Hedge ratio

Therefore, the discount rate is a risk-free rate rf or risky rate depending on the hedge ratio ... 

If the hedge ratio is 1, we have a risk-free rate. In this point, the position of the investor is long above the underlying share price and also receiving a coupon. At a hedge ratio of 1, the option will move identically to the underlying asset. 

If the hedge ratio is 0, according to formula (9.22) we obtain a risky rate. At a hedge ratio of 0, the option does not follow the moves of underlying share price. 

If the hedge ratio stays in the middle, the discount rate is stiU a risky rate. 

The change in share price determines a change in the option value. For instance, at the highest value of the share price or 11.51 the option value is 8.91. If the share price changes to 11.45, the new option value is 8.85. Therefore, the hedge ratio for that node is equal to 1 as follows ... 

Conversely, at the lowest node, the hedge ratio is 0 because the option is out of money or 0. This means that in the first case the bond trade like the equity, while in the second case like a conventional bond. Therefore when the share price increases the delta approaches unity, implying that the option is deeply in the money. In contrast, when the share price is low relative to the conversion price, the sensitivity of the convertible and therefore of the embedded option is low. 

In the first case the right discount rate to apply is the risk-free rate equal to 1.04%, while in the second case is the risky rate equal to 8.04%. Figure 9.30 shows the hedge ratio at each node. 

Depending on the hedge ratio, the discount rate changes according to the formula (9.22). This is shown in Figure 9.31. 

Questions such as the uses to which European bond futures can be put, contract specifications and trading volumes are discussed with illustrative examples. Technical issues, which surround the use of bond futures, are also examined and presented with numerical examples. The issues include the calculation of gross and net basis, identifying the cheapest-to-deliver (CTD) cash market bond, different approaches to measuring relative volatility, calculating hedge ratios, and portfolio duration adjustment. Bloomberg screen output is used to provide a real world flavour to the topics covered. 

The Euro-Bund contract specifies a contract size of 100,000. To establish an appropriate hedge ratio, a natural starting point is to divide the value of the underlying portfolio by the contract size ... 

This, however, is only part of the picture. Even though the bond held in the portfolio is the CTD, the cash market bond s behaviour relative to the notional bond described in the futures contract needs to be taken into account. The following equation illustrates how, in general, the appropriately adjusted hedge ratio can be estimated. 

Substituting the estimated BPVs into equation (16.11) provides and an adjust hedge ratio based on the relative volatility estimated to exist between the cash market bond and the futures contract. 

At the time, the June 2003 Euro-BUND future is trading at 116.42, and the cheapest-to-deliver (CTD) bond for the June 2003 futures contract is the 5% bond maturing on 4 January 2012, and having a conversion factor of 0.9341. The PVBP (or DVOl) for the CTD bond is 0.0788, while that for the bond actually held is 0.0840. The hedge ratio for the bond portfolio is therefore... 

The main issue with this hedge ratio is to hedge the position for small movements in the CDS curve. The hedge ratio is such that there is no profit or loss for a small parallel shift in the CDS curve. 

Therefore, to hedge against profit and loss (P8cL) movements for small changes in the credit curve, we would use the following hedge ratio. 

For example, let us assume that an investor holds a bond with a price, Pbond that the recovery on this bond in the event of default is bond- The loss to the investor is (Pbond bond) result of the default. Flowever, the investor can hedge the bond position by buying CDS protection, for an amount equal to the default neutral hedge ratio multiplied by the standard CDS contract. 

The payoff from the standard CDS contract is (1 - PcDs) nd the default neutral hedge ratio multiplied by (1 - Rqds) would provide the hedging cash flow to offset the loss on the bond position. 

For bond positions hedged via CDSs in a default neutral hedge ratio, the profit and loss generated will depend on the recovery values actually obtained for the bond and the CDS contract. The assumptions regarding recovery are crucial for hedge effectiveness. 

The notional value of the CDS protection is equal to the market price of the bond. This is effectively the same as the default neutral hedge ratio with the recovery equal to zero. 

The primary risk measure required when using a swap to hedge is the present value of a basis point. PVBP, known in the U.S. market as the dollar value of a basis point, or DVBP indicates how much a swap s value will move for each basis point change in interest rates and is employed to calculate the hedge ratio. PVBP is derived using equation (7 23). 

The correct nominal amount of the swap is established using the PVBP hedge ratio, shown as equation (7.27). Though the market still uses this method, its assumption of parallel yield-curve shifts can lead to significant hedging error. 

The notional maturity of a long-bond contract is always given as a range for the contract on the 10-year note, for example, it is six to ten years. The duration used to calculate the hedge ratio would be that of the cheapest-to-deliver bond. 

Asymmetric Optimal Hedge Ratio with an Application... 

The interested reader can refer to the following literature on the optimal hedge ratio [3,6-8, 10, 11, 15, 17]. 

The rest of the paper is stmctured as follows. Section 2 makes a brief discussion of the optimal hedge ratio including a mathematical derivation of this ratio. Section 3 describes the underlying methodology for estimating the asymmetric OHR, and it also proves mathematically the asymmetric property of the OHR. Section 4 provides an empirical application, and the last section concludes the paper. 

Thus, h is the hedge ratio, which can be obtained as the slope parameter in a regression of the price of the spot instrument on the price of the future (hedging) instrument. This can be demonstrated mathematically also. Let us substitute (19.3) into (19.2) that yields the following equation ... 

Methodology for Estimating Asymmetric Optimal Hedge Ratio... 

Table 19.1 The estimated hedge ratios (the standard errors are presented in the parentheses)...

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