Black–Scholes Theta, One-Day Repricing, and Volatility Surface Rolls

# Question about option theta


# Question about option theta












I have a question about a option theta.

When I evaluate the option theta of near expiry put option using Black-Scholes formula given the data as follow:

- Index Level = 20,500

- Strike Price = 20,000

- Interest Rate = 0.14%

- Dividend Yield = 3.41%

- Volatility = 23.64%

- Time to maturity = 0.01 (2 days)

Using data as above, the put option price is \$12.97, but the theta per one day is -\$13.39. That means after one day, the put option price becomes negative given the above parameters (except time to maturity) do not change. I wonder the reason of put option price is smaller than the absolute value of theta.

I guess the reason is that the theta tell us the linear change of put option price against the time, but the relationship of option price and time to maturity is not linear, then the above case may happen. Is my conjecture correct? And is there any financial interpretataion on the above phonomena?

## Answer by Christian Fries (score 3)

https://quant.stackexchange.com/a/8865

Disclaimer: I did not check your example, i.e., that "theta * 1 day" will predict a negative option price.

Theta is the derivative with respect to time-to-maturity. It is the change of the option price with respect to an infinitessimal change in time and not with respect to a change of one day - even if the derivative is "scaled" towards a time scale having 1 day as its unit.

In other words $\int_{t}^{t+d} \frac{\partial V}{\partial \tau} d \tau = V(t+d) - V(t)$ and in general $\int_{t}^{t+d} \frac{\partial V}{\partial \tau} d \tau \neq \frac{\partial V}{\partial \tau}(t) \cdot d$, however, the latter is an approximation. Here $V(t)$ is the option value, evaluated at time $t$.

## Answer by AKdemy (score 2)

https://quant.stackexchange.com/a/82181

It can happen for model theta to be larger than the option value itself.

Closed form theta in BS (see for example the Wikipedia Greeks Finance) is the change per unit time (the change after one year). In other words, mathematically the result of the formula for theta is expressed in value per year. Many professional pricing engines actually display it as 1 day theta (computed as BS_Theta / 365).

More often though, finite difference (FD) theta is actually computed as a true 1 day bump and reprice theta (shifting the evaluation date one day forward and repricing). A complete replication of Bloomberg's OVML and Quantlib can be found in this answer. Using FD theta has at least two advantages:

- BS theta can exceed actual market value of an option if the time to expiry is short (see below for an example)

- Holidays and weekends can easily be included in the computation (Friday will be a 3 day theta, provided Monday is a working day)

For example, the value from the calculator is a 1 year theta, which is easy to show with the following Julia code:

```
using Distributions, DataFrames

N(x) = cdf(Normal(0,1),x)
n(x) = pdf(Normal(0,1),x)
"""
https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks
"""
function BSM(S,K,t,r,d,σ, cp)
  d1 = ( log(S/K) + (r - d + 1/2*σ^2)*t ) / (σ*sqrt(t))
  d2 = d1 - σ*sqrt(t)
  opt  = cp*exp(-d*t)S*N(cp*d1) - cp*exp(-r*t)*K*N(cp*d2)
  theta_c = (-(S * exp(-d*t)*n(d1)* σ )/ (2 * sqrt(t)) - r * K * exp(-r*t) * N(d2) + d * S * exp(-d*t)*N(d1))
  theta_p = (-(S * exp(-d*t)*n(d1)* σ )/ (2 * sqrt(t)) + r * K * exp(-r*t) * N(-d2) - d * S * exp(-d*t)*N(-d1))
  return opt, theta_c, theta_p
end

S, K, r, t, σ = 50, 50, 0.12, 0.25, 0.3
res = BSM(S, K, t, r, 0, σ, 1)
DataFrame(Call = res[1], Theta = res[2] )
```

This theta value is in line with the proposed calculator (this is also assuming rates are continuous).

However, it is rather useless from a practical perspective. What one would usually do is to look at what happens to the option price with one less day to expiry. You get this value by dividing BS theta by 365. FD theta is the result of repricing the model with one less day to expiry, keeping all else equal and simply looking at the price difference between the two option values.

```
res2 = BSM(S, K, t - 1/365, r, 0, σ, 1)
day_theta = res[2]/365
fd_theta = res2[1]- res[1]
DataFrame(Symbol("Call") => res[1], Symbol("Call -1 day") => res2[1], Symbol("Theta") => res[2], Symbol("1 Day Theta") => day_theta, Symbol("FD Theta") => fd_theta)
```

We can use the below example, which prices a OTM option with 5 days to expiry and 1 million notional, to show why FD theta is often preferred. BS theta would actually result in a negative option value in this case.

```
r1 = BSM(45, 50, 5/365, r, 0, σ, 1) .* 1000000
r2 = BSM(45, 50, 4/365, r, 0, σ, 1) .* 1000000
DataFrame(Symbol("Call") => r1[1], Symbol("Call -1 day") => r2[1], Symbol("1 Day Theta") => r1[2]/365, Symbol("FD Theta") => r2[1] - r1[1])
```

## Answer by Abhay Nainan (score 1)

https://quant.stackexchange.com/a/82330

When computing Finite-Difference theta where you shift the evaluation date one day forward, you are essentially answering the question of how do the pricing parameters of a model roll on the next day.

For example, is your new O/N vol the same as yesterday's O/N vol, or does it roll to what yesterday's forward O/N vol implied? Ofcourse event-weights from yesterday's O/N vol must be removed, but will you just re-use the ex-event term-structure of yesterday, or will you roll to the 1d-forward implied term structure?

You also have to think about the roll of your RiskReversal and Fly parameters, since they too have a term-structure of their own. Assuming the RR/Fly parameters are upward sloping, you earn more theta on a short strike if the new day's surface term-structure is the same as yesterday's term-structure(since your RR/Fly for the same date has now rolled down a notch).

In FX vol, some currency pairs have very steep forward curves. An ATMF option struck at yesterday's forward rate, may no longer have the same delta if the forward curve is unchanged, because the new ATMF forward for that date will be a touch lower. If your system has VolXDelta mapping, this will lead your strike to be marked at a different vol because the delta for your expiry date has changed. This may be immaterial for EURUSD, but an important consideration for USDTRY.

These considerations may not have a material impact on your 1d-theta, but if you sell a 2m 25d strike on the premium side of the distribution, and the ATM,RR,Fly,Swap term structures are all upward sloping, then assuming a) spot unchanged, and b) input term structures unchanged, your 5d-theta for that option will be meaningfully higher than what just bumping your time-parameter 5 days would imply, since you are rolling to a lower vol on the ATM,RR,Fly term sturctures, marginally offset by rolling to a more OTM delta (further into the premium side of the distribution) due to the swap term-structure.