Touch-Then-No-Touch Events Depend on the Expiry Condition
Summary
The note distinguishes two barrier-event definitions that can look similar when expressed as a difference of probabilities. A one-touch-down, no-touch-up claim pays at expiry when the lower barrier is reached by expiry and the upper barrier is never reached by expiry. Its event is therefore a lower hit by expiry together with an upper barrier survival through expiry.
A different contract pays only if the lower barrier is reached before the upper barrier, even if the upper barrier is touched later. The answer formalizes both events using first hitting times and explains that the first event’s price under risk-neutral valuation is the discounted probability of its specified payoff. It also notes that paying the rebate at the lower-barrier hit time instead of expiry changes the discounting and complicates valuation. The distinction matters because “touch A before B” does not by itself mean “touch A and avoid B for the rest of the trade.”
Key ideas
- A lower-barrier hit followed by no upper-barrier hit through expiry differs from hitting the lower barrier first.
- The ordering of first-passage times captures whether the lower barrier was reached before the upper one.
- The expiry-only event requires the upper barrier to remain untouched for the entire period through expiry.
- Risk-neutral pricing discounts an expiry-paid unit rebate by the expiry discount factor.
- Paying the rebate at the hitting time makes discounting path-dependent.
Tags
Full text
# Does time remaining matter in NO Touch-ONE Touch probabilities?
# Does time remaining matter in NO Touch-ONE Touch probabilities?
I asked a question some days back and got an answer which I understand and make sense: Probability of touching short call strike and not touching touching short put strike of a short strangle?
However, the answer to that question brought up another related question to my mind.
With the probability of that event happening given as P(A ∩ Bcomplement)=P(A)−P(A∩B):
Is that the probability for the event with or without time still left to expiration?
Or the probability of the event only without time left in the trade?
NOTE: Because if there is still time in the trade, it can still touch B before the trade expires even though it touched A first and satisfied the condition
## Answer by ir7 (score 1)
https://quant.stackexchange.com/a/65511
Referring to one touch down no touch up (OTD-NTU) option with expiry $T$ as interpreted in your previous question, it might help to formalize the payoff stated in the respective answer.
The option pays rebate \$1 at expiry $T$ if
$$ \boxed{ \tau_L \leq T \; {\rm and} \; \tau_L < \tau^H }$$ where:
$$ \tau_L = \min \; \{t \geq 0 : S_t \leq L \} $$
and
$$ \tau^H = \min \; \{t \geq 0 : S_t \geq H \}, $$
with $L< S_0 < H$.
Its price amounts to calculating (under $Q$ probability measure)
$$ E^Q\left[e^{-rT}1_{\{\tau_L \leq T\} \cap \{\tau_L < \tau^H\}}\right]=e^{-rT} Q(\{\tau_L \leq T\} \cap \{\tau_L < \tau^H\}), $$
where $1_A$ is $1$ if event $A$ takes place, and $0$ otherwise, and $r$ is a flat risk-free discount rate.
The \$1 rebate pay can be made at touching (hitting) time $\tau_L$ too, in which case the price of the option is:
$$ E^Q\left[e^{-r\tau_L} 1_{\{\tau_L \leq T\} \cap \{\tau_L < \tau^H\}}\right]. $$
Its calculation is more complex (pay timing is random).
Back to your $A$ and $B$ events in the question, they are:
$$ A = \{\tau_L \leq T \}, \; B = \{\tau^H \leq T \}.$$
So, the payoff for which
$$Q(A\cap B^c) = Q(\{\tau_L \leq T\} \cap \{\tau^H > T\})$$
would be its price is the one that would pay \$1 at expiry $T$ if
$$ \boxed{ \tau_L \leq T \; {\rm and} \; \tau^H > T.} $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.