People who repeatedly risk money, whether they be traders for financial institutions, corporate executives, day traders, or sports bettors, sometimes appear to do better than chance only because the risk of large losses is hidden or overlooked. As students of casino gambling know, one way to obscure the risk of large losses is to bet more when you are losing and less when you are winning. In 19th century casinos, betting strategies that did this were called martingales. Following such strategies, whether deliberately or unwittingly, was called martingaling. Traders in financial instruments often martingale; in fact, they are martingaling whenever they respond to a margin call. A businessperson who doubles down on an apparently losing investment is martingaling. Opinionated sports bettors easily fall into martingaling. The martingale index, defined in this paper, measures the portion of the apparent success of a betting, trading, or investment strategy that can be attributed to martingaling. We calculate the martingale index for some popular casino strategies and also for some strategies that model random trading in S&P 500 futures and in stocks. And we discuss how educating the public about the martingale index might help both businesses and individuals avoid the temptations of martingaling.

In this paper we present many investigations into the results of simulating the process of hedging a vanilla option at discrete times. We consider mainly a ‘maxi’ option (paying Max(A, B)), though calls, puts and ‘minis’ are also considered. We show the sensitivity of the variability of the hedging error to the actual investment strategy adopted, and to the many ways in which the simulated real world can diverge from the assumed option pricing model. We show how prudential reserves can be calculated, using conditional tail expectations, and how net premiums or fair values (which we present as the same) can be calculated, allowing for the necessary prudential reserves. We use two bond models, the very simple Black-Scholes one and a less unrealistic one. We also use the Wilkie model as an even more realistic real-world model, allowing for many complications in it to make it more realistic. We make observations on the important difference between real-world models and option pricing models, and emphasise the latter as the way of getting hedging quantities, and not just option prices.

A formal definition of investment risk in actuarial investigations is given. Case studies estimating the investment risk associated with different investment strategies for defined benefit pension funds using historic market data are presented. It is shown that a few decades ago, when bond markets only extended in depth to 20-year maturities, the investment risk of investing in equities was of the same order of magnitude as the investment risk introduced by the duration mismatch from investing in bonds for immature schemes. It is shown that now, with the extension of the term of bond markets and introduction of strippable bonds, the least risk portfolio for the same pension liability is a bond portfolio of suitable duration. It is argued that investment risk voluntarily undertaken in defined benefit pension plans has grown markedly in recent decades, at a time when the ability to bear the investment risk has diminished. Investment risk in pension funds is quite different to investment risk for other investors, which leads to the possibility that current portfolios are not optimised — that is, there exist portfolios which increase the expected surplus without increasing risk. The formalising of our intuitive concept of investment risk in actuarial applications is a first step in the identification of more efficient portfolios.