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 ε 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. [...]
[...] This gap is due to the number of steps (1000), the use of the historical volatility rather than the implied volatility and the assumption that the volatility is constant over the time Compare your calculation of the stock volatility to the volatility implied by the Black-Scholes Model We determine the volatilities implied by option prices observed in the market, when K = $120, S0 = $ risk-free rate = and T-t = In other words we suppose that the value of the call option is 3.190 and the one of the put option is 6.700 (Yahoo figures), and assuming that the Black-Scholes correctly prices the option, we can use an iterative search procedure to infer the implied volatility of the underlying asset from the market prices of the options. [...]
[...] Besides being hedged through delta and gamma, we can also be hedged through a v-neutral position Rho ρ Rho represents the sensitivity of the value of a portfolio to interest rates. In our case it means that for a change in the interest rates, the value of the options increase by $ 0.059 and 0.108 for the call and put respectively. Appendices 1. Data 2. Binomial Model 1. Determination of d and p Date Price St rt=ln(St)-ln( u 1,0403154 d (down) 0,9612470 p 0,4977214 1-p 0,5022786 2. [...]
[...] 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. Consequently, the value of the option is equal to f = [p*fu + * exp With our example we obtained the following values for the derivatives: The differences (respectively and are essentially due to errors linked to the approach is constant over the period; the number of steps is assumed and low in our example In option pricing, there is an important general principle known as Risk- Neutral Valuation. [...]
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