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# TIL: Put-Call Parity, Synthetic Futures, Rho, and Cost of Carry

- Authors
- Name
- Teddy Xinyuan Chen

## Table of Contents

## Cost of Carry (Opportunity Cost)

If you long futures, you give up the interest you could have earned on the cash you used to buy the futures.

```
spot price + cost of carry = futures price
```

### Slightly More Advanced: FX Futures CoC via Interest Rate Differential

https://www.youtube.com/watch?v=osT6i81ahr8

## Rho

The Greek that no one seems to care about.

https://www.optionseducation.org/advancedconcepts/rho

Rho is derivative of the contract price with respect to the interest rate.

When interest rates rise, call prices rise (since opportunity cost of receiving interest rises) and put prices fall.

## Put-Call Parity

https://www.cmegroup.com/education/courses/introduction-to-options/put-call-parity.html

The equation seems confusing at first, strike price is involved, but not the spot price.

```
FURICES PRICE - CALL PRICE + PUT PRICE - STRIKE PRICE = ZERO
```

This means `f - s = c - p = cost (net debit) of the synthetic futures = cost of longing the futures at strike s`

.

What does f - s mean?

### Intuitive Explanation 1 - Lock-In Price

There's a more intuitive way to think about this:

```
Futures Price = Strike Price + Cost of Synthetic Futures (AKA the price to "lock in", to commit to buy at the strike price)
```

Through the *price discovery* process of the derivatives involved, we have the parity.

But I feel the equation would be more precise if we also include the cost of carry:

```
Futures Price = Strike Price + Cost of Synthetic Futures + Cost of Carry
```

Or,

```
Futures Price = Strike Price + Cost of Locking into Strike Price + Cost of Carry
```

Or,

Very precise definition:

Simply stated, the difference between the call price and put price for European options with the same exercise price and expiration date must be equal to the present value of the difference between the forward price and exercise price. This relationship, one of the most important in option pricing, goes by various names. In textbooks, it is commonly referred to as put-call parity. Traders may also refer to it as the combo value, the synthetic relationship, or the conversion market.

### Intuitive Explanation 2 - Arbitrage

If we don't have `Futures Price = Strike Price + Cost of Synthetic Futures`

, arbitrage opportunity arises.

If the futures price is higher than the sum of the strike price and the cost of the synthetic futures, we can short the futures and long the synthetic futures then immediately exercise (if it's American style), and vice versa.

### Example: SPX

Maybe I should be using /ES synthetic futures instead of SPX, but it doesn't matter much for this example. The math is not off.

```
f=5773.75 # /ESZ24, forward price
synth=23.65 # cost of long synthetic futures, without the 100 multiplier
strike=5750
strike + synth ~= f
```

### Example: /VX (VIX Futures)

Volitility, a different asset class.

```
f=18.030 # /VXV24
synth=1.04 # cost of long synthetic futures, without the 100 multiplier
strike=17
strike + synth ~= f
```

### f: Futures Price, or Forward Price?

https://www.interactivebrokers.com/campus/trading-lessons/understanding-put-call-parity-2/

Accorining to this IBKR post,

```
Forward value (future value of the underlying assuming price does not move)
= (Current value) x (1 + interest rate * days until expiration/365) – dividends
```

But I think futures price is a better measure of "forward value"