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# WIFEWeek in Risk (April 14th)

#### David Harper CFA FRM

##### David Harper CFA FRM
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New Practice Questions
In the Forum

1. When is it optimal to early exercise? David Kim found a problem with one of my question's choices (here at https://trtl.bz/2GcAnr2) about the optimality of early option exercise. Our forum has discussed this interesting dilemna for years; ie., when exactly is it optimal to exercise?. The feature can be viewed as an option embedded in an option. As is so often the case, precise language is important. The thread includes a small model I built that quantifies the trade-off. In the case of an American call on a dividend-paying stock, the choice is between the exercising early to collect dividends versus the value of waiting. The value of waiting is classic finance: Strike minus PV(Strike) = K*[1-exp(rT)].

2. Convexity: We had a couple of conversations above convexity, including Malz's characterization of a securitization's equity tranche as positively convex, a senior bond tranche as negatively convex, and the mezzanine tranche as mixed. See https://trtl.bz/2Gc1vGD. In the context of the vanilla bond asset class, @abhinavkhanna and I had a clarifying conversation about convexity https://trtl.bz/2Ggksru. Convexity is a second derivative such that positive convexity is when the first derivative (aka, slope of the tangent line) is increasing. Hopefully, you can visualize, per the curvature of the classic price/factor relationship, how vanilla bonds (i.e., bonds with embedded options) and options exhibit positive convexity. The slope of the bond's tangent (to the price/yield curve) is negative but increasing with yield; hence its positive convexity! This slope is mathematically simple: ∂P/∂Y = -D*P. If negate this, we get dollar duration = D*P. Then if we divded dollar duration by the number of basis points in one unit (100.0%), we get DV01 = D*P/10,000.

3. Credit value adjustment (CVA): @nansverma asked some good questions about Jon Gregory's credit value adjustment (CVA): https://trtl.bz/2UELCm7 Did you know Jon Gregory is a forum member? There were some superficial formula changes between his second and third edition, with respect to CVA. But it should not throw off a fundamental understanding of CVA, which translates the fundamental expected loss formula, EL = EAD*PD*LGD, into an expression over time. Per one of the good questions, the default probability in the expression in unconditional. We were early to understand this feature: we informed an edit to one of GARP's practice exam questions on CVA.

4. Additionally: Also interesting this week: @Jaskarn on the multiple ways (I called them layers) in which counterparty credit risk also involves market risk. @Branislav on Stulz's risk failure typology (a typology that challenges all of us!) https://trtl.bz/2UyLiVM. @Merlinius with a provocative question about a seemingly straightforward CAPM discount rate question https://trtl.bz/2UBKX4W. Mansoor correcting my imprecise reference to Dowd's bid/ask spread as additionally endogenous https://trtl.bz/2UFGLAX. @Jaskarn on Crouhy's distinction between traditional bank models and originate-to-distribute (OTD) models https://trtl.bz/2UyLowC

External

1. Consolidated Basel Framework: The Basel Committee has collected its global banking standards into one "consolidated" sub-site at https://www.bis.org/basel_framework/. For those of us who have studied the regulations over the years, this is an overdue but welcome development. Previously, to lookup regulations (in order to answer technical subscriber questions), I often had to link-jump through a chain of consecutive standards. BIS even recorded a YouTube video explainer at https://www.bis.org/bcbs/publ/d462.htm. (yes, that's correct, BIS has a YouTube channel, so, um, anything is possible?).

2. Stationarity explained: On the knowledge area of Time Series Analysis, the FRM assigns four chapters from Diebold. These are notoriously difficult because they lack scaffolding: ideally, you should read the four first unassigned chapters in order to follow some of the terminology. In particular, the definitions of stationarity in Chapter 7 assume prerequisites. Last week, Shay Palachy of Toward Data Science wrote one of the better introductions to stationarity that I've read: https://trtl.bz/2GiKHgT e.g., "In the most intuitive sense, stationarity means that the statistical properties of a process generating a time series do not change over time. It does not mean that the series does not change over time, just that the way it changes does not itself change over time." His simple images, in particular Figure 6, are really helpful.

3. Who is hedging climate change and disrupting insurance: Finally, I wanted to share two provocative articles from Sunday's New York Times: How Big Business is Hedging Against the Apocalypse https://trtl.bz/2GbsXnO and: A.I. Is Changing Insurance https://trtl.bz/2GfxwgY.

4. Elsewhere: ISO 31000 VS. COSO: Comparing and Contrasting the World's Leading Risk Management Standards (Carol Williams) https://trtl.bz/2Gfy6Lx; Progress on the Transition to Risk-Free Rates (Federal Reserve) https://trtl.bz/2GiogIX; Mathematicians Discover the Perfect Way to Multiply https://trtl.bz/2Gfxq8X

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