金融代写|金融模型代写Modelling in finance代考|Numerical examples

In this section we give some numerical examples of the impact of the futures features on the price and risks in the case of constant spread hypothesis $\mathrm{S}^{\mathrm{CPN}}$. The analysis is done in a Hull-White one-factor model. We compare the futures to the underlying swap. The comparison is on the price (for market making) and the curve risk (for hedging). The comparisons are done with a flat curve at $3 \%$ (continuously compounded) rate and a $2 \%$ Hull-White constant volatility.

From Equation (6.2), it is clear that the impact on the value is twofold. First is the multiplication by the common factor $1 / P^D\left(0, t_0\right)$. This factor is always larger than 1 when interest is positive. This represents the fact that the profit is paid immediately (through the margin) and not at final settlement. As the futures usually have an expiry up to three or six months in the future, the impact is up to $0.5$ times the short term rate. With the very low rates prevalent at the time of writing this impact is very small: below $0.10 \%$. For higher rates, the impact can be non-negligible. When interest is at $3 \%$, the impact on a six month future is around $1.5 \%$.

The differences between the futures price (with the 1 subtracted) and the underlying swap present value are displayed in Table 6.4. The difference clearly depends on three items: expiry, tenor and moneyness. The moneyness is the difference between the current forward swap rate and the fixed rate embedded in the futures. If markets have moved significantly between the issuance time and the current level, the swaps can be far off-the-money.

On the curve risk side, the derivative with respect to the rate will be roughly $t_n-t_0$ for the future and $t_n$ for the swap. The ratio swap notional/future notional will be impacted by this. Using this very crude approximation, for a three month futures, the ratio is around $1.125$ for a 2 year tenor and $1.01$ for a 30 year tenor. Actual ratios for different scenarios are given in Table 6.5. The ratios strongly depend on the future expiry and underlying swap tenor but very weakly on the moneyness.

金融代写|金融模型代写Modelling in finance代考|Portfolio hedging

This section is not really an extra instrument but describes an approach to computing hedges with different instruments. The material could have been in another chapter but is a little bit short to be a chapter by itself. For that reason I have appended it to the end of the instruments chapter.

When computing the sensitivities of a portfolio to the market quotes used in the curve construction, one implicitly computes the quantity of the instruments used in the curve construction required to hedge the portfolio. In some cases one would like to compute the (optimal) hedge with a different set of reference securities. One possibility in that case is to compute the synthetic market quotes of the reference securities, reconstruct the curves from those instruments and recompute the sensitivities.

Another possibility is to use the original sensitivities and the sensitivities of the hedging instruments and find a good hedge, optimal in a sense to be described. A method to do that in the case where the original sensitivities are considered independent is described in (Andersen and Piterbarg, 2010, Section 6.4). Similar techniques are used in a different context to minimise delta/normal VaR using the covariance matrix to indicate the importance of and interaction between the sensitivities. In the book by Andersen and Piterbarg (2010), the method is referred to as the Jacobian method. As we suggest systematically computing the Jacobian matrices of curve construction and other processes, we will refrain from using that terminology, which can be confusing, and refer to it instead as the hedging with reference securities method.

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金融代写|金融模型代写在金融建模代考|投资组合对冲

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