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Is there a Greek that describes the sensitivity of an option's time value to the strike price? Or is option time value independent of strike?

It's obvious that strike doesn't change once an option is bought/sold, but it is useful to know the relationship between time pemium of options of different strikes when you make purchase/sale decision so you can pick the right one.

2 Answers 2

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The Greeks are used evaluate an option's sensitivity to change in price, time, volatility, interest rates (delta, theta, vega,rho).

Gamma measures the sensitivity of a delta to change in price.

The strike price is a fixed item in the contract. It does not vary so there is no Greek for it.

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    Baeker: It's obvious that strike doesn't change once an option is bought/sold, but it is useful to know the relationship between time pemium of option of different strikes when you make purchase/sale decision.
    – techie11
    Commented Oct 4, 2018 at 18:50
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    teckie: If it's obvious that the strike price doesn't change once an option is bought/sold then it should be obvious that there will be no Greek as such. You could generate a hypothetical price distribution curve across strikes using the same IV for all options for an underlying but that would not be accurate since since IV is not constant across all strikes (volatility smile?). If you want to see the relationship of premium across strikes, use an option pricing model. Or if you really want simple, drop the option chain into Excel and calculate intrinsic and extrinsic. Commented Oct 4, 2018 at 19:11
  • Bob Baeker: the simple method (use option chain) sounds good to me. But another question arises from your mention of volatility smille: Since in Black-Scholes model, the \sigma (standard deviation) is the underlying's rate of return which is same for options across strikes. does it mean standard deviation is the historical volatilty? all textbook says IV instead of historical volatility is the 6 input parameters of the BS equation.
    – techie11
    Commented Oct 4, 2018 at 21:27
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    Here's an explanation of the difference between historic and implied volatility: investopedia.com/articles/investing-strategy/071616/… Commented Oct 4, 2018 at 22:40
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    @techie11 To add on to Bob's answer - Implied Volatility is the result of inserting the actual live price of a particular option in the market into the Black Scholes model. If you have all other variables, you can calculate the IV yourself, it is merely shown on an options chain as a matter of comparative convenience. Historical Volatility, however, is a calculation based on past results of the underlying. Commented Nov 22, 2023 at 15:43
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I disagree with the given answer.

Greeks are just sensitivities to the input parameters in Black Scholes. Since the strike is an input you can compute the derivative of the option value with respect to the strike, denoted by ∂C/∂K.

In fact, this is frequently done and called Dual Delta, which is also the probability of an option finishing in the money. It is also a key building block for Local Vol and used in the Breeden-Litzenberger (1978) formula which is used for the extraction of a risk-neutral distribution from observed option prices:

  • The first derivative of the call option price wrt the strike is the risk neutral distribution function and its
  • second partial derivative is the risk neutral density function.

This logic is also used in the method shown by Malz in the Fed Staff Report No. 677 on June 2014 A Simple and Reliable Way to Compute Option-Based Risk-Neutral Distributions. For anyone interested, I show a gif for a modified version that I coded that allows for strikes that do not lie on a uniform grid (they do not have constant spacing) here.

Edit

As @nanoman commented, the question may actually be about the time value, which for a call is C - max(0, S-K). If S > K (ITM), you have intrinsic value and time value combined. Insofar, it is correct that for ITM one needs to add + 1 to Dual Delta.

However, I am not sure that this was what the question really intended to ask because in general one never computes the sensitivity of an options time value. Greeks are either always the sensitivity of the option value (time value plus intrinsic value) to changes in inputs; or the sensitivity of Greeks to changes in inputs for higher order Greeks.

Also, the other answer writes that

The Greeks are used [to] evaluate an option's sensitivity to change in [input(s)]

This value does exist and is frequently used as well.

TL;DR

It like to utilize computers to demonstrate topics. Black Scholes Merton (including Dual Delta) can be computed in Julia with the following code:

using Distributions, DataFrames
N(x) = cdf(Normal(0,1),x)

function BSM(S,K,t,rf,d,σ)
  d1 = ( log(S/K) + (rf - d + 1/2*σ^2)*t ) / (σ*sqrt(t))
  d2 = d1 - σ*sqrt(t)
  return exp(-d*t)S*N(d1) - exp(-rf*t)*K*N(d2), -exp(-rf*t)*N(d2)
end

Computing it for OTM, ITM and ATM looks like this:

S_OTM = 90
S_ATM = 100
S_ITM = 110
K, t, rf, d, σ = 100,1,0,0,0.4

df = DataFrame("Call Moneyness" => ["OTM: S = $(S_OTM) < K = $K", "ATM: S = $(S_ATM) = K = $K", "ITM: S = $(S_ITM) > K = $K", "ITM: S = $(S_ITM) > K = $(K*1.01)"], 
    "Call" => [BSM(S_OTM,K, t, rf, d, σ)[1] ,
                                    BSM(S_ATM, K, t, rf, d, σ)[1],
                                    BSM(S_ITM, K, t, rf, d, σ)[1],
                                    BSM(S_ITM, K*1.01, t, rf, d, σ)[1]], 
    "Intrinsic Value" => [max(0,S_OTM-K),
                                    max(0,S_ATM-K),
                                    max(0,S_ITM-K),
                                    max(0,S_ITM-K*1.01)],
    "Time Value (TV)" => [BSM(S_OTM,K, t, rf, d, σ)[1] - max(0,S_OTM-K),
                                    BSM(S_ATM, K, t, rf, d, σ)[1] - max(0,S_ATM-K),
                                    BSM(S_ITM, K, t, rf, d, σ)[1] - max(0,S_ITM-K),
                                    BSM(S_ITM, K*1.01, t, rf, d, σ)[1] - max(0,S_ITM-K*1.01)],
               
                 
                  "Dual Delta" =>  [BSM(S_OTM, K, t, rf, d, σ)[2] ,
                                    BSM(S_ATM, K, t, rf, d, σ)[2],
                                    BSM(S_ITM, K, t, rf, d, σ)[2],
                                    BSM(S_ITM, K*1.01, t, rf, d, σ)[2]],
          "Strike Bumped (K ↑)" => [BSM(S_OTM, K*1.01, t, rf, d, σ)[1] - BSM(S_OTM,K, t, rf, d, σ)[1],
                                    BSM(S_ATM, K*1.01, t, rf, d, σ)[1] - BSM(S_ATM,K, t, rf, d, σ)[1],
                                    BSM(S_ITM, K*1.01, t, rf, d, σ)[1] - BSM(S_ITM,K, t, rf, d, σ)[1],
                                    "N/A"],
    "K ↑ change TV" => [(BSM(S_OTM, K*1.01, t, rf, d, σ)[1] - max(0,S_OTM-K*1.01)) - (BSM(S_OTM,K, t, rf, d, σ)[1] - max(0,S_OTM-K)),
                                    (BSM(S_ATM, K*1.01, t, rf, d, σ)[1] - max(0,S_ATM-K*1.01)) - (BSM(S_ATM,K, t, rf, d, σ)[1] - max(0,S_ATM-K)),
                                    (BSM(S_ITM, K*1.01, t, rf, d, σ)[1] - max(0,S_ITM-K*1.01)) - (BSM(S_ITM,K, t, rf, d, σ)[1] - max(0,S_ITM-K)),
    "N/A"]
)
  • The Call Option Value is the Black Scholes Merton price
  • Intrinsic Value = max(0, S-K)
  • Time Value (TV) is just the Call option price minus intrinsic value
  • Dual Delta is the closed form solution from the function
  • Strike Bumped (K ↑) is the bump and reprice Dual Delta (technically with forward difference for simplicity; usually central difference would be used for most Greeks as shown here
  • K ↑ change TV is the change in Time Value due to change in Strike. It is clearly visible that the value is +1 for the ITM option.
  • The last line shows the actual values of bumping strike by 1% (from 100 to 101). Time value moves from 12.11 to 12.6 which corresponds to the computed change in theoretical value and ~ Dual Delta +1.

The resulting dataframe (I omitted the code for PrettyTables formatting looks like this: enter image description here

ATM has the largest Time Value. Therefore, increasing strike results in an increase in time value for ITM options because we move closer to ATM. The opposite holds for OTM options where S < K. Below is a GIF showing this relationship and how TV is changing.

enter image description here

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    This answer is not quite right. The question is about the sensitivity of time value (premium minus intrinsic value). So ∂C/∂K would be correct for out-of-the-money calls, but for in-the-money calls, the answer would be ∂C/∂K + 1 (because the time value is C - S + K).
    – nanoman
    Commented Nov 22, 2023 at 16:52

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