# Options, futures and other derivatives

- Evaluation of the two IBM's options (one call and one put).
- The Black-Scholes model.
- The Monte Carlo simulation.

- Comparision between the stock volatility to the volatility implied by the Black-Scholes Model.
- Interpret the values of the Greeks as given by the binomial tree.
- Delta.
- Gamma.
- Theta.
- Vega.
- Rho.

- Appendices.
- Bibliography.

The binomial model is used to describe stock price movements through consecutive periods of time over the life of the option and to determine the actual price of this derivative. Each period is an independent trial. The binomial formula describes a process in which stock return volatility is constant through time (= 23.64% in our example). Thus, the stock can move up with constant probability p, called the up transition probability (p = 49.77 % in our example). If it moves up, u is equal to 1 plus the rate of return for an upward movement (u = 1.0403 in our example). Or, the stock can move down with constant probability (1-p), called the down transition probability (1-p = 50.23 % in our example). If it moves down, d is equal to 1 plus the rate of return for a downward movement (d = 0.9613 in our example). Thus, the variable p and (1-p) can be interpreted as the risk-neutral probabilities of an upward and downward movement in the stock price. Graphically, by taking into account all of these movements we obtain a binomial tree. Each boxed value from which there are successive moves (two branches) is called a node. Each node gives us the potential value for the stock and option price at a specified time.

[...] Results Option prices (in Binomial Model Actual price (Yahoo) Gap 3. Black and Scholes Model Determination of d1, d2, N(d1) et N(d2) Cumulative Normal Distribution Results Option prices With Black-Scholes pricing Actual price (Yahoo) Gap formulas 4. Monte Carlo Estimations Estimates Call (in Put (in Iterations Iterati St ? ST Call PV(Call) Put PV(Put) on Results Option prices (in With Monte-Carlo simulation Actual price (Yahoo) Gap 5. Implied Volatility Procedure Iterative search procedure Bibliography 1. CFA Book, Level 1 (2008), Volume John [...]

[...] We obtain the following results: N = 0.3972 and N = 0.3635 In other words, there is a probability of that N would be less than - 0.2606 and a probability of that N would be less than - Finally, the preceding Black-Sholes formulas lead us to the following results: Thus, comparing both call and put prices with their respective actual prices, we observe an error equal to for the call and for the put. This gap is mainly due to the fact that we use here the historical volatilities (estimated from a history of the stock price) to value the options, whereas in practice, traders usually use the implied volatilities. [...]

[...] Using the call as our example, to achieve gamma neutrality of a portfolio made up of IBM options and other options, the weight of the IBM options should be such that : -?/?IBM The drawback of having a ? neutral portfolio is that you must re-adjust the delta hedge Theta ? Rate of change of the value of the portfolio with respect to the passage of time: Over the passage of time, the call option and the value of the put option fall by $ 0.045 and $ 0.039 per calendar day respectively, ceteris paribus. [...]