Options, futures and other derivatives

Résumé du document

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.

Extraits

[...] we assume that ( = r = in our case). However, as we mentioned above, it is important to notice that the solutions obtained by the BSM differential equation are valid in all worlds, not just those where investors are risk-neutral. Indeed, when we move from a risk-neural world to a risk-adverse world, two phenomena occur: the expected growth rate in the stock price changes and the discount rate that must be used also changes. It happens that these two changes always offset each other exactly. [...]

[...] Implied Volatility Procedure Bibliography 1. CFA Book, Level 1 (2008), Volume John C. Hull, Options, Futures and Other Derivatives, 6th edition, Prentice Hall 3. www.investopedia.com 4. www.wikipedia.com All the details of the calculation are given in the appendices. [...]

[...] Assuming that we sell 10 call option contracts, we would be hedged by buying 386 shares. Indeed, considering that St is $116 and K is $120, we will not exercise and the loss generated by the loss of the premium is offset by the gain realised in having bought the shares at $116. Vice versa, in the case of a put option, we will exercise and the gain generated by selling at $120 offsets the loss generated by the fact that we sold 614 shares at $116 for the purpose of the hedging Gamma Γ Having a low gamma, from the perspective of the call writer, is a welcomed sign. [...]

[...] 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. To calculate the value of the derivative[1] in a risk neutral world, we need to discount the expected payoffs given by the following equation: p*fu + at the risk-free rate. In fact, in a risk neutral world, all individuals are indifferent to risk, thus they do not require any risk premium and the expected return on all securities is the risk-free interest rate. [...]

[...] Options, futures and other derivatives 1. Obtain your own valuation of the two IBM's options (one call and one put) using the three following methods Assumptions : Stock will not pay any dividends until the maturity of the options The options are European options Strike price is equal to $ The binomial model Assumptions : 5 steps Risk neutral valuation The binomial model is used to describe stock price movements on consecutive periods of time over the life of the option and to determine the actual price of this derivative. [...]