4
$\begingroup$

Is there a way to calculate the price of a binary option (i.e., an option that pays out 1 dollar when the stock price hits $x$ amount) using market call/put option prices, forward prices, etc. for a stock? Assume no interest rate.

Provided that the company never goes bankrupt, shouldn't the value of this option be 1 dollar? If the stock price follows a random walk, it will hit $x$ in finite time with probability 1.

Improve this question
$\endgroup$
  • $\begingroup$ This is not the definition of the binary option. You are critically missing some maturity $T$ which limits the life of this option. If I understand well, you're trying to value the option using a replicating portfolio is that right? $\endgroup$ – SRKX Feb 11 '15 at 8:11
  • $\begingroup$ Yes that is correct. Assume there is no expiration date $T$. $\endgroup$ – FinForFun Feb 11 '15 at 22:28
  • $\begingroup$ But if there is no $T$ your option doesn't really make sense. In what context are you seeing that? If you assume the stock follows a GBM and $\sigma > 0$, then your stock will indeed for sure hit $x$ at some point in the future. If there is no discount, then you know that this option is worth 1 today indeed. $\endgroup$ – SRKX Feb 12 '15 at 1:29
1
$\begingroup$

Sorry to disagree but if interest rates is 0, the binary is still not worth $1 now.

Suppose spot $S(0) = 100$, assume $x = 110$ and upon touch (whenever it happens as the option has no maturity) you receive one dollar.

Suppose I buy 1 stock. If the barrier hits, i sell the stock and receive 110 USD. What if I buy N stocks at t=0? upon hit of barrier i sell my stock and pocket N*110. Now pick N = $\frac{1}{110}$. When i sell my N stocks upon hit, I will hold $N*110 = 1$ USD.

So this strategy of holding $N = 1/110$ stocks replicates the payout. Furthermore at iniation it costs me $N*S(0) = 100/110 = 0.909090$. Note that this is cheaper than your price of 1 USD.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.