Implied volatility

Applying a principal component analysis, they show that there is evidence for a factor driving the implied volatilities of caps and swaptions, that do not drive the term structure. 

An equilibrium model of the term structure, of which we reviewed three in the previous section, is a model that is derived from (or consistent with) a general equilibrium model of the economy. They use generally constant parameters, including most crucially a constant volatility, and the actual parameters used are often calculated from historical time series data. Banks commonly also use parameters that are calculated from actual data and implied volatilities, which are obtained from the prices of exchange-traded option contracts. 

The different models can lend themselves to a particular calibration method. In the Ho-Lee model, rally parallel yield curve shifts are captured and the current yield curve is a direct input therefore, a constant volatility parameter is used. This implies that all the forward rate implied volatilities are identical. In practice, this is not necessarily realistic, as long-dated bond prices often experience lower volatility than short-dated bond prices. The model also assumes... 

Conventionally, investment banks price convertible securities using the implied volatility rather historical volatility. The implied volatility is different to the historical volatility in which the first one refers to volatility going forward. In other words, banks at issue given all parameters, determine the volatility as a goal seek into the model. 

During the book runner phase, banks go in the market with a range of coupons and conversion premium in which each value in the range has a different implied volatility, representing the market sentiment. Thus, the market supply... 

Consider the hypothetical bond explained in Section 9.3.1. The convertible security has a price of 107.5 that includes a historical volatility of 35%. However, the convertible is valued at par (100) using the implied volatility. In practice, given all parameters and given the price of convertible bond (100), the implied volatility is a result of a goal seek value in which the percentage volatility level obtained is 25% (Figure 9.12). 

The reason to use implied volatility is that market anticipates mean reversion and uses the implied volatility to gauge the volatility of individual assets relative to the market. Implied volatility represents a market option about the underlying asset and therefore is forward looking. However, the estimate of implied volatility is conditioned by the choice of other inputs in particular, the credit spread applied in the option-free bond and the conversion premium of the tmderlying asset (Example 9.2). 

Consider the example of Beni Stabili SpA. On 11 July 2014 the convertible of Beni Stabili quoted to 112.229. Analysing Figure 9.13 we see that the option-free bond value is 98.5, while the option value is 613.7. In practical terms, the option value can be seen as the difference between market price of convertible and option-free bond (112.2 — 98.5). However, the option value is found through the implied volatility that is equal to 20.6%. 

Finally, the implied volatility can be compared with the historical volatility of the underlying asset. Going to market, if the implied volatility is less than volatility of the underlying share, it means that an investor might speculate and make profit. 

As introduced, maintaining fixed all parameters, the implied volatility is found by a goal seek procedure. In the same way, the implied spread depends... 

These models are two more general families of models incorporating Vasicek model and CIR model, respectively. The first one is used more often as it can be calibrated to the observable term structure of interest rates and the volatility term structure of spot or forward rates. However, its implied volatility structures may be unrealistic. Hence, it may be wise to use a constant coefficient P(t) = P and a constant volatility parameter a(t) = a and then calibrate the model using only the term structure of market interest rates. It is still theoretically possible that the short rate r may go negative. The risk-neutral probability for the occurrence of such an event is... 

Cap prices can also be valued analytically using the Hull-White model. The cap prices calculated using the implied volatilities of interest rate caps and the Black-Scholes model serve as the calibrating instruments. After the Hull-White model has been calibrated, the parameters a and o that minimize a goodness-of-fit measure can be used to solve for the convexity bias. 

The estimate of the volatility of the spread by using the implied volatility of the reference asset yield, impUed volatility of the benchmark yield and a suitable forward looking estimate of the correlation between the returns on the reference asset yield and benchmark asset yield. 

In pricing an option that expires in the future, however, the relevant factor is not historical but future volatility, which, by definition, cannot be measured directly. Market makers get around this problem by reversing the process that derives option prices from volatility and other parameters. Given an option price, they calculate the implied volatility. The implied volatilities of options that are either deeply in or deeply out of the money tend to be high. 

Calculate the price of a call option written with strike price 21 and a maturity of three months written on a non-dividend-paying stock whose current share price is 25 and whose implied volatility is 23 percent, given a short-term risk-free interest rate of 5 percent. 

The market uses implied volatilities to gauge the volatility of individual assets relative to the market. The price volatility of an asset is not constant. It fluctuates with the overall volatility of the market, and for reasons specific to the asset itself When deriving implied volatility from exchange-traded options, market makers compute more than one value, because different options on the same asset will imply different volatilities depending on how close to at the money the option is. The price of an at-the-money option is more sensitive to volatility than that of a deeply in- or out-of-the-money one. 

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