Find probability that payoff function is in $[10,20]$

Question

In moment $t=0$ we bought option with expiration date $T=2$. The payoff function of this option is given by:

$$f=(\max_{t\in[0,T]} S_t -110)^{+}$$

where $S_t$ satisfies

$$dS_t=15dW_t$$ $$S_0=95$$

Find probability $P(f\in[10,20])$. Use cumulative density function of standard normal distribution to express it.

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Obviously

$$S_t=95+15W_t$$

But i don't have any idea how can I move further. Any help please?

Answer 1

Generally:

$P(f\in[10,20]) = P(120 \leq S_t \leq 130) = P(S_t \leq 130) - P(S_t \leq 120)$ That is, the probability that the option is between 10 and twenty is the same that the stock is between 120 and 130. The probability that the stock is between 120 and 130 is the probability the stock is less than 130 minus the probability that it is less than 120. If you are a more visual person consider this picture:

Source MathWave

If we know that $W_t$ follows a driftless-Wiener process that:

$W_t - W_0 \sim N(0, 15^2t) \rightarrow S_t \sim N(95, 15^2t)$

If $S_t \sim N(95, 15^2t)$ then $P(S_t \leq 130) = $ prob. that a normal random variable with mean 95 and std. deviation 15t which is just the cumulative distribution function (CDF) of that normal random variable.

Here, that CDF is $\Phi( (x-95) / (15 \sqrt(t)))$, where $\Phi$ is the CDF of the standard normal. So the answer is $$\Phi( (130-95) / (15 \sqrt t)) - \Phi( (120-95) / (15 \sqrt t))$$

Update: I thought that this was a simple option payoff but this is actually a trickier problem because it is a look back option taking the maximum stock value from 0 to 2. The general idea with this more complex problem is we are studying the maximum of a Wiener process motion rather than the Wiener process itself. Stochastic Processes and Related Distributions by Daniel Herlemont, section 2 should be of some help. But I don't have a solution to this problem at this time.

Answer 2

Maybe I'm familiar with a completely different background of mathematical finance, but I'll drop my two cents anyways.

First, note that

$$ \int d Wt $$

is a stochastic integral and not easily integrable with standard methods. I hope you are preset with a tool of stochastic calculus.

The following method based on no-arbitrage was first introduced by Merton.

• We look for a derivative price based on the price of $S_t$, denoted by $\Pi(S_0)$
• You form a portfolio based on the underlying $S$ and the derivative $f$, with portfolio dynamics $dV_t = V_t (\omega_t^s \frac{dS_t}{S_t} + \omega_t^f \frac{df}{f} )$, where $\omega$ denotes the two weights on the portfolio
• These weights are chosen such that we "kill" the stochastic component $dW$
• We now have $dV_t = V_t k dt$
• Due to no-arbitrage, $k$ must be zero (there is no growth component on $S_t$ either)
• solve for $f$