Let's take an intuitive look at the key factors that drive option premiums and how they affect the price of calls and puts.

You'll notice that some factors impact both call and put prices in the same way while others have opposite effects.

Understanding why comes down to how these factors influence either the intrinsic value, the time value, or both.

Let's start with the spot price of the underlying assets.

When the spot price increases, call option prices go up while put option prices go down.

This makes sense. If we think about intrinsic value, a higher spot price makes call options more likely to be in the money or moves them further into the money, increasing their intrinsic value or the potential for it.

On the other hand, for puts a rising spot, price could move them out of the money, or at least closer to being outta the money, reducing their intrinsic value or the chance of gaining intrinsic value before expiring.

So the spot price directly affects the intrinsic value, which is why its impact on calls and puts is in opposite directions.

Next, we have time to expiry.

Increasing the time to expiry raises the price of both calls and puts.

This might seem less intuitive at first, but it becomes clear when we think about time value.

The longer the time to expiry, the more time there is for the underlying price to move, increasing the uncertainty or increasing the potential for the option to gain intrinsic value.

This uncertainty is valuable for both call and put buyers because the underlying price could move up or down, which is why the premium for both calls and puts is higher with a longer time to maturity.

Moving on to volatility, this also increases the price of both calls and puts.

This is because volatility amplifies the potential for large price movements in either direction.

Higher volatility means a greater chance that the option will end up significantly in the money, whether it's a call or a put.

It also increases the chance of being further outta the money, but that has no impact.

Since there's not any further loss for being further out of the money, the option just won't be exercised.

This increases the time value of both types of options as greater price swings create more uncertainty and time value is essentially the price of uncertainty.

Now, let's consider interest rates.

When interest rates rise, call option prices tend to increase while put option prices decrease.

To understand this, think about how interest rates affect the cost of carrying or financing the underlying assets.

Higher interest rates mean a greater benefit to deferring payment for the assets because the money you would've spent on the underlying assets can instead be invested at a higher interest rate until option expiry.

This makes call options more valuable as they allow you to delay payment while keeping the upside potential for put options.

The logic flips when interest rates are high.

It's more attractive to sell the underlying asset immediately and invest the proceeds at the higher interest rates rather than holding onto the asset and waiting for the option to potentially move into the money.

This reduces the value of put options because the opportunity cost of holding the asset increases.

In other words, higher interest rates make immediate liquidation more appealing, which lowers the value of having the right to sell later.

Finally, we have the underlying assets yield such as dividend yields for stock.

An increase in yield tends to decrease, call option prices and increase put option prices.

This is because higher yields increase the attractiveness of holding the assets directly.

You receive more income from dividends or yields.

This reduces the relative value of call options because owning the option doesn't give you the right to collect those yields since you don't own the underlying assets yet.

For put options though it's the opposite.

Higher yields make it less attractive to sell the asset immediately because you'd be giving up the future income from those yields.

A put option, however, allows you to hold onto the assets to continue receiving the yield while still having the right to sell it later if necessary.

This flexibility increases the value of put options when asset yields are high.

Now, if we take a step back, you'll notice a pattern.

The factors that affect both calls and puts in the same way, like time to expiry and volatility, they Primarily influence the time value of the option.

This makes sense because both calls and puts gain from increased uncertainty and more time for potential price movements.

In contrast, the factors that have opposite effects on calls and puts like spot price, interest rates, and asset yield, they primarily influence the intrinsic value or the present value of potential payoffs.

Since calls and puts are mirror images in terms of their payoffs, changes in these factors naturally affect them in opposite directions.