# Buy and Hold P&L for options

I want to calculate the buy and hold P&L for an option in the following "extreme" scenario:

• Valuation date = t
• Calculate B&H P&L from t to t+1
• Option maturity: t+2
• ATM option

Due to the option being ATM and close to maturity, we have significant Theta effects. However, I want to exclude theta effects from B&H P&L. The B&H P&L should be calculated as of valuation date t but with market data input from t+1.

Now, the problem I see is that while the underlying spot level in t+1 may be easy (together with other market data input into the BS formula), I wonder how to treat implied vol. My first idea is to back out the impl. vol from the market price in t+1 and plug this into the BS formula with valuation date t. However, the backed out implied vol probably contains significant theta in t+1, but I want to exclude theta.

Do you have any idea how to calculate the B&H P&L excluding Theta in this case in a full revaluation framework?

Say the value of your option on day $t$ is $V(S_t, \sigma_t, \tau)$ where $S_t$ is the spot price, $\sigma_t$ is the implied volatility and $\tau$ is the number of days to expiry (it also depends on the strike price, interest rate etc but I've ignored these for simplicity).

Your P&L from $t$ to $t+1$ is

$$V(S_{t+1}, \sigma_{t+1}, \tau-1) - V(S_t,\sigma_t,\tau)$$

By adding and subtracting terms, you can try to isolate particular components of P&L,

$$\underbrace{V(S_{t+1}, \sigma_{t+1}, \tau-1) - V(S_{t+1}, \sigma_{t+1}, \tau)}_{\text{Theta P&L}} + \underbrace{V(S_{t+1},\sigma_{t+1}, \tau) - V(S_t,\sigma_t,\tau)}_{\text{P&L excluding theta}}$$

The second term is what you wanted to isolate (P&L excluding theta effects) and the first term contains the residual (in this case, it is mostly theta).

However, $\sigma_{t+1}$ and $\sigma_t$ are implied volatilities for options with a different number of days to maturity, and different moneyness - therefore they can include some theta/delta/gamma effects as well (actually, these are. You can handle this by adding/subtracting more terms,

$$\underbrace{V(S_{t+1}, \sigma_{t+1}, \tau-1) - V(S_{t+1}, \sigma_{t+1}, \tau)}_{\text{Theta P&L}} + \underbrace{V(S_{t+1},\sigma_{t+1}, \tau) - V(S_{t+1},\sigma^*_{t+1},\tau)}_{\text{Rolldown P&L}} + \underbrace{V(S_{t+1},\sigma^*_{t+1}, \tau) - V(S_t,\sigma_t,\tau)}_{\text{P&L excluding theta and rolldown}}$$

where $\sigma^*_{t+1}$ is an implied volatility chosen to minimise the effect of rolling down the volatility surface. For example, you might pick $\sigma^*$ to be the implied volatility of an option with the same moneyness as the option you are valuing, but with two days to maturity rather than one day.

Bear in mind that this is always going to be something of an approximation - the idea that option P&L decomposes neatly into components like delta, gamma, theta, vega etc is a fiction. There is always a residual, and the residual can be large, especially when the inputs are changing rapidly (e.g. when you are close to expiry).

• Thanks for your answer. Let me address some comments/questions: - In your definition of Theta P&L, shouldn't it be minus instead of plus? - How to pick sigma* in your example make sense to me. Is the following recipe correct for this impl. vol: Take the same moneyness of the option in t+1 as it was in t and back out the corresponding impl. vol with T2M of 2 days (which in my example would be t+3). However, as the option expires in t+2, I would need some interpolation between two quoted maturities right? (t+2 and the next quoted maturity in the market) – alpha Sep 9 '18 at 14:26
• I would use the implied vol for an option with the same moneyness that your original option has on t+1 (presumably you want to incorporate the effect of moving along the volatility surface in strike space, while factoring out the effect of moving along the vol surface in T2M space). Yes, if there is no option quoted to expire two days after t+1 then you will need to interpolate. – Chris Taylor Sep 9 '18 at 18:10
• The moneyness in t+1 for my original option with strike K depends on the new underlying spot level in t+1. Based on the new underlying level, I will get a new moneyness in t+1 and take the impl. vol for this moneyness? Or fix the moneyness from the previous day and take the impl. vol w.r.t this previous moneyness – alpha Sep 10 '18 at 7:57