Determine fair price of a digital option?

Question

>A ***digital option*** pays one dollar at time $t = T$ if the asset price isabove a fixed level (strike) $K$ and is worthless otherwise. >Consider the followingmodel, with $r = 0$:>\\begin{array}{|c|c|c|}\\hline\\omega& S(0) & S(1) & S(2) \\\\ \\hline \\omega_1&6 & 10&12 \\\\ \\hline \\omega_2&6 &10 & 7\\\\ \\hline \\omega_3&6 &4 & 7\\\\ \\hline \\omega_4&6 &4 & 3\\\\ \\hline\\end{array}>Evaluate $\\Bbb E_\\Bbb Q[X]$ and determine the fair price of the digital option struck at $4$.What I have done for a part that preceded this was find the risk neutral probabilities $\\Bbb Q = (p, \\frac{1}{3}-p,\\frac{5}{12}-\\frac{5}{4}p,\\frac{1}{4}+\\frac{5}{4}p)$ with a restriction on $0<p<\\frac{1}{3}$ using systems of equations. I am not sure how I can find the expectation based on this and what the random variable $X$ represents exactly. I guess the expected value that I am supposed to find will be the fair price?

Answer 1

I have no idea what you are coding in. It doesn't matter.

Let S= Spot Price and F= Forward price and σ be Implied Vol. and K = strike. With Lognormal we have the Probability S will be above K at Maturity is

p = N(((lnF/K)-.5σ²)/σ) which is the probabilty you will get $1 at maturity. This has to get discounted back to present value. So the fair value is

e^-rtN(((lnF/K)-.5σ²)/σ) w Where N is the cumulative standard normal distribution

I have been intimately involved with options for more years than I care to mention. I have very large Vanilla and exotic option portfolios. If you need help. pleas feel free to reach out.