Real world exposure and CVA simulation
The risk-neutral approach assumes that asset prices follow stochastic process with drift coinciding with the short rate r(t) being risk-free interest rate. dS(t)=S(t)[rdt+σ(S(t),t) 〖dW〗^Q (t)]
Instead, in real-world measure they follow more complex process, which embodies time and risk aversion of investors, namely: dS(t)=S(t)[μ(S(t),t)dt+σ(S(t),t) 〖dW〗^R (t)] or, equivalently, a process with real-world stochastic discount factors which depend on risk-free interest rates but also on asset prices itself.
The form of this process with almost arbitrary process’s drift term complicates the implementation; for example, makes it difficult in practice to simulate asset prices through standard analytic or quasi-analytic approaches of measure transformation via Girsanov theorem. Let alone the case of imperfect replication with infinite number of no-arbitrage consistent measures, the absence of the unambiguous common real-world measure, as opposed to risk-neutral ones, make real-world simulation much more difficult to implement that risk-neutral one.
Having a limited supply of analytic or quasi-analytic shortcuts and unique no-arbitrage consistent measure, practitioners often choose a brute force approach for risk management purposes such as real-world PFE and CVA simulations, which both require full simulation of exposure distributions for any other-than-vanilla instrument. This brute force approach, known as nested Monte Carlo on Monte Carlo, comprises the following two steps repeated in a loop: drawing real-world yield curves and other relevant market factors at all forward model steps using explicit model assumptions (equivalently, assumptions on drift terms) for a subset of stochastic paths,
...
... middle of paper ...
...k-neutral market models.
Nested Monte Carlo schema requires calculation of instrument
There are a few shortcuts for vanilla instruments. One can omit next nested Monte Carlo step and just price vanilla instruments off the forward curves at each model step t_i. Similarly, one can reduce the pricing of vanilla caps or swaptions to interpolation mechanism of forward volatilities/prices, which is an effective but valid way to avoid building nested evolution model.
For weak path-dependent instruments such as Bermudan callable instruments, which permit representation of conditional exposures as nonlinear function of some market states or non-callable underlying exposures, there is a way to avoid by reusing standard risk-neutral American Monte Carlo results.
Works Cited
Solvency II and Nested Simulations - a Least-Squares Monte Carlo Approach
Staum
Riccardo Rebonato
Money related derivatives empower companies to exchange particular monetary dangers, (for example, premium rate hazard, cash, value and product value hazard, and credit hazard, and so ...
The financial challenge in the managing risk simulation was to balance between preserving capital and capital appreciation in the investment of funds based on a persons’ risk tolerance. The simulation targeted the stock mix for a client’s aversion to risk and the ability of the investment portfolio to have an expected rate of return. The prediction of fund future prices acted as a hedge and had an impact on the rate of return depending on the changing financial landscape of a company. The overall effect was to juggle the mix based on past history and predict a future outcome.
We have devoted our study to apply statistical methods to stochastic differential equations, initially to estimate by the historical method, which uses the property of independence and normality of the outputs. The Black-Scholes model and its alternatives are largely used by the professionals. For that, the estimate of its parameters deserves that we interested in other techniques more adapted: discrete method. The discrete method makes it possible to estimate the parameters of Black-Scholes model in the case of the discrete paths. In this method, it is necessary to observe the process during a certain interval of time i.e. to use all the observations of the paths. The discrete method being based on the criterion which minimizes the variances of the estimators and the small errors with the true values of the share price of gold(Khaldi Khaled, 2010).
After evalutating both the Black-Scholes Model and the Brownian Motion, we have come to know that the Black-Scholes Model is quite predictive as it gets close to the observed price. We found that with the Brownian motion it may take on negative values which results certain modelling prices to be frowned upon , hence making Black-Scholes Model more realistic. As we ventured in this study, we found that there is still more research to be done since many of the modern option models stems from the Black-Scholes model. Thus making the modern option pricing models more
Even though most of us may not realized it, interest rate actually play an important role in our everyday lives due to its great effect on the buying power. For instances, if the interest rate is higher, people tend to reduce their spending and rather save it in the deposit account due to the large interest that they can gained. However, if the interest rate is lower, they rather spend it than keeping it in the deposit account. The reason for this is because the ups and down of the interest rates have a significant impact on their personal income. Furthermore, since interest rate have a major impact on investment it is important for the investors to keep track on these interest rate’s trend before making any decision.
[14]. G. William Schwert, 2002. Anomalies and market efficient, NBER Working paper No. 9277, Oct 2002. JEL No. G14, G12, G34, G32
Asset prices instantly and completely reflect all information of the previous prices. This means future price variations can’t be foreseen by using preceding prices.
There are perfect market conditions; no transaction costs, no taxes’ (Da, Guo and Jagannathan 2012).
Ritter, Lawrence R., Silber, William L., Udell, Gregory F. 2000, Money, banking, and Financial Markets, 10th edn, USA.
After the financial crisis of the late 1990s, the demands for risk management tools have increased. The investors have been effectively utilizing such products as KOSPI 200 futures and options, 3-Year KTB futures and USD futures to meet their hedging needs.
Chapter 11 closes our discussion with several insights into the efficient market theory. There have been many attempts to discredit the random walk theory, but none of the theories hold against empirical evidence. Any pattern that is noticed by investors will disappear as investors try to exploit it and the valuation methods of growth rate are far too difficult to predict. As we said before the random walk concludes that no patterns exist in the market, pricing is accurate and all information available is already incorporated into the stock price. Therefore the market is efficient. Even if errors do occur in short-run pricing, they will correct themselves in the long run. The random walk suggest that short-term prices cannot be predicted and to buy stocks for the long run. Malkiel concludes the best way to consistently be profitable is to buy and hold a broad based market index fund. As the market rises so will the investors returns since historically the market continues to rise as a whole.
Answer: Monte Carlo simulation is a very flexible technique and could easily be adapted or extended. Usually, when it is difficult or, sometimes, even impossible to obtain a closed-form expression of certain results or attributes, it becomes very useful [1]. Because, through repeated random sampling, we might be able to obtain approximate values of our desired results or attributes.
Investment theory is based upon some simple concepts. Investors should want to maximize their return while minimizing their risk at the same time. In order to accomplish this goal investors should diversify their portfolios based upon expected returns and standard deviations of individual securities. Investment theory assumes that investors are risk averse, which means that they will choose a portfolio with a smaller standard deviation. (Alexander, Sharpe, and Bailey, 1998). It is also assumed that wealth has marginal utility, which basically means that a dollar potentially lost has more perceived value than a dollar potentially gained. An indifference curve is a term that represents a combination of risk and expected return that has an equal amount of utility to an investor. A two dimensional figure that provides us with return measurements on the vertical axis and risk measurements (std. deviation) on the horizontal axis will show indifference curves starting at a point and moving higher up the vertical axis the further along the horizontal axis it moves. Therefore a risk averse investor will choose an indifference curve that lies the furthest to the northwest because this would r...
The International Fisher Effect – Dealing with interest rate differentials and expected change in spot foreign exchange rates
Using the Modern Portfolio Theory, overtime risk assets will provide a higher expected rate of return, as compensation to the investors for accepting a high risk. The high risk will eventually lower collecting asset classes to the portfolio, thus reducing the volatile risk, and increasing the expected rates of return. Furthermore the purpose of this theory is to develop the most optimal investments portfolio which would yield the highest rate of return while ascertaining the risk for the individual or corporate investor.