![]() If dP is the unit-normal distribution (i.e., the underlying pdf is normally distributed with zero mean and standard deviation of unity,Īnd the equivalent risk-neutral measure for this example, for a small neighbourhoodĬlearly Z is selected to give the desired mean to the equivalent measure. 210-214)Īnd P are equivalent probability measures and are related by (, pp. Is a random variable for each point in time t) that has a drift rate μ and is driven by a Wiener process The possibilities that not-equivalent measures describe are different. , then the event B is impossible in measure P whereas event B is possible in measure Q, and the two measures P and Q are not equivalent. This means that all events that are possible under the measure P are possible under the measure Q, and vice versa. ![]() Presumably fluctuations about the mean value are described well by the shape and scale parameters of the distribution and one should use the best available estimate of the location parameter of the distribution (i.e., the mean drift rate, α) to price an option.Ĭonsider a normal distribution with mean μ and variance σ 2. , for various strike prices and values of α, givenĬorresponds to prices obtained via risk-neutral pricing, i.e., by the Black-Scholes formula. The “success” of risk-neutral pricing owes to the fact that the magnitude of the random fluctuations are typically significantly greater than the magnitude of the risk premium, i.e., The seller has, on average, an advantage under risk-neutral pricing whenĪnd overestimates the price of a European call option when Thus the risk-neutral pricing underestimates the value of the call option and gives, on average, an advantage to the option buyer when The pdf and CDF for a normal distribution with a mean of zero and a variance of σ 2. Is the cost of a European call option using risk-neutral pricing (i.e., using the same assumptions that yield the Black-Scholes option pricing formula). The probability that a measurement of the random variable Let the probability density function (pdf) for a random variable With α the true drift rate of the underlying option and r the risk-free rate.įirst some notation and background information. Is the risk premium and σ 2 is the variance of the one-day return of the asset that underlies the call option. It is found that risk-neutral pricing used in the pricing of European call options is a specious concept that is only approximately correct and that ignores terms of Next we will consider a third method of pricing an option, that of replication.The concept of risk-neutral pricing of European call options is investigated from a mathematical approach. How does this affect the expected value of the stock in tomorrow's world? Well, $\mathbb(S)=S_1$. The only value of $p$ which causes the option value $C$ to agree with the price obtained from the hedging argument is $p=0.5$. This leads to the expected value of the option price $C$ to be: Thus, the expected value of our stock $S$ tomorrow, is given by: that both probabilities sum to unity and thus one of the events must occur. ![]() Since these are the only two probabilities, we can see that $p + (1-p) = 1$, i.e. Let us assume that the probability of the stock going up to 110 is given by $p$ and that the probability of it falling is given by $1-p$. We will use a probability argument for this particular technique, which is known as risk neutral pricing. ![]() Our task as an insurance firm is to price a call option struck at $K = 100$ such that all risk is eliminated from the sale of this option to a purchaser. Click for Part 1 and Part 2.Ĭonsider the same world as before, which has a stock valued today at $S$ equal to 100, with the possibility of a rise in price to 110 or a fall in price to 90. Note: It will be necessary to read the prior articles on the Binomial Trees in order to familiarise yourself with the example of the stock and option before proceeding. We will now utilise a probability argument and show that the value $C$ of the call-option is achieved. The most surprising consequence of the argument was that the probability of the stock going up or down did not factor into the discussion. This was guaranteed by the principle of no arbitrage. In our last article on Hedging the sale of a Call Option with a Two-State Tree we showed that there was one unique price for a call option on an underlying stock, in a world with two-future states.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |