Maximum Price Distribution for a Binary Option Under GBM
Summary
The document asks how to describe the distribution of the highest price reached by a cash-or-nothing digital call before expiration. It assumes the underlying follows geometric Brownian motion with constant volatility and uses the Black-style probability formula for the option’s value at each time. The central idea in the proposed answer is to relate the option’s maximum value to the underlying’s maximum price through a one-to-one mapping, then apply a reflection argument for the underlying’s running maximum.
The answer gives a candidate cumulative distribution in terms of the normal cumulative distribution and its inverse, but explicitly expresses doubt about the algebra. It provides no derivation, validation, numerical example, or discussion of boundary cases. The reflection result also needs careful attention to the drift and to the threshold event before it can support a reliable formula. Treat the expression as an unverified proposal rather than an established result.
Key ideas
- The question concerns the running maximum of a digital call’s value before expiration.
- The option value is expressed as a function of the underlying price and remaining time under constant-volatility geometric Brownian motion.
- The proposed approach maps the option-value maximum to an underlying-price maximum.
- The answer invokes a reflection argument but flags its own formula as potentially incorrect.
- A derivation and checks would be needed before using the stated distribution.
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Full text
# Probability distribution of maximum value of binary option?
# Probability distribution of maximum value of binary option?
A binary option with payout \$0/\$100 is trading at \$30 with 12 hours to expiration.
Assuming the underlying follows a geometric Brownian motion (hence volatility remains constant), what probability distribution describes the option's maximum price between now and expiration?
I'm looking for a generic "formula". Even though I used price and expiration, I'm assuming the generic formula is a function of volatility (of course, price and expiration determine volatility).
More concretely:
Assume short time to expiry and hence null interest rates and dividends are null.
The time $t$ Black price (underlying $S_t$ is a GBM $dS_t = \sigma S_t dW_t$, $\sigma>0$, constant number) of a $K$-strike cash-or-nothing binary (digital) call option paying $1_{\{S_T>K\}}$ dollars at expiry time $T$ is $$P_t\triangleq \Phi\left(\frac{\ln(S_t/K)-0.5\sigma^2(T-t)}{\sigma\sqrt{T-t}}\right).$$
We are interested in the distribution (or just time $0$ expectation) of the variable:
$$\max_{t\in[0,T]} P_t, $$ (with fixed $T$, $K$ and $\sigma$) much like one is interested in the distribution (or just time $0$ expectation) of $$\max_{t\in[0,T]} S_t.$$
## Answer by dm63 (score 2)
https://quant.stackexchange.com/a/22037
I believe this can be solved using the reflection theorem: $$P(\max S_t > x) = 2 P (S_T > x)$$ Hence the required densities can be obtained solely from the distribution of $S_T$.
There is a one to one correspondence between $\max P_t$ and $\max S_t$, so that $$P (\max P_t < y) = P (\max S_t < g(y) )$$ where the function $g$ is the inverse of the function for $P_t$ in terms of $S_t$ given in the OP.
Continuing that logic I get for the final answer
$$P (\max P_t < y) = 1 - 2 N \left[ \frac{\ln(S_0/K)}{ \sigma \sqrt{T}} - InvN(y) - \sigma \sqrt{T}\right]$$
where $N[]$ is the cumulative normal distribution, $InvN[]$ is its inverse, $S_0$ is the stock price today.
Have to say I'm not 100pct confident of the algebra.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.