Replicating a Put Option with a Delta-Based Stock Position
Summary
The document explains how a synthetic put can be approximated through delta-based adjustments to a stock portfolio. A put’s delta measures how its value changes for a small change in the underlying stock price; because put delta is nonpositive, the hedge involves reducing stock exposure. The response gives a numerical illustration in which a portfolio holding stock sells a fraction of its shares corresponding to the put’s negative delta. It compares the portfolio’s change in value after a stock-price decline with the change in a portfolio that includes the put.
The example shows the local, first-order relationship between put value and the underlying price, and why a short stock position can offset some downside exposure. The document also notes that a more complete replication invests the sale proceeds in a risk-free asset. This is not a static, exact replication: option delta changes as market conditions and time change, so maintaining the hedge generally requires rebalancing. The illustration omits transaction costs, financing details, and other portfolio constraints.
Key ideas
- Put delta is the sensitivity of put value to a change in the underlying stock price.
- A put has nonpositive delta, so a delta hedge reduces stock exposure by selling shares.
- The short stock position can approximate the put’s price response for a small underlying move.
- A fuller replication places the proceeds from stock sales in a risk-free asset.
- Because delta changes over time and with market conditions, the hedge generally needs rebalancing.
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Full text
# How to create a synthetic put?
# How to create a synthetic put?
I have been reading into Hull's section on portfolio insurance through synthetic puts.
My understanding is that in order to replicate a put we should replicate it's delta. Proceeding, Hull states that
> To create a synthetic put option synthetically, the fund manager should ensure that at any given time a proportion $$e^{qT}[1-N(d_{1})]$$ (delta) of the stocks in the original portfolio has been sold and the proceeds invested in the riskless assets.
I am unsure as to how this replicates the delta and furthermore, the put option.
If someone could explain further than what has been written in Hull with an example, it would be greatly appreciated.
## Answer by SRKX (score 5, accepted)
https://quant.stackexchange.com/a/27623
As you can see from the wiki page, the delta of a put is
$$\Delta = -e^{-qT}N(-d_1)= -e^{-qT} \left(1-N(d_1)\right)$$
Recall that this $\Delta$ is the derivative of the value of the put $p$ with respect to the value of the underlying stock $S$: $\frac{\partial p}{\partial S}$.
So this means that if the underlying goes up by 1, the price of the put change by $\Delta$. Clearly, you see that for puts $\Delta \leq 0$, which means that if the value of the underlying goes up, the put value goes down.
Let's say you have asset $S_0=100$ and you want to replicate a put which would have a $\Delta=-0.25$, then you sell you $0.25$ the stock.
Say the value next day is $S_1 = 80$:
Assume you have a portfolio of $S$ and a put $p$:
- Value at $t=0$: $S_0 + p_0 = 100 + p_0$
- Value at $t=1$: $S_1 + p_1 = S_1 + (S_1 - S_0 )\Delta + p_0 = 80 + (-20) \cdot (-0.25) + p_0 = 85+p_0$
- The PnL is $(85+p_0) - (100 + p_0) = -15$.
Assume now you don't buy the put but you replicate by investing $\Delta$ of the stock:
- Value at $t=0$: $S_0 + \Delta S_0 = (1+\Delta)S_0 = 0.75 \cdot 100 = 75$
- Value at $t=1$: $(1+\Delta)S_1 = 0.75 \cdot 80 = 60$
- The PnL is $-15$ as well
You replicated the option (that's the idea to be perfectly correct you indeed need to invest the proceeds at risk free indeed, that's essentially because $p_0$ doesn't move exactly by $\Delta$).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.