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By no-arbitrage, the price of a vanilla European call with $K=0$ should be that of the underlying stock (as selling the call is perfectly hedged by buying the stock). However, is this true in practice?

More concretely, if you ran an options desk and I called you asking me to make me a market on $50M worth of 0-strike calls, would your market be identical to your market on the stock itself? If the answer is no, what factors contribute to your altered quote and in what ways?

The most I can think of is that there's an opportunity cost to hedging a short call because you aren't earning the risk-free rate on the cash you used to buy stock and hold it. Likewise, with the long call, you're paying borrow from the time you sell the stock (to hedge) until the time you cover with the delivered stock at maturity. Is this actually a consideration? Are there any others?

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  • $\begingroup$ Buying a call option gives you the right to buy the stock in the future. Assuming European options for simplicity, you will need to think of the price at expiry (the no arbitrage forward), not today's spot. That is also the result you get if you run a simple Black Scholes calculator and set strike to 0 $\endgroup$
    – AKdemy
    2 days ago
  • $\begingroup$ @AKdemy . I am assuming you have in mind that the stock pays dividends. $\endgroup$
    – Kurt G.
    yesterday
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    $\begingroup$ @actinidia Have you seen in practice an options market that quotes prices of far ITM and far OTM options ? If so please post some details . $\endgroup$
    – Kurt G.
    yesterday
  • $\begingroup$ @Kurt G., yes, I had dividends in mind. Since you do not yet own the stock with an option, you forgo the dividend income you could get by buying ths stock immediately. $\endgroup$
    – AKdemy
    yesterday
  • $\begingroup$ In theory, there is no difference between theory and practice. In practice, there is. Ps: zero-strike equity call options don't really trade in practice. $\endgroup$ 13 mins ago

2 Answers 2

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I do not think anyone actually trades this but in any case, it would follow standard pricing logics.

If you have a strike of truly 0, volatility does not matter anymore. The expected future value of unconditionally receiving the stock equals the forward. However, with options, the stock is received conditionally on the probability N(d2), the (risk neutral) probability of the option expiring in the money. In the case of a zero strike option, this probability is 1, no matter what. You can check out this answer for some related details.

So you know you will use this right for sure (you can buy the underlying for "free"). What matters is the forward at that time (which you discount to get the current options price).

In case of zero dividend and interest rates, the forward will be idenical to spot, and your option is worth the spot value. In case of dividends, you have to look at the forward. Below is a bit of Julia code to showcase this.

# load packages
using Pkg,  Distributions, DataFrames, Plots, PlotThemes
# define cdf
N(x) = cdf(Normal(0,1),x)
# generic Black Scholes pricer
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)
    c  = exp(-d*t)S*N(d1) - exp(-rf*t)*K*N(d2) 
  return c
end 

# inputs
s = 4000.00        # spot
k = 4000.00        # strike
σ = 15/100         # IVOL 
t = 90 / 365       # time to maturity in years
r = 0.1 / 100      # interest rates
r = log(1+r*t)/t   # continous rates
d = 1.5/100        # dividends 
d = log(1+d*t)/t   # cont. divs

DataFrame(Call = BSM(s,k,t,r,d,σ)[1])

This value should match pricers like Bloomberg's OVME exactly.

enter image description here

Now, if you set set k = 0 you get the discounted fwd.

enter image description here

The following lines compute the option value for different values of IVOL

theme(:juno)
k = 0
σ = 0.01:0.01:30  #  0.1 is 10% (meaning 30 is very unrealistically high)
plot(σ, BSM.(s,k,t,r,d,σ), 
                        legend = false, 
                        title = "ATM Option value for different IVOL",
                        xlabel = "IVOL",
                        ylabel = "Option value")
ylims!((0,5000))

enter image description here

compared to the value of an ATM option (see here for an explanation of what happens to call and put options if IVOL tends to infinity). enter image description here

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  • $\begingroup$ Does this mean that the market on the call (with nonzero dividends) will differ from the spot market insofar as the market maker has uncertainty about future dividends? Practically, will this impact on the spread be substantial when moving large size? From what I know, unlike the situation when one wants to hedge forward interest rates, there's no liquid market for hedging divs in arbitrary names. $\endgroup$
    – actinidia
    13 hours ago
  • $\begingroup$ To be honest, I thought this is a purely theoretical question but seems there are actually markets like the Australien Stock exchange where LEPOs trade. The ASX explains they are like a forward purchase of shares which is in line with what I showed (ignoring that mathematically speaking K=0 it is not defined in BS model, as @Frido Rolloos pointed out). $\endgroup$
    – AKdemy
    11 hours ago
  • $\begingroup$ If you have questions about an actual market, I will not be able to help but it would help if you add a lot of details in this case. That way someone trading in that specific market will be able to better understand your specific issue. Otherwise, I would say the question lacks clarity and is not focused enough. The explanatory booklet from ASX is already 28 pages for example. $\endgroup$
    – AKdemy
    11 hours ago
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The (call) price is undefined for $K=0$. You can only speak about the price as $K \downarrow 0$. Not the same thing as $\infty$ is not a number.

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