How to create a synthetic put?

Question

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 1

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 $\Delta$).