Actuarial Notes

Hedging

Study guides: Exam ALTAM, Exam INV-201

Hedging is taking a position in one or more instruments whose value moves opposite to an existing exposure, so that the combined portfolio is insensitive to a chosen risk factor. A hedge is static if it is set once and held (for example, buying the matching put), or dynamic if it is rebalanced as sensitivities change (delta hedging); it is perfect only when the hedge instrument and the exposure move together exactly, otherwise basis risk remains.

\frac{\partial}{\partial S}\Big(V + \sum_i n_i H_i\Big) = 0

\frac{\partial^2}{\partial S^2}\Big(V + \sum_i n_i H_i\Big) = 0

h^* = \rho\,\frac{\sigma_S}{\sigma_F}

  • The first two blocks are delta-neutral and gamma-neutral conditions: V is the exposure, H_i the hedge instruments and n_i the quantities held. Each condition needs its own instrument, and the underlying itself has zero gamma, so gamma is neutralized with options first and the delta is then fixed with the underlying; vega and rho neutrality are added the same way
  • h^* is the minimum-variance hedge ratio for hedging an exposure with futures: \sigma_S and \sigma_F are the standard deviations of the exposure's and the futures' returns and \rho their correlation; the hedge removes the fraction \rho^2 of the variance (the hedge effectiveness)
  • An insurer hedging an embedded option in a liability (a variable annuity GMxB, an indexed annuity crediting) can hedge equity, interest-rate and volatility risk with traded instruments; mortality, policyholder behaviour, basis, counterparty and correlation risk can be hedged only partly or not at all
  • Hedges fail when their assumptions fail: discrete rebalancing, jumps and gaps, realized volatility above the hedge's, liquidity, model and counterparty risk

Making a Guarantee Book Delta- and Gamma-Neutral An insurer has written maturity guarantees (puts) on variable annuity funds. Measured per unit of the equity index, its book has delta +450 and gamma -30. It can trade the index and a listed put on the index with delta -0.45 and gamma 0.04. Find the hedge that makes the book delta- and gamma-neutral.

Answer Gamma first, since only the option carries gamma:

\begin{align*}
-30 + 0.04\,n &= 0 \\
n &= 750 \text{ puts}
\end{align*}

The puts add delta 750(-0.45) = -337.5, leaving

\begin{align*}
\Delta &= 450 - 337.5 \\
       &= 112.5
\end{align*}

so the insurer buys 750 puts and shorts 112.5 units of the index. Fixing delta first and then adding the options would undo the delta hedge.

Minimum-Variance Hedge with Basis Risk A \$50 million variable annuity fund tracks a managed fund with return volatility 18\%. The insurer hedges it with index futures whose returns have volatility 20\% and correlation 0.92 with the fund. Each contract is \$250 times an index level of 5{,}000. Find the number of contracts to short and the volatility left after hedging.

Answer

\begin{align*}
h^* &= 0.92 \times \frac{0.18}{0.20} \\
    &= 0.828 \\
N^* &= \frac{0.828 \times 50{,}000{,}000}{250 \times 5{,}000} \\
    &= 33.12
\end{align*}

Short about 33 contracts. The hedge removes \rho^2 = 84.64\% of the variance; the residual volatility is

\begin{align*}
\sigma_{\text{resid}} &= 0.18\sqrt{1 - 0.92^2} \\
                     &= 0.0705
\end{align*}

or about \$3.53 million on the fund. The 7.05\% that remains is basis risk, the part of the fund's return the index does not explain.

How it appears on the exam

  • INV-201 ("the Greeks of derivatives"; "static and dynamic hedging"): given the Greeks of a liability and of available instruments, find the positions that neutralize two or three of delta, gamma and vega, then evaluate what the hedge leaves exposed.
  • INV-201 (hedging "embedded options in liabilities"): describe a hedge program for a GMxB, an indexed annuity or a RILA, separating hedgeable equity, rate and volatility risk from the partly hedgeable mortality, lapse, basis and counterparty risks, as a written answer.
  • INV-201 ("how hedge strategies may fail"): diagnose a hedge program's losses from given hedge results (gaps, realized volatility, basis, model error).

Exam traps

  • Solving delta before gamma. The underlying has no gamma but the options have delta; neutralize gamma with options, then rebalance delta with the underlying, or the delta hedge is broken by the option trade.
  • Sign of the liability's Greeks. An insurer writing a guarantee is short the option. A GMMB is a put, with negative delta, so the insurer's book has positive delta and negative gamma; hedging the put's own Greeks instead of the book's trades in the wrong direction and doubles the exposure.
  • Treating the hedge ratio as the correlation or the volatility ratio alone. h^* = \rho\,\sigma_S/\sigma_F; using \rho or \sigma_S/\sigma_F by itself over- or under-hedges.
  • Claiming mortality or lapse risk is hedged by the equity hedge. The hedge quantities assume expected survivorship and lapses; any experience different from that becomes unhedged P&L.