Why can’t delta’s be used to price double no touch options?

Question

Here is the link to a MATLAB one touch option pricing calculator I used:OT

I tried several inputs and I noticed that the one touch option price is approximately twice the delta of an equivalent vanilla option(where the barrier of the one touch option is used to represent the strike of the vanilla option).

For example:

AssetPrice = 100;

Rate = 0.00;

Volatility = 0.3;

Settle ='01-Jan-2018';

Maturity ='10-Jan-2018';

DividendYield = Rate - 0.0;

Barrier = 103;

Payoff = 100;

If I use the data above, then a vanilla option pricing calculator{options} would yield a delta of around 0.27 for the 103 strike option and the one touch calculator yields a price of around 54%(i.e 0.27*2).

However, I noticed this math does not work for the double no touch options. Here is a link to the double no touch option pricing calculator for MATLAB that I used: DNT

For example using the same data above but with an extra lower barrier:

AssetPrice = 100; Rate = 0.00;

Volatility = 0.3;

Settle ='01-Jan-2018';

Maturity ='10-Jan-2018';

DividendYield = Rate - 0.0;

Barrier1 = 103;

Barrier2 = 97;

Payoff = 100;

The double no touch options calculator gives a price of around 6.2%. Now if I was to compare the equivalent vanilla option{options}, I would get two options(because of the 2 barriers representing two different strikes) each with delta around 0.27. Using probability, because it’s 2 no touch options we get the compliment of the 2 one touch options. So I expected the answer to be (1-2x0.27)(1-2x0.27) =21.16%

The 21.16% form the vanilla option pricing calculator is way larger than the 6.2% from the double no touch option pricing calculator.

Question: Why can’t I estimate the price of a double no touch option by using deltas just like for the one touch options?

Answer 1

Using formulas from [1] and the fact that interest rates are assumed to be zero the one-touch option has value \begin{align}\tag{1} 1-N(\alpha)+\frac{S}{B}N(\beta) \end{align} where \begin{align} \alpha&=\frac{\log(B/S)+\sigma^2T/2}{\sigma\sqrt{T}}\,\quad\quad\beta=\frac{-\log(B/S)+\sigma^2T/2}{\sigma\sqrt{T}}\,. \end{align} Eq. (1) can be written as \begin{align} N(-\alpha)+\frac{S}{B}N(\beta)\,. \end{align} The delta of a plain vanilla call option with strike $B$ however is $$ N(\beta)\,. $$ It is easy to see that your experimental observation that OT equals two times the call delta holds only when the following conditions are satisfied:

• $S\approx B$
• $\sigma^2T$ is small

I would rather not draw too many conclusions from your accidental observation esp. when going from the single to the double barrier case.

[1] E. G. Haug, The Complete Guide to Option Pricing Formulas. McGraw-Hill, 1998.

Answer 2

The calculation (1-2x0.27)(1-2x0.27) for not hitting either barrier does not work, because the events {A=not hitting upper barrier,B= not hitting lower barrier} are not independent. Rather one needs to calculate $$p(A and B)= p(A)p(B|A)$$. Now p(B|A) is the probability of not hitting the upper barrier given that we have not hit the lower barrier. But this is smaller than p(B) because knowing that you did not hit the lower barrier means that you are considering paths that did not go down much - but those are more likely than average to hit the upper barrier.