and S. You should use the Vasicek model to compute the market value of the bonds in your portfolio after 1 day. (ii) The 1 day 95% Expected Shortfall of the hedging portfolio at time 0 (iii) The expected profit and loss of the strategy (iv) The variance of the profit and loss of the strategy Demonstrate numerically that as dt tends to 0, the variance of the strategy tends to zero, and estimate the rate of convergence. Demonstrate numerically that if the trader is also allowed to use a bond with maturity U to hedge the risk, it is possible to achieve lower Value at Risk and to achieve a faster rate of convergence. You may find that the benefits of this approach depend heavily upon the value of s. To help us mark your work, you should include a table of results formatted as shown in Table 2 in addition to your own choice of graphs and analysis: Part 3: Advanced issues (to be done after part 2) Some possible suggestions of topics you might consider in this section are ? Back-testing hedging with real market data; ? More sophisticated interest rate models; ? Hedging in more sophisticated equity models; ? Superhedging and subhedging in incomplete markets; ? Hedging in markets with transaction costs; 3 Hedging with S Bond 1 day VaR 1 day Expected Shortfall Expected Profit and Loss Variance of Profit and Loss Hedging with S Bond U bond 1 day VaR 1 day Expected Shortfall Expected Profit and Loss Variance of Profit and Loss Table 2: Summary of numerical results ? Indifference pricing as an alternative to risk neutral pricing in incomplete markets; ? Hedging in rough volatility models; References [1] F. Black and M. Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81(3):637654, 1973. [2] D. Filipovic. Term-Structure Models. A Graduate Course. Springer, 2009. [3] J. M. Harrison and D. M. Kreps. Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20(3):381408, 1979. [4] J. M. Harrison and S. R. Pliska. Martingales and stochastic integrals in the theory of continuous trading. Stochastic processes and their applications, 11(3):215260, 1981. [5] O. Vasicek. An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2):177188, 1977. 4

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