Estimating Barrier-Hit Probabilities with a Price Process Model
Summary
The document asks how to estimate the chance that the euro falls below a specified exchange-rate barrier at any point before a deadline. It includes an attempted calculation using an American digital option formula and market inputs such as spot price, volatility, interest rate, and time. The response clarifies that the event is a barrier crossing: an option analogy would be a down-and-in barrier, which activates when the underlying falls through the barrier.
It also points out that an option-pricing formula is not essential for estimating the probability. Instead, one can specify a stochastic model for the exchange rate and derive the crossing probability from that model; geometric Brownian motion is offered as an example with a closed-form treatment. The discussion provides no derivation, validation, or assessment of whether the model assumptions fit exchange rates. The resulting probability is therefore model-dependent, and the document does not establish that the displayed spreadsheet calculation is correct.
Key ideas
- A probability of crossing a lower price threshold is a barrier-hitting problem.
- A down-and-in barrier option provides a useful conceptual analogy for a downward crossing.
- A stochastic price model can estimate the event probability without pricing an option first.
- Geometric Brownian motion is suggested as a tractable model, but its assumptions are not evaluated.
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Full text
# Calculating the probability of a price change using an options pricing formula
# Calculating the probability of a price change using an options pricing formula
I don't know if I'm doing this right and I'd greatly appreciate help. I'm trying to use an option pricing formula to backout the likelihood of the Euro dropping below $1.27, even for a minute, at any time by April 10. My calculations are in an excel worksheet labeled "Current" here: https://www.dropbox.com/s/sggti4iji5tjfne/binary-american-option.xlsx
The formula is from here: http://www.matthiasthul.com/joomla/attachments/article/70/American%20Digital.pdf
```
3/25/2013 Today's Date
4/10/2013 End Date
B= 1.27 strike price
S0= 1.28652 spot price
T= 0.043835616 time to strike date (years)
sigma= 0.0896 volatility of underlying, measured in std dev of annual % change
r= -0.001 risk free interest rate
below this line calculated automatically
alpha= -0.00501408
beta= 0.00301408
ratio= 0.987159158
logratio= -0.012923998
zscore1= -0.695973397
zscore2= -0.681887312
cumnormal1 0.243222745
cumnormal2 0.247655105
factor1 1.003224855
factor2 1.013007874
term1 0.244007103
term2 0.250876571
price 0.494883674
```
## Answer by SRKX (score 3)
https://quant.stackexchange.com/a/7613
I you want to look at an option-pricing method, you would have to look at a down-and-in barrier option:
> Down-and-in: spot price starts above the barrier level and has to move down for the option to become activated.
But the thing is, you do not have to look at an option pricing formula, you just need a model and a handle on probability theory.
So, if you assume the classic Geometric Brownian Motion, then you will be able to find a close form solution in this article.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.