Implied Probability
- Implied Probability is how often a bet needs to win on average to break even over the long haul. When betting at even money, if we win half our bets and loses half our bets we’d net even. Therefore the implied probability at even money is 50%. If a bookie was giving us 2.
- In addition to the showing equivalent odds across a variety of formats, the calculator also provides a very handy Implied Probability calculation. What this does is take the odds that have been entered in by the user and then works out from those odds, the exact probability that.
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The prices of forwards, futures and options contain a great deal of information about what the market thinks or fears about future market conditions.
The prices of forwards, futures and options contain a great deal of information about what the market thinks or fears about future market conditions. Some familiar examples are the use of the forward exchange premium or the spread between a long- and a short-term interest rate as a market forecast of the change in future exchange rate or the short-term interest rate. In fact by combining forward and option prices, an estimate of the entire probability distribution of the future asset price can be obtained.
A common application is the use of the option implied volatility as an estimate of the standard deviation of returns on the underlying asset over the life of the option. Here, one adopts the viewpoint of the Black-Scholes model that asset returns, or percent changes in interest rates, are normally distributed. It is then easy to plot an implied probability distribution of the asset returns, say, one month hence: it is the normal distribution with a mean equal to the current forward premium and a standard deviation equal to the implied volatility.
If several option prices on the same asset and with the same maturity, but with different exercise prices, are observed, the implied probability distribution of returns can be estimated without adopting a particular option pricing model. The resulting estimated distribution may display significant departures--particularly skewness and kurtosis--from the normal distributions at the center of the Black-Scholes view of the world.
Implied Probability is simply the process of converting betting odds into probabilities. When a Bookie sets the odds on a bet, he must first determine what chance he gives that bet of winning. This process is called “capping.” The Bookie must work out the probability that he believes a bet will win. He then sets the odds based on this probability.
To illustrate the value of implied probability distributions in interpreting market behavior, we can use that tool to follow one recent event in the exchange markets. In common with many emerging market currencies, the Mexican peso came under severe pressure in the second half of 1997, eventually giving way and depreciating sharply over the subsequent year--albeit less sharply than many Asian currencies, since the peso had left its peg years earlier, in December 1994. The evolution of its implied distribution, plotted in Figure 1, illustrates several typical features of option-based asset price forecasts.
The implied distributions are estimated by passing an interpolating polynomial through the implied volatilities of peso options with five different exercise prices--the at-the-money and four out-of-the-money puts and calls. The option maturity and thus the forecast horizon are one month. This procedure does not involve any model assumption about the probability distribution of the exchange rate one month hence: we are simply letting the market tell us what it thinks the distribution is.
Each estimate based on all five options is juxtaposed for comparison with an implied distribution based only on the at-the-money implied volatility and the Black-Scholes assumption of normal returns. At each date, both implied distributions have the same mean, equal to the contemporaneous one-month forward rate.
Implied Probability Odds
On September 30, 1997, as peso depreciation fears grew pronounced, the implied distribution is bimodal. The right hump represents the probability of a sharp depreciation. The markets at this point appear to have believed that the peso would either fall sharply, or the pressure would dissipate and the peso would bounce back modestly, with few alternatives in between. This is the classic jump or event risk situation, in which either a large drop in price occurs, with low probability, or a small rise in price, with high likelihood. The forward rate appears as an average of these two possibilities. Note that the dispersion of the peso's value in the event of devaluation is greater than that in the event of a recovery, as one would expect.
In the event, the peso fell far more sharply over the subsequent month than the market's worst expectations. This led to the next phase, illustrated by the implied distribution on October 31, 1997. The general level of implied volatility has now skyrocketed from under 10% to over 30% per annum. Uncertainty dominates the implied distribution: at these levels of implied volatility, skewness and kurtosis are swamped by high dispersion (note the wide range of exchange rates with a positive probability). The graph depicts a state of extreme anxiety and uncertainty, with the market eager to protect against both a sharp further depreciation, of which examples abounded in the exchange markets, and a sharp rally back to fundamental values. Both extremes, and any intermediate outcome, seemed plausible. The realized rate on November 30, 1997 was in fact quite close to its forward rate expected value.
For the time being, the peso depreciation had run its course, and the markets calmed a bit, while remaining wary. The pace of depreciation began to pick up again in mid-1998, in tandem with other emerging market currencies. This situation is reflected in an implied distribution with a more typical volatility of about 10%, but with a large right-hand tail, as on June 30, 1998. The market appears to have been protecting itself against some significant peso depreciation over the ensuing month, without having a strong view on whether it would be a sharp jump or a continuation of the peso's slow grind lower. And, in the event, the peso was little changed a month later.
Implied Probability Definition
The final panel depicts the risk-neutral distribution at the end of July 1998. Market anxiety, the dispersion of the expected future value of the peso, and the skewness of the distribution have all abated somewhat: the implied distribution is less disperse now, but market anxiety is still clearly reflected in the long right tail. In the following month, contagion from the largely unanticipated Russian debt and currency crises brought the peso sharply lower, far exceeding its forward rate forecast.
Implied Probability Of Default
This week's Learning Curve was written byAllan M. Malz, partner inThe RiskMetrics Groupin New York.