26

A quick google search retrieves the syllabus for the Stanford STATS 242 class. You can find it here. Just in case it's taken down at some point I'll copy-paste the source material. Keep in mind that I have no idea if this material is good or bad -- I didn't make this list. Also keep in mind that it contains treatments of what does and does not work. With ...


20

These are all examples on Ito Formula in its general form (with quadratic variations):


20

You can't make any concrete statements about the monotonicity, convexity or even sign of the yield curve. Yields are almost always positive, and in the past (2007 and earlier) you could find people who would argue that yields must be positive, typically using a no-arbitrage argument. But recent history has shown us that it is possible for even 10Y yields to ...


15

Find the topic of model-independent properties of option prices very interesting as well. Here are some results that I am aware of and the respective references in the literature. Some are already contained in your initial list as well. Plain Vanilla Prices are Convex in the Strike Theorem 4 in Merton (1973). Delta is Bounded by the Slopes of the Payoff ...


12

Here are couple references. Especially the first link to Andy Lo's paper contains a list of Sharpe ratios of popular mutual and hedge funds: The Statistics of Sharpe Ratios Dow Jones Credit Suisse Hedge Fund Index Generalized Sharpe Ratios and Portfolio Performance Evaluation I would go with the first paper.


12

Along with Gatheral's book, I'd recommend reading Lorenzo Bergomi's "Stochastic Volatility Modelling". The first 2 chapters are available for download on his website. That being said, let me try to give you the basic picture. Below we assume that the equity forward curve $F(0,t)=\Bbb{E}_0^\Bbb{Q}[S_t]$ is given for all $t$ smaller than some relevant ...


11

The answer your are looking for might be the story in "Benchmarking Measures of Investment Performance with Perfect-Foresight and Bankrupt Asset Allocation Strategies", by Grauer (Journal of Portfolio Management). While this work main concerns are the differential ranking of various performance measures and with negative betas for market timing strategies, ...


9

In general, quantitative finance requires mathematics, finance, and numerical programming. The mix of the three and the areas of focus within the three will depend on the particular area you intend to work in. For example, option pricing, risk, and asset management are all related but derivative modeling would draw more on stochastic processes and ...


8

By definition, the average investor holds the market portfolio. Risk aversion can be measured as the slope (i.e. ratio of expected returns to volatility) on the efficient frontier. Therefore, the risk aversion of the average investor assuming the S&P500 is the proxy for the market portfolio is the expected returns of the S&P 500 divided by the ...


8

Stochastics are usually applied in the field of derivatives pricing. In this setting the task is to price a derivative such that it fits into the landscape of tradable instruments (no-arbitrage). We work using the risk-neutral measure - usually denoted by $Q$. The measure is derived from other traded instruments. In risk analysis (e.g. calculate the VaR, ES ...


7

The first book that comes to mind that is written in the style of Definition - Proposition - Proof is: Bjork - Arbitrage Theory in Continuous Time It's pretty well written and can get quite technical. Probably a more common reference is the two-volume set: Shreve - Stochastic Calculus for Finance I & II The first part deals with the binomial model, ...


7

Tsay's Analysis of Financial Time Series should be what you're looking for.


7

A lot has happened since Markowitz and Sharpe. While their work is still considered foundational, the empirical/practical relevance of their models has been questioned by later work. Here are a few more recent articles about portfolio theory, in no particular order (all accessible online): Jorion: Bayes-Stein Estimation for Portfolio Analysis, JFQA, 1986 ...


6

Theta Calculus, a system for representation of complex financial instruments. Kupper & Drapeau's unification of risk concepts. Several papers by Schmid, Bodnar, Okhrin on optimal portfolio weights and tests of same. For example, A test for the weights of the global minimum variance portfolio in an elliptical model. Similarly, Kan and Smith's work on the ...


6

Check Noncommutative Geometry and Stochastic Calculus: Applications in Mathematical Finance


6

I think a good book to start in your case is: Attilio Meucci: Risk and Asset Allocation I once had a seminar held by Attilio that was based on the book and it blew my mind. The book is very intuitive yet rigorous.


6

Elements of Statistical Learning by Hastie, Tibshirani and Friedman is one of the most-cited books for your purpose. Although it does not have any direct applications to Finance, this is definitely a good book to have in your professional library and can be used as a reference for most topics. If you want to use a book with more financial applications, I ...


6

One of the best pieces ever written on this topic is Salomon's "Principles of Principal Components," which is readily available on the Internet. I won't go into the details, since this paper is ridiculously comprehensive, but the fundamental idea is straightforward -- if you run a PCA based on yields, the first three components capture most of the variances, ...


6

MF is linked with physics mostly because it solves the same PDEs (Black-Scholes equation is a certain type of Schrödinger equation for instance). As for the specific links you mentioned : Lie Algebra : Magnus expansion (to build fast approximation of time dependent ODEs like those arising in credit risk) Differential geometry : link with Varadhan ...


6

I think that "An Introduction to Statistical Learning: with Applications in R (Springer Texts in Statistics)" suggested by KarolisR could be useful but too much machine learning oriented. Moreover, such a book is for beginners. As a thorough book (PhD level) on statistics, I suggest "Statistical Inference" by Casella and Berger.


6

1: Follow the calculations in The Complete Guide to Option Pricing Formulas. The book has many formulas, sample values and outputs. Highly recommended for validating your results. Apparently, this is one of most popular books used by real-world quants (simple and fast). 2: You can still use QuantLib to price with year fractions. I have an example: ...


6

Note that, as in this question, for $s\ge t\ge 0$, \begin{align*} n_s = e^{-a_n(s-t)}n_t + \int_t^s \theta_n(u)e^{-a_n(s-u)} du + \int_t^s \sigma_n e^{-a_n(s-u)} dW^n_u, \end{align*} and \begin{align*} r_s = e^{-a_r(s-t)}r_t + \int_t^s (\theta_r(u) -\rho_{r,n}\sigma_n\sigma_r) e^{-a_r(s-u)} du + \int_t^s \sigma_r e^{-a_r(s-u)} dW^r_u. \end{align*} Moreover,...


5

I would even stick to the original paper by Sharpe (1966): Mutual Fund Performance. The Journal of Business Vol. 39, No. 1, Part 2 pp.119--138 If you look at the numbers on Page 6 you can see that the funds sharpe ratios roughly are between $0$ and $1$. Since the Sharpe ratio already adjusts for the risk-free rate, you cannot really argue about its ...


5

I would start with explaining random walk (this should be fairly simple) and then making a connection to heat equation in discrete time. This paper is doing exactly this and by leaving out technicalities you should make this pretty intuitive for students. Basically the intuition is as follows: At each integer time unit, the heat at each point is spread ...


5

A concrete example of negative forward rates is provided by the 3M CHF LIBOR futures. They're all trading above a price of 100, which implies negative forward rates. See the prices here. Despite the prices of the forwards, CHF libor hasn't actually fixed negative yet. But the forwards are certainly all below zero. Also, your formula for the forward rate ...


5

Forward interest rates are negative whenever the yield curve is negatively sloped. The US term structure was inverted most recently around 2007. Hard to find bank deposits that have negative yields (find countries experiencing deflation and you may find it), however, treasury bills during recent times of financial stress have yielded a negative rate. The ...


5

I thought this was an interesting example to add. It concerns a "ratio model" of habit (as opposed to a "difference" model of habit). See, for example, Abel (1990, American Economic Review). Let $$ x_t = \lambda \int_{-\infty}^t e^{-\lambda(t-s)} c_s ds. $$ (For context, $x_t$ is a log habit index that is given by a geometric average of past consumption, ...


5

There are plenty of books on portfolio issues built according to formula "some theory + some R code (or Matlab, or S - which is very similar to R)". See for example Pfaff B. Financial Risk Modelling and Portfolio Optimization with R.// 2013. Best M.J. Portfolio Optimization. Chapman & Hall, 2010. Würtz D. et al. Portfolio Optimization with R/Rmetrics. ...


5

Introductory: "Introduction to the Mathematics of Financial Derivatives"; Salih Neftci More mathematical: "Stochastic Calculus for Finance I and II"; Steven Shreve


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