Portfolio insurance

As we have already conjectured in the beginning, options can be used in order to provide an insurance towards adverse movements of the market. If one manages a well diversified portfolio that mimics the index [with beta equal to one and dividend yield equal to the market dividend yield], then the procedure is straightforward. However, sometimes the traded options required are either not available or not sufficiently liquid. In such cases one can create options synthetically. There are two ways in achieving this.

Using the put-call parity relationship. This stems from the fact that

$$S_{T}-K=\max \left[ S_{T}-K,0\right] -\max \left[ K-S_{T},0\right]$$   ,

suggesting that under risk neutrality

$$\mathbf{E}_{Q}\left[ S_{T}-K\right] =\mathbf{E}_{Q}\max \left[ S_{T}-K,0\right] -\mathbf{E}_{Q}\max \left[ K-S_{T},0\right]$$   .

In addition, under risk neutrality the stock price dynamics [with $\delta$ the dividend yield] are given by the following SDE

$$\frac{dS\left( t\right) }{S\left( t\right) }=\left[ r\left( t\right) -\delta \left( t\right) \right] dt+\sigma \left( t\right) dB\left( t\right)$$   ,

giving that $\mathbf{E}_{Q}S_{T}=Se^{\left( r-\delta \right) T}$. We also have that

$$= e^{-rT}\mathbf{E}_{Q}\max \left[ S_{T}-K,0\right]$$ Call

$$= e^{-rT}\mathbf{E}_{Q}\max \left[ K-S_{T},0\right]$$ Put

The above expressions imply the put-call parity relationship

$$Se^{-\delta T}+$$Put$$=$$Call$$+Ke^{-rT}$$.

This relationship allows one to replicate a call [or put] by using the corresponding put [or call] and the money market account.

Another approach would be to construct an option completely synthetically, using the delta hedging methodology and the money market alone. In this context one wants to create a position which will match the delta of any given option. A put option on the portfolio, for example, has delta equal to

$$\Delta _{P} = e^{-\delta T}\left[ \mathcal{N}\left( d_{1}\right) -1\right]$$

$$d_{1} = \frac{\ln \left( \Pi /K_{\Pi }\right) +\left( r-\delta +\frac{%% \sigma _{\Pi }^{2}}{2}\right) T}{\sigma _{\Pi }\sqrt{T}}\text{,}$$

where now $K_{\Pi }$ is the strike price [for the insurance] and $\sigma _{\Pi }$ is the volatility of the portfolio [which is beta time the market volatility, although in this case $\beta _{\Pi }=1$]. Remember that the BS methodology suggests that a portfolio composed by

$$\left. \begin{tabular}{l} $-1$ derivative [with price $P$] \\ $+\Delta _{P}$ shares of the underlying \end{tabular}\right\} \Rightarrow$$   gives the risk free rate of return$$.$$

Apparently, in order to replicate a derivative, one has to take the inverse position to the one that hedges it. Therefore, by employing a dynamic strategy where a fraction $-\Delta _{P}$ of the original portfolio is always invested in riskless assets, the fund manager ensures that the overall position will match the given delta.

The above are illustrated in the following

Example 45 (Portfolio insurance I)   Say that a fund manager has a well-diversified portfolio, which mirrors the FTSE250 index [ $\beta _{\Pi }=1$]. The value of the portfolio is $%% \Pi =\pounds 360m$, whereas the value of the index is $S=1200$ pts. The manager wants to insure that the portfolio she is holding will not decline by more than $5\%$ over a six month period. Say that the risk free rate is $%% r=6\%$ p.a., and the dividend yield for both FTSE250 and the portfolio is $\delta _{S}=\delta _{\Pi }=3\%$ p.a. The volatility of the index is $\sigma _{S}=30\%$ p.a. Since $\beta _{\Pi }=1$ and the dividend yields are the same, if the portfolio looses $5\%$ of its value, it is implied that the FTSE250 index has lost $5\%$ of its value too. Therefore the fund manager wants to insure against the index falling below $%% S^{\star }=1140$ pts. The strategy that he should employ would be buying six month put options with strike price $K=S^{\star }=1140$. Since today the value of the portfolio is $\frac{360m}{1200}=300k$ times the index, she should buy $300k$ such options.

Now we turn into examining what the price of such an insurance would be.

Example 46 (cont. The value of the insurance I)   Suppose that the BS formula is correct. The value of a put option with the above characteristics would be given by

$$P = Ke^{-rT}\mathcal{N}\left( -d_{2}\right) -Se^{-\delta T}\mathcal{N}%% \left( -d_{1}\right)$$

$$d_{1} = \frac{\ln \left( S/K\right) +\left( r-\delta _{S}+\sigma _{S}^{2}/2\right) T}{\sigma _{S}\sqrt{T}}\text{,}$$

$$d_{2} = d_{1}-\sigma _{S}\sqrt{T}$$

implying that $P=\pounds 63.40$. Since the fund manager wants to buy $300k$ of these contracts, the value of the insurance is

$$300k\times \pounds 63.40=\pounds 19,020k\approx \pounds 19m$$.

Put options can be replicated using calls and the put call parity. In this setting

Example 47 (cont. Using calls I)   The fund manager can also create a synthetic put, using call option and the put call parity

$$Se^{-\delta T}+ = +Ke^{-rT}$$

$$= +Ke^{-rT}-Se^{-\delta T}$$ Put

This strategy would include

$$\begin{tabular}{l} Shortsell $Se^{-\delta T}=\pounds 354.6m$ wor... ... \\ Invest the remaining cash [or borrow] at the risk free rate. \end{tabular}$$

This strategy would have the same value as the one involved put options.

A delta hedging strategy implies that in fact the put option can be approximated by borrowing or lending part of the portfolio at the risk free rate of return. This would be very handy if for example options with strike price $K=1140$ did not exist.

Example 48 (cont. Using $\Delta$-hedging I)   The delta of a put option on a dividend paying asset such as the index is

$$\Delta _{P}=e^{-\delta T}\left[ \mathcal{N}\left( d_{1}\right) -1\right]$$   ,

which in this case is $\Delta _{P}=-33.27\%$. This implies that the put option is approximated by selling one third of the portfolio and invest this amount [ $\approx \pounds 120m$] in risk free securities.

When the portfolio's beta and/or the dividend yield do not match the market, the procedure is slightly more complicated. One has to utilize the CAPM in order to match the insurance required for the portfolio with the equivalent market moves. In addition, it can be shown that the optimal hedging ratio that we first encountered when we discussed the futures contracts still holds: if the value of the portfolio is $\nu$ times the index, then $\beta _{\Pi }\times \nu$ put contracts have to be bought. One can restate the above in the following way: one has to by $\beta _{\Pi }$ times the contracts she would buy if beta was equal to one. Now suppose that the initial setting of the above example is slightly changed to illustrate these differences.

Example 49 (Portfolio insurance II)   Now suppose that the beta of the portfolio is $\beta _{\Pi }=1.5$, and that the portfolio's dividend yield is $\delta _{\Pi }=4\%$ p.a. Since $\delta _{\Pi }=4\%$ p.a. or $2\%$ over six months, if the value of the portfolio declines by $5\%$ over the next six months, the total return will be

$$-5\%+2\%=-3\%$$, or $$6\%$$ p.a.

CAPM will now give us the corresponding market total return

$$r_{\Pi }-r = \beta _{\Pi }\left( r_{S}-r\right)$$

$$r_{S} = -2\% -1\%$$

This is the total return. Since the FTSE250 dividend yield is $\delta _{S}=3\%$ p.a., or $1.5\%$ over six months, we expect the index to have fallen by $2.5\%$ over the six month period, down to $S^{\star \star }=1170$ pts. This means that in order to insure against a $5\%$ drop of the portfolio value, the fund manager needs to use six month put options with exercise price $K=S^{\star \star }=1170$. Today the value of the portfolio is $%% \frac{360m}{1200}=300k$ times the index, therefore the optimal hedge ratio dictates that she should buy $\beta _{\Pi }\times 300k=450k$ such options.

We will now see that the cost of the insurance has increased significantly

Example 50 (cont. The value of the insurance II)   The value of a put option with the new characteristics would now be $%% P=\pounds 76.28$. Since the fund manager wants to buy now $450k$ of these contracts, the value of the insurance is

$$450k\times \pounds 76.28=\pounds 34,326k\approx \pounds 34.3m$$.

The replicating procedure remains the same

Example 51 (cont. Using calls II)   The strategy creating a synthetic put, using a call option and the put call parity is the following

$$\begin{tabular}{l} Shortsell $Se^{-\delta T}=\pounds 354.6m$ wor... ... \\ Invest the remaining cash [or borrow] at the risk free rate. \end{tabular}$$

This strategy would have the same value as the one involved put options.

If the fund manager wants to consider a delta hedging strategy, she must calculate the delta of the portfolio. This is not as straightforward as in the previous case.^8.2

Example 52 (cont. Using $\Delta$-hedging II)   The insurance can be seen as a put option on the portfolio instead of the index, with strike price $K=342m$. In addition, the CAPM implies that the volatility of the well diversified portfolio is $\sigma _{\Pi }=\beta _{\Pi }\sigma _{S}=45\%$ p.a. In this context the delta of the portfolio would be

$$\Delta _{\Pi }=e^{-\delta T}\left[ \mathcal{N}\left( d_{1}\right) -1\right]$$   ,

where now

$$d_{1}=\frac{\ln \left( 360/342\right) +\left[ 6\%-4\%+\left( 45\%\right) ^{2}/2\right] \times 0.5}{45\%\times 0.5}\text{,}$$

giving $\Delta _{\Pi }=-35.17\%$. The put option can be approximated by selling $35\%$ of the portfolio and by investing this amount [ $\pounds 128m$] in risk free assets.