• Stock Price of Apple + Prediction of Future Price \(\rightarrow\) Stock Investing Strategy \(\rightarrow\) Profit

• Stock Price of Apple + Option \(\rightarrow\) Option Investing Strategy \(\rightarrow\) Profit

• Data from Yahoo Finance.

• Apple Inc historical stock price from 11/01/2012 to 10/31/2013, total 252 records.

• Option Pool:

The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is:

\[ C(S,t)= N(d_1)S - N(d_2)Ke^{-r(T-t)} \] \[ d_1 = \frac{ln(\frac{S}{K}) + (r+\frac{\sigma^2}{2})(T-t)}{\sigma \sqrt{T-t} } \] \[ d_2 = \frac{ln(\frac{S}{K}) + (r-\frac{\sigma^2}{2})(T-t)}{\sigma \sqrt{T-t} } \]

• \(S\) = Stock price, \(X\) = Strike price of option
• \(r\) = Risk-free interest rate, \(T\) = Time to expiration in years
• \(\sigma\) = Volatility of the relative price change of the underlying stock price
• \(N(x)\) = The cumulative normal distribution function

2013/11/4 + 134 = 2014/5/18 (modified)

\(K = 90\)

\(S_0=\) $73.27

Implied Volatitlity = 33.08%

The option price is:

• \(c = 2.15\)

• \(p= 18.75\)

• Stock Price of Apple + Option Price \(\rightarrow\) Option Investing Strategy \(\rightarrow\) Profit

• Stock Price of Apple + Option Price + Prediction of Future Price \(\rightarrow\) Option Investing Strategy \(\rightarrow\) Profit

A geometric Brownian motion (GBM) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion with drift.

\[ dS_t = \mu S_t dt + \sigma S_t d W_t \]

We define a jump process as a type of stochastic process that has discrete movements, called jumps, rather than small continuous movements, where the notion of jump is common in mathematics.

\[ N_t = \sum \limits^{\infty}_{k=1} 1_{[T_k, \infty)}(t), where\ t \in \Re^{+} \]

Where note the \(N_{t_1}- N_{t_2}\) has Possion Distribution with parameters $\lambda (t - s). In general, we have the following

\[ P(N_t = k) = \frac{(\lambda t)^k}{k !} e^{-\lambda t} \]

The general solution of the stochastic differential equation of GBM with Jumps is given by the following.

\[ S_t = S_0 exp((\mu - \frac{1}{2}\sigma^2)t +\sigma W_t)(1-\sigma)^{N_t} \]

Predicted Price at 2014/5/18 = 88.21143

Long Call: -2.15 \(\rightarrow\) Rejected

Short Call: +2.15 > 0 \(\rightarrow\) Selected

Long put: -18.75 + (90 - 88.21143) < 0 \(\rightarrow\) Rejected

Short Put: + 18.75 - (90 - 88.21143) > 0 \(\rightarrow\) Selected

Actual Price = 85.57

Long Call: - 2.15 \(\rightarrow\) Rejected

Short Call: + 2.15 > 0 \(\rightarrow\) Selected

Long Put: -18.75 + (90 - 85.57) \(\rightarrow\) Rejected

Short Put: + 18.75 - (90 - 85.57) > 0 \(\rightarrow\) Selected

Butterfly Spread, Bull Spread…

Predict with whole setting( 1-year range, 12 adjustments at the 1st of every month)