For a vanilla option, I know that the probability of the option expiring in the money is simply the delta of the option... but how would I calculate the probability, without doing monte carlo, of the underlying touching the strike at some time at or before maturity?
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There is a simple solution if there is no drift, as the probability $p(x,t)$ obeys a simple diffusion equation: $\mathrm{d}(p)/\mathrm{d}t = \frac{1}{2} \sigma^2 \frac{\mathrm{d}(\mathrm{d}(p))}{\mathrm{d}x^2}$, here $x$ is the price difference $\text{price}(t) - \text{price}(t=0)$. Of course there is a simple solution to the diffusion equation (using scaling as a method to solve the PDE): |
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Allow me to disagree with Jaydles' proposal ; his methodology is valid only if the events of touching the barrier on each were independent. If you are working within the standard Black-Scholes framework, you're looking for the probability of a drifted Brownian motion hitting a fixed level before a fixed time ; this probability is derived in most stochastic calculus texts, see for example Karatzas-Shreve or Chesney-Jeanblanc-Yor. Another way of seeing it : you're trying to price a knock-in digital option with 0 interest rate, or knock-in zero bond. You can find formulae for these in Peter Carr's work on barrier options. |
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This surely isn't the most efficient way, but if you want something quick and dirty: You could run a vanilla model that calcs delta for each expiration date between now and expiration, and grab the delta for each. That would give you the likelihood that it's in the money at the close on any day. From that, you can pretty easily calculate the odds that it's not in the money each day (just subtract the delta from one), multiply them all together, and subract the product from one to determine the likelihood that it closes above the strike between now and expiration. This does require running the formula to calc delta many times, and it ignores the risk of an intra-day touch, but it doesn't require writing something to calc the exotic you're describing. |
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First of all, the delta is not the probability of the option in the black scholes model, it is instead the closely related N(d2) (binary probability) Secondly, the black scholes model gives risk neutral probabilities - for a binary event this is ok, but it gives no correct measure of, say, how far you would be through Thirdly, the options you are interested in are traded in the market - they are called binary no touch or one touch options ... there are several mechanisms to price, depending on your model for volatility ... black scholes pricers for them are available online, eg here http://www.volopta.com/Matlab.html |
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I believe you can modify my https://github.com/barrycarter/bcapps/blob/master/box-option-value.m to do this. You're effectively looking for the distribution of the maximum (or minimum) of the price for a given period of time. |
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An asymptotic answer is $$ p(t > \tau) \sim \mathrm{e}^{(-t/\tau)} \qquad \text{where $\tau = x^2/(\frac{\sigma^2}{2} \pi^2)$}; $$ However, as in much of statistics, asymptotic answers = "In the long-run" are not usually helpful to traders, or even investors. Example: say initial price $x_0=\$20$, and you wish the probability of reaching say, $x= \$25$ before $x = \$15$ with an annual $\sigma$ of say $0.2$. Convert $X$ to fractions, $X=(\$25-\$15)/\$15$, and calculate $\tau = 1.126$ years. That is without some drift and for reasonable values of the volatility, you are going to have to wait a long time. |
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