What determines an option holder's daily P/L? Part 3

Level 1 - Zero to Options Hero - A few more odds and ends related to the greeks and explaining an option's P/L

So part 1 and part 2 of this series will be enough to understand the bulk (~95%) of daily P/L changes in your option portfolio, but there are a few loose ends that I wanted to wrap up here in part 3 which are smaller in magnitude but still important to understand when the goal is to understand the drivers of P/L in your options portfolio. Let’s dive in!

The cost of carry, forward rate to maturity, and expected value of the underlying at maturity and their relation to option pricing

I am going to explain the last major greek in the Black Scholes option pricing model that I haven’t gone into yet, which is rho. But to do this, I think the most intuitive thing is to first explain the concepts of cost of carry and forward rates to option maturity.

The forward rate is the fair market rate at which someone who wanted to buy the underlying asset at a future date and someone who wanted to sell the underlying asset at a future date would agree to transact. Because all financial instruments rely on the concept of efficient complete markets and no-arbitrage pricing, this rate should incorporate all the costs and benefits to the forward seller of financing the purchase and then carrying the underlying asset to that future date. For example, the 1-month forward rate on WTI oil should include the cost of borrowing US dollars to purchase the barrels of oil, and the cost of transporting those barrels of oil from Cushing, Oklahoma to the most cost-efficient storage location, the cost of 1 month of storage, and then transporting those barrels of oil back to Cushing, Oklahoma.¹

The forward rate is essential to pricing options under Black Scholes because it is the forward rate of the underlying asset at the option’s maturity (not the spot rate) which is used to calculate the value of the option premium. One way to think about this is that Black Scholes assumes that the most likely rate of the underlying asset at the maturity of the option is the current forward rate. That is why the forward rate is used in the equations in order to gauge the likelihood that the option will expire in the money.²

Rho is a greek that takes this cost of carry that impacts the forward rate of the underlying into account. In particular, rho represents the risk-free rate or interest rate that the seller of the forward contract can borrow at in order to finance the purchase of the underlying asset. Note that for assets denominated in USD, this would be a US interest rate (i.e. the SOFR rate), and for other assets, it would be their respective currency’s interest rate. If the interest rate goes up, the cost to the forward contract seller of fulfilling the contract goes up (i.e. the cost of carry goes up) and hence the forward rate should go up to compensate the forward contract seller for this. Therefore, if the interest rate goes up, call options should be worth more and put options should be worth less, all else equal. So rho is positive for call options and negative for put options.

Dividend yield, borrow yield, convenience yield, and Rho_foreign_currency

Whew, I think that was the hard part! There are a few other greeks that options traders look at which are similar to rho. They are specific to different asset classes, so let’s quickly go through them one by one!

For the forward contract seller in equities, the impact of rho is offset due to the fact that owners of equities often accumulate 1) periodic dividends (which are modeled by traders as an annualized yield and 2) borrow fees that short sellers will pay to borrow stocks in order to short them (again modeled by traders as an annualized yield).

For commodities, there is a similar concept called convenience yield which takes into account the costs and benefits of having the physical commodity in storage which can fluctuate depending on overall supply/demand dynamics for the physical commodity.

For foreign exchange options, there is a concept of rho_foreign which is the interest rate received by the forward contract seller. For example, a seller of an EUR/USD forward contract sells EUR forward and buys USD forward, which means they are paying USD interest and receiving EUR interest during the lifetime of the forward contract. So rho_foreign in this case would be the sensitivity to the EUR interest rate curve.

The main points to take away from this post are:

1. Option prices and P/L will fluctuate with the cost of carry, hence the existence of rho (and other asset class-dependent greeks).

2. Rho is positive for call options and negative for put options. Rho P/L can be calculated daily depending on the day to day changes in interest rate yield curves.

3. Rho of an option is linear to the time to maturity. This makes sense as the longer the time to maturity, the more the cost of carry in a proportional fashion (i.e. the cost of carry in $ terms for 2 months should be about double the cost of carry for 1 month assuming constant interest rates/flat interest rate curves). Options with longer maturities will thus have a larger sensitivity to interest rates (similar to the concept of bond duration).
   
   
   
   

That’s it for this 3 part series on understanding options P/L in the Black Scholes world! Happy to answer any questions and hope you all look forward to the next post in the Zero to Option Hero series.

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1. I used an example of physical barrels of oil to make it easier to picture, WTI (West Texas Intermediate) oil futures contracts which are one of the most actively traded oil contracts (futures contracts are quite similar to forwards contracts with a few additional mathematical wrinkles) are physically settled in Cushing, Oklahoma, USA hence that location in the example.

2. One thing drilled into junior traders and quants on Wall Street either by their seniors or in school is that the forward rate is used to calculate the expected value and probability of the option being in the money not in the real world, but in something called the risk neutral measure. This is actually something to do with fancy maths such as stochastic calculus so not really necessary to understand to profitably trade options, but nice to know as bar trivia perhaps!