Super Bowl XLVII was held at the Superdome in New Orleans on February 3, 2013 and featured the San Francisco 49ers against the Baltimore Ravens. Going into Super Bowl XLVII the San Francisco 49ers were favorites to win which was not surprising following their impressive season. The Ravens opened up a substantial lead, 28–6, early in the third quarter. This was followed by a rather unusual event – the stadium suffered a 34 min power outage. When power was restored the 49ers began to come back. The game’s outcome was in doubt late in the final quarter when the 49ers were close to scoring a go-ahead touchdown. The 49ers failed to score however and the Ravens held on for a 34–31 upset victory.

We applied the model of Section 2 from the viewpoint of the Ravens so that *X*(*t*) corresponds to the Raven’s lead over the 49ers at time *t*. provides the score at the end of each quarter.

Table 3 Super Bowl XLVII by quarter.

The parameters of our model were obtained using information from the pregame betting markets. The initial *point spread* established the Ravens as a 4 point underdog so we set the mean of our outcome, *X*(1), as

$$\mu =\mathbb{E}\mathrm{(}X\mathrm{(}1\mathrm{)}\mathrm{)}=-4.$$

The markets’ pregame assessment of the probability that the Ravens would win is determined from the pregame *money-line* odds. These odds were quoted as San Francisco −175 and Baltimore +155. This means that a bettor would have to bet $175 to win $100 on the 49ers while a bet of $100 on the Ravens would lead to a win of $155. We converted both of these money lines to *implied probabilities* of each team winning, using the equations

$${p}_{SF}=\frac{175}{100+175}=0.686\text{\hspace{0.17em}and\hspace{0.17em}}{p}_{Bal}=\frac{100}{100+155}=0.392$$

The resulting probabilities do not sum to one. This “excess” probability is in fact the mechanism by which the oddsmakers derive their compensation. The probabilities actually sum to one plus a quantity known as the “market vig”, also known as the bookmaker’s edge. In this example,

$${p}_{SF}+{p}_{Bal}=0.686+0.392=1.078$$

providing a 7.8% edge for the bookmakers. Put differently, if bettors place money proportionally across the two teams then the bookies will make 7.8% of the total staked no matter the outcome of the game. To account for this edge in our model, we used the mid-point of the two implied probabilities that Baltimore would win to determine *p*. This yielded

$$p=\frac{1}{2}{p}_{Bal}+\frac{1}{2}\mathrm{(}1-{p}_{SF}\mathrm{)}=0.353$$

Thus, from the Ravens perspective we took the initial probability of a win to be *p*=P(*X*(1)>0)=0.353.

Figure 2 shows the evolution of the markets’ assessment of the conditional probability of the Ravens winning as the game progresses, which we denote by $${p}_{t}^{mkt}.$$ These probabilities are derived from betting prices obtained from the online betting website `TradeSports.com`. The contract volumes (amount of betting) are displayed along the *x*-axis. Baltimore’s win probability started trading at $${p}_{0}^{mkt}=0.38$$ (similar to the initial estimate derived from the money odds above) and the dynamic probabilities fluctuated dramatically throughout the course of an exciting game. The Ravens took a commanding 21–6 lead at half time and the market was trading at $${p}_{\text{0}\text{.5}}^{mkt}\approx \mathrm{0.90.}$$ The market-implied probability increased to 0.95 when Baltimore extended its lead by returning the second-half kickoff for a touchdown. This was followed by the power blackout. During the blackout 42,760 contracts changed hands with Baltimore’s win probability ticking down from 0.95 to 0.94. There was considerable movement in the market probability as the game became closer in the fourth quarter. Near the end of the fourth quarter, the 49ers had the ball in position to take the lead (the point labelled “49ers 1st & Goal” in the figure) and Baltimore’s win probability had dropped to just above 30%. The 49ers failed to score in the course of the next four plays and the win probability jumped back up near one.

Figure 2: Betting data for Super Bowl XLVII between the Ravens and 49ers obtained from the `TradeSports.com` website. The solid line shows the dynamic market probabilities of the Ravens winning with the vertical axis giving the probability scale (numbers expressed as percentages). The number of contracts traded at each point in time is represented by the vertical bars along the bottom of the figure; these can also be interpreted with reference to the vertical axis with the numbers now representing thousands of units traded.

Our model allows us to provide answers for a number of important questions:

*What implied volatility is consistent with pregame market expectations?*

The implied volatility of the Super Bowl is calculated by substituting the pair (*μ*, *p*)=(−4, 0.353) into our definition and solving for *σ*_{IV}. We obtained

$${\sigma}_{IV}=\frac{\mu}{{\Phi}^{-1}\mathrm{(}p\mathrm{)}}=\frac{-4}{-0.377}=10.60$$

(the value we have been using throughout the article) where we have used Φ^{−1}(*p*)=*qnorm*(0.353)=−0.377. So on a volatility scale the 4 point pregame advantage assessed for the 49ers is less than $$\text{0}\text{.5}\sigma .$$ The inferred *σ*=10.6 is historically low, as the typical volatility of an NFL game is somewhere between 13 and 14 points (see Stern, 1991). One possible explanation is that for a competitive game like this, one might expect a lower than usual volatility. (Of course another possibility is that the two market assessments, the point spread and the win probability, are inconsistent for some reason.) The outcome *X*(1)=3 was within one standard deviation of the pregame model which had an expectation of *μ*=−4 and volatility of *σ*=10.6. As the game progresses our framework can be used to address a variety of additional questions.

*What’s the probability of the Ravens winning given their lead at half time?*

At half time Baltimore led by 15 points, 21 to 6. There are two ways to estimate the probability that Baltimore will win the game given this half time lead. Stern (1994) assumes the volatility parameter is constant during the course of the game (and equal to the pre-game value). Using this assumption we applied the formula derived earlier with conditional mean for the final outcome 15+0.5*(−4)=13 and conditional volatility $$10.6\sqrt{1-0.5}=\mathrm{7.5.}$$ These yielded a probability of .96 for Baltimore to win the game.

A second estimate of the probability of Baltimore winning given their half time lead can be obtained directly from the betting market. From the online betting market we have traded contracts on `TradeSports.com` that yield a half time probability of $${p}_{\text{0}\text{.5}}^{mkt}=\mathrm{0.90.}$$ There is a notable difference in the two estimates. One possible explanation for the difference is that the market assesses time varying volatility and the market price (probability) reflects a more accurate underlying probability. This leads to further study of the implied volatility.

*What’s the implied volatility for the second half of the game?*

We determined the market’s assessment of implied volatility at half time by assuming that $${p}_{t}^{mkt}$$ reflects all available information. We applied the formula of Section 2.3 using *t*=0.5 and obtained

$${\sigma}_{IV\mathrm{,}\text{\hspace{0.17em}}t=\text{0}\text{.5}}=\frac{l+\mu \mathrm{(}1-t\mathrm{)}}{{\Phi}^{-1}\mathrm{(}{p}_{t}\mathrm{)}\sqrt{1-t}}=\frac{15-2}{{\Phi}^{-1}\mathrm{(}0.9\mathrm{)}\sqrt{0.5}}=14$$

where *qnorm*(0.9)=1.28. As 14>10.6, the market was expecting a more volatile second half – possibly anticipating a comeback from the 49ers. It is interesting to note that the implied half time volatility parameter is more consistent with the value typical for regular season games.

*How can we use this framework to form a valid betting strategy?*

The model provides a method for identifying good betting opportunities. If you believe that the initial money line and point spread identify an appropriate value for the volatility (10.6 points in the Super Bowl example) – and this stays constant throughout the game – opportunities arise when there’s a difference between the dynamic market probabilities and the probabilites obtained from the Brownian motion model. This can be a reasonable assumption unless there has been some new material information such as a key injury. For example, given the initial implied volatility *σ*=10.6, at half time with the Ravens having a *l*+*μ*(1−*t*)=13 points edge we would assess a

$${p}_{\text{0}\text{.5}}=\Phi \mathrm{(}13/\mathrm{(}10.6/\sqrt{2}\mathrm{)}\mathrm{)}=0.96$$

probability of winning versus the $${p}_{\text{0}\text{.5}}^{mkt}=0.90$$ rate. This suggests a bet on the Ravens is in order. To determine our optimal bet size, *ω*_{bet}, we might appeal to the Kelly criterion (Kelly, 1956) which would yield

$${\omega}_{bet}={p}_{\text{0}\text{.5}}-\frac{{q}_{\text{0}\text{.5}}}{{O}^{mkt}}=0.96-\frac{0.1}{1/9}=0.60$$

where $${p}_{\text{0}\text{.5}}$$ is our assessment of the probability of winning at half time (0.96), $${q}_{\text{0}\text{.5}}=1-{p}_{\text{0}\text{.5}}$$ and the market offered odds are given by

$${O}^{mkt}=\mathrm{(}1-{p}^{mkt}\mathrm{)}/{p}^{mkt}=\mathrm{(}1-0.9\mathrm{)}/0.9=1/9.$$

This would imply a bet of 60% of capital. The Kelly criterion is the optimal bet size for long-term capital appreciation. In this setting the bet size seems quite high. It is mathematically correct since we believe we have a 96% chance of earning a 10% return but the bet size is very sensitive to our estimated probability of winning. Given that the Brownian motion model is just an approximation we may wish to be more conservative. It is common in such situations to bet less than the optimal Kelly bet size. The fractional Kelly criterion scales one’s bet by a risk aversion parameter, *γ*, often *γ*=2 or *γ*=3. Here with *γ*=3, the bet size would then be 0.60/3=0.20, or 20% of capital.

Alternate strategies for assessing betting opportunties are possible. The assumption of constant volatility in the previous paragraph is often reasonable but not always. There may be new information available such as a change in weather conditions or an expected change in team strategy (e.g., given the deficit the 49ers might be expected to take more chances). If so, the implied volatility calculation demonstrated above provides a useful way for bettors to evaluate their opportunities. In the Super Bowl one might judge that the half-time implied volatility (14 points) is in fact more realistic than the pre-game value (10.6 points). This would lead one to conclude that the market probability $${p}_{\text{0}\text{.5}}^{mkt}$$ is the more accurate estimate and there would not be a betting opportunity at half time of the game.

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