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I believe the answer is no, as minimum value of call option is S - PV(K), which can never be below S-K.

The reason for the question is this paragraph in Natenberg, pg 109:

Is it ever possible for an option to have a positive theta such that if nothing changes, the option will be worth more tomorrow than it is today? In fact, this can happen because of the depressing effect of interest rates. Consider a 60 call on an underlying contract that is currently trading at 100. How much might this call be worth if we know that at expiration the underlying contract will still be at 100? At expiration, the option will be worth 40, its intrinsic value. How- ever, if the option is subject to stock-type settlement, today it will only be worth the present value of 40, perhaps 39. If the underlying price remains at 100, as time passes, the value of the option must rise from 39 (its value today) to 40 (its intrinsic value at expiration). The option in effect has negative time value and therefore a positive theta. It will be worth slightly more as each day passes. This is shown in Figure 7-9.enter image description here

The reason why I think Natenberg is incorrect is by constructing a portfolio where I am long the ITM option and short the stock, the proceeds have present value S=100. Now if the stock doesn't move and I exercise the call at some future time, I pay K in the future, whose present value is PV(60) which is less than 60. So the call option value = 100-PV(60) > 40.

Another way to disprove Natenberg's claim is by looking at the formula for call theta based on BS. With positive risk free rate there is no way to make theta positive.

From an intuitive point of view it seems that when we take a partial derivative of option value wrt time, we don't really hold S constant. Instead S does go up by S X r X dt. Is this intuition correct?

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    $\begingroup$ You wrote " looking at the formula for call theta based on BS. With positive risk free rate there is no way to make theta positive". However in BSM, with a big enough dividend rate $q$ it is possible. $\endgroup$
    – nbbo2
    Commented Oct 27, 2022 at 15:31
  • $\begingroup$ How does your last paragraph relate to the question, specifically your examples? $\endgroup$
    – AKdemy
    Commented Oct 27, 2022 at 16:32
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    $\begingroup$ nbbo2 is correct, the key difference lies in the dividends (or repo margins). The graph you point to clearly corresponds to a negative equity carry cost, that cost being defined as the (risk-free) rate minus dividend yield minus repo margins. $\endgroup$
    – Quantuple
    Commented Oct 28, 2022 at 6:23

1 Answer 1

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@nbbo2 and @Quantuple already answered the question in their comments but if in doubt, I always think computer coding is very helpful because you can simply try it out and run a lot of calculations in one go. Since option prices and Greeks (assuming Black Scholes) are well defined, it is straightforward to set this up, and replicate the chart from the book.

Using Julia, and typing the wikipedia formulas into code, would look like this:

   using Plots, Distributions, PlotThemes, DataFrames
    theme(:juno)
    #theme(:solarized)
    Plots.scalefontsizes(1.3)
    
    # define cdf and pdf
    N(x) = cdf(Normal(0,1),x)
    n(x) = pdf(Normal(0,1),x)
    
    # generic put call pricer
    function BSM(S,K,t,r,q,σ)
        d1 = ( log(S/K) + (r - q + 1/2*σ^2)*t ) / (σ*sqrt(t))
        d2 = d1 - σ*sqrt(t)
        c  = exp(-q*t)S*N(d1) - exp(-r*t)*K*N(d2)
        theta = (-(S*exp(-q*t)*n(d1)*σ)/(2*sqrt(t)) + q*exp(-q*t)*S*N(d1) - r*exp(-r*t)*K*N(d2))/365
      return c, theta 
    end
    
    # define inputs
    spot = 0.2:0.1:15 # underlying value
    k = 5             # strike
    r = 0.03          # interest rate: 3%
    q = 0.09          # dividend = 9%
    q2 = 0.12        
    t = 1             # time to expiry in years
    σ = 0.3           # implied volatility
    
    # compute option 
    opt_val = BSM.(spot,k,t,r,q,σ)
    df = DataFrame(BSM.(spot,k,t,r,q,σ))
    rename!(df, :2 => :theta)
    theta = df[!,:theta]
    
    df1 = DataFrame(BSM.(spot,k,t,r,q2,σ))
    rename!(df1, :2 => :theta1)
    theta1 = df1[!,:theta1]
    
    x = [i < 0 ? i : 0 for i in  [val[1] for val in opt_val].-[val for val in max.(spot.-k,0)]]
    area = [val[1] for val in opt_val].-x
    
# plot option
    plot(spot,[val[1] for val in opt_val], 
                label = "Theoretical Value with dividend = $(round(q*100, digits = 4))%", 
                title = "Natenberg replica", 
                size = (900,600),
                xlabel = "Underlying Price",
                ylabel = "Option Value",
    linwidth = 2,
    linecolor = :lightgreen)

# plot intrinsic value
    plot!(spot,[val for val in max.(spot.-k,0)], label = "Intrinsic Value", legend=:topleft,  linestyle = :dash)

## color the area where there is negative time value 
    plot!(spot, [val[1] for val in opt_val],
                            fillrange = area, fillalpha = 0.35, c = 3, label = "Negative Time Value with dividend = $(round(q*100,digits=4))%" )

# plot a second option with a different dividend
    plot!(spot,[val[1] for val in BSM.(spot,k,t,r,q2,σ)], label = "Theoretical Value with dividend = $(round(q2*100,digits=4))%", title = "Natenberg replica")

# shade area where theta is positive    
vspan!([spot[length(opt_val) - length(theta[theta .>= 0])], spot[length(spot)]], 
            linecolor = :grey, fillcolor = :blue, opacity = 0.05, 
            label ="Positive θ for dividend = $(round(q*100,digits=4))%")
    vspan!([spot[length(opt_val) - length(theta1[theta1 .>= 0])], spot[length(spot)]], 
            linecolor = :grey, fillcolor = :yellow, opacity = 0.03, 
            label ="Positive θ for dividend = $(round(q2*100,digits=4))%")

It is not necessary to understand the code in full but there are a few things to note:

  • the Black Scholes model and theta is the same formula as shown in Wikipedia
  • the shaded area in the curve in green shows where there is negative time value for the option with dividend set to q (9% in this example)
  • the blue curve is the payoff for the same option but with dividends set to 12%
  • the shaded bars show the areas where Black Scholes theta is positive (blue for q = 9% and yellow for q2 = 12%) - note that the blue bar overlays parts of the yellow bar

enter image description here

  • It closely resembles the chart from Natenberg, and given the impact of r and q on the option (and forward), this shape makes intuitive sense
  • Theta itself is also closely related to that area as is visible by the colored bars starting more or less at the intersection of the option value with the intrinsic value

Adding a few lines of code, similar to this answer, allows us to make this chart interactive (quality cannot be much better due to size limit):

enter image description here

It is not a formal prove but in my opinion, often intuition and charts are also very useful. Last but not least, one can also plot theta as a function of spot to see that it indeed turns positive for (deep) ITM options with positive dividends:

enter image description here

A related question for puts is here.

Edit

I wanted to comment with regards to the comments below but it turned out a bit lengthy.

I am not a 100% sure why the focus is on the depressing effect of interest rates. Copying the answer I linked above, there are 2 circumstances that can lead to the value of an european option being lower than intrinsic value

  • deep ITM puts in presence of positive interest rates r>0
  • deep ITM calls in presence of positive dividend yield q>0

which also coincides with the 2 circumstances under which it makes sense for an american option to be exercised early.

Insofar @nbbo2 may be right and the mixed parts of a section about puts into this section. The simplest explanation is with American puts, where stock price is zero. If the strike is 60, exercising early results in an immediate gain of 60. Due to interest rates (time value of money), receiving 60 now is preferable to later (you cannot gain more by waiting, because negative stock prices are impossible).

However, it's clearly written in terms of discounting the PV of the intrinsic value of the call, which kind of rules out that it was intended for puts. This argument is sloppy imho because it by definition rules out any time value of money. It will always be worth at least the PV of its intrinsic value (plus time value, which can be negative).

If it’s not mixing arguments for puts and calls, I think it may be because the book uses mainly intuition and simple examples as opposed to mathematical rigor. He also writes elsewhere in the book that

Theta Risk is the opposite side of gamma risk ... negative gamma always goes hand in hand with a positive theta

Theta for deep ITM calls can be positive as we have seen. Yet, Delta will tend towards 1, which means Gamma will tend towards 0. Hence, it is perfectly reasonable to have positive Theta, with a non-negative (0) Gamma as shown below.

enter image description here

He also writes that delta is approximately equal to the probability that the option will finish in the money, but he moves on to explain that delta is only an approximation of the probability because interest considerations and dividends may distort this interpretation. While rates and divs will impact delta as shown here, this is not the reason delta is not the true probability. See Understanding N(d1) and N(d2): Risk-Adjusted Probabilities in the Black-Scholes Model by Lars Tyge Nielsen for a detailed explanation. Only if time to maturity and vol is low, $d1 \approx d2$ and delta will be closer to the risk-adjusted probability of the event that the option will finish in the money, which is $P(S_T > X) = N(d2)$.

While some parts may not be 100% correct, it is probably still much more useful than "correct" expositions that require a lot more detail in a book like this. It's a hands on approach with a commonsense point of view. It is sufficient to know theta can be positive, that delta is approximately equal to the probability and that usually gamma and theta have opposite signs. Anything more complicated than the concepts mentioned in the books will likely not help you much on a trading floor. You can have a look at the discussions and answers to this question to see that thinking in detail can actually be quite cumbersome, while the answer to the question likely adds no value to an option trader.

In the words of Euan Sinclair on P.XV of the preface in his book called Option Trading,

Many successful traders have had only a basic understanding of the mathematics behind BSM and certainly have no idea of the differences between calculus and stochastic calculus.

That's not meant to say Natenberg was "only" a successful trader but that the book was likely written in a way that builds the intuitions needed to be a successful trader, not a quant.

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  • $\begingroup$ Accepting the answer because the code is very helpful. And indeed setting positive dividend or short stock rate does give rise to the graph shown in Natenberg. But the explanation given in the textbook does not mention dividends. For example consider this text "In fact, this can happen because of the depressing effect of interest rates. Consider a 60 call on an underlying contract that is currently trading at 100. How much might this call be worth if we know that at expiration the underlying contract will still be at 100? At expiration, the option will be worth 40, its intrinsic value." $\endgroup$
    – Shreyans
    Commented Oct 29, 2022 at 14:02
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    $\begingroup$ Yes, it seems to me there was an editing error in Natenberg's book and that some wording about puts accidentally got inserted in this section on calls, or perhaps this section was initially about puts and got changed but not completely. It would be worth correcting in a new edition... $\endgroup$
    – nbbo2
    Commented Oct 29, 2022 at 16:00

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