A pension fund is sitting on a huge, diversified equity portfolio and is nervous about a crash. Selling the stock outright fixes the problem but is drastic — it gives up all the upside, incurs transaction costs and taxes on a scale that matters, and can be politically awkward for a fund whose mandate is "stay invested in equities." What the fund manager actually wants is insurance: keep the portfolio, but put a floor under how much it can lose. Having just seen how a desk manages every Greek of an options book, we now have exactly the machinery needed to build that floor — and, as we'll see, to accidentally make the crash worse when everyone builds the same floor at once.

Two ways to buy the floor

The floor is a protective put: hold the stock portfolio, and also hold a put option on it with some strike K below today's value. However far the market falls, the put's payoff makes up the difference, so the combined position is worth at least K at expiry. There are two ways to get there.

• Buy the real put. Pay an explicit premium today for an index put (or a basket of puts) with the strike and expiry you want. Simple, and the cost is known upfront — but exchange-listed puts in the size and tenor a huge fund needs may not exist, or may be expensive and illiquid.
• Manufacture the put yourself — dynamically. Don't buy a put contract at all. Instead, continuously shift the portfolio's split between equities and cash so that the combined position's value tracks what a real protective put would be worth, at every possible level the market could reach. This is exactly the delta-hedging machinery from this module's first two lessons, aimed at replicating an option that was never actually bought — hence synthetic, or dynamic, portfolio insurance.

The replication weight

A put's delta is \Delta_{\text{put}} = \Phi(d_1) - 1, always negative. A protective put — long stock plus long that put — therefore has combined delta 1 + \Delta_{\text{put}} = \Phi(d_1). That's the fraction of wealth the synthetic strategy should hold in equities at any given index level, with the rest parked in T-bills:

As S falls, d_1 falls, and \Phi(d_1) falls with it — the strategy mechanically sells equities and buys T-bills as the market drops, moving money into safety exactly as things get scarier. As the market recovers, it mechanically buys back in. Done continuously and frictionlessly, this reproduces the protective put's payoff without ever paying an explicit premium — the "cost" of the insurance shows up instead as the same buy-high, sell-low slippage from rebalancing that gamma and the limits of delta hedging described for any delta hedge.

Worked example — de-risking a $100M fund as the market drops

A fund runs synthetic portfolio insurance on a \$100M equity portfolio, targeting a floor near 5\% below today's index level. As the index falls, the equity weight w = \Phi(d_1) and the dollar allocation are recomputed and traded to:

Notice the size of the trades: as the market falls further, the fund doesn't just keep selling — it sells more, in bigger dollar chunks, precisely because \Phi(d_1) is steepest near the money (the same gamma effect from two lessons ago). A strategy designed to protect one fund, followed faithfully, demands heavier and heavier selling exactly when the market is already falling hardest.

Seeing the de-risking curve

Below, w(S) = \Phi(d_1(S)) plots the fraction of the portfolio kept in equities against the index level, for a floor strike near K = 95. Reading right to left is reading the market falling: the curve slides down from "mostly equities" toward "mostly cash," steepest in the middle — the same S-curve shape as a call's delta, now relabeled as an allocation rule. Shorten the time remaining and the curve steepens further, meaning larger trades triggered by smaller moves as expiry (or the review date) approaches.