How does an option's time value depend on moneyness?

Question

How does an option's time value (also known as extrinsic or instrumental value) depend on how far it is in the money or out of the money? In other words, how does the time value change as the underlying price changes?

Answer 1

By definition, an option's premia is the sum of intrinsic value and time value. The time value of premia declines as the option goes more ITM ("in-the-money") or OTM ("out-of-the-money"), ceteris paribus.

An intuitive explanation for this can be found by thinking of time value as the expected P&L of a long option position dynamically hedged by going short (long) Delta units of a call (put). Delta is the sensitivity of the option premia to the price of the underlying. It is a value between 0 and 1 and increases with moneyness. Delta of an ATM ("at-the-money") option is approximately 0.5. Suppose that you are long 100 ATM calls struck at X and delta-hedged by being short 50 units of the underlying at price S=X. If S goes up to S', your option is ITM and delta goes up as well--let's say to 0.6. Now, to be hedged, you have to sell 10 more units of the underlying at S'. Now suppose that the price goes back from S' to S. Delta decreases back to 0.5 and you have to buy back 10 units of the underlying to be hedged again. You have made a S' - S profit on 10 units by holding a market neutral position. Over the interval of time between re-hedges, your delta-hedged position gets long in an up market and short in a down market, and as you re-hedge you are realising little profits from the re-hedges as the underlying goes up and down in price.

This is a valuable property to have in a position that by construction takes no market risk. But there is no free lunch, and time value can be thought of as the price you have to pay in order to have exposure to this delta-hedging P&L over the life of your option position. (It is also why time value increases the more volatile the market expects the underlying to be--Implied Volatility--and time value erodes as the option's expiry gets closer. The sensitivity of option's premia--and therefore time value--to those two factors are called Vega and Theta, but that is another discussion.)

So why does time value decline the more ITM or OTM the option is? The reason is that the change in Delta given a change in underlying price is not linear, but is rather a convex function of price. The change in delta is greatest when the option is ATM, but as the option goes more and more ITM (OTM), Delta gets closer and closer to 1 (0) and the changes in Delta get smaller and smaller. Therefore, so do the little re-hedging profits and so does time value. Delta-hedging a deep ITM or OTM option doesn't generate much P&L for the simple reason that Delta doesn't change very much.

The sensitivity of Delta to a change in underlying price is called Gamma, and Gamma is the reason why time value is worth so little in very ITM option but worth allot in an ATM option. A more formal answer can be given to your question, but given the nature of the question, I hope that this intuitive sketch will be more illuminating.

Answer 2

Here is a plot of the time value for a typical call option as a function of spot. The image below was produced for a call option with the following parameters: Strike = 100, risk free rate = 0, volatility = 30%, Time to maturity = 0.5.

Here, by the way, is theta, the time decay (derivative of total value with respect to time) versus spot using the same parameters:

R Code for theta:

    BSOption <- function(S,K,r,v,T,type){
      d1 = (log(S/K)+(r+.5*v*v)*T)/(v*sqrt(T))
      d2 = d1 - v*sqrt(T)
      if(type == "call"||type == "Call"||type == "C"||type == "c"||type == 1)
      {
        delta = pnorm(d1)
        theta = (-S*dnorm(d1)*v/(2*sqrt(T))) - 
          r*K*exp(-r*T)*pnorm(d2);
        price = S*pnorm(d1)- K*exp(-r*T)*pnorm(d2)
      }
      else{
        delta = -pnorm(-d1)
        theta = (-S*dnorm(d1)*v/(2*sqrt(T))) + 
          r*K*exp(-r*T)*pnorm(-d2);
        price = K*exp(-r*T)*pnorm(-d2)-S*pnorm(-d1)
      }
      gamma = dnorm(d1)/(S*v*sqrt(T))
      vega = S*dnorm(d1)*sqrt(T)
      theta = theta/365
      BSoption = data.frame(price, delta, gamma, vega, theta)
      return(BSoption)
    }

S = seq(80,120,1); 
plot(S,BSOption(S,100,0,.3,.5,1)$theta,ylab="Theta",xlab="Spot"); title("Theta vs Spot")

R code for time value:

S = seq(80,120,1); 
intrinsic = array(0,c(length(S),1))
for(i in 1:length(intrinsic)){
    intrinsic[i] = max(S[i] - 100,0)
}
plot(S,BSOption(S,100,0,.3,.5,1)$price-intrinsic,ylab="Time Value",xlab="Spot");
title("Time Value vs Spot")

Answer 3

The options value is dependent on the moneyness of the option. Taking the partial derivative of Theta w.r.t. term S/K (whole term) will give the sensitivity of Theta to moneyness.

S/K is the moneyness.