This is not a formal document. It contains information (ramblings) about features of the Wilberforce application that have been considered for implementation, or have been implemented. Some attempt has been made to ensure that this document matches with the actual development of Wilberforce, but it is almost certain that from time to time, it will not.
Wilberforce consists of single form split into LHS and RHS panes.
Yield curves supply discount factor information. What is a discount factor? £100 in the hand now is worth more that a promise to pay £100 in a years time. Why? Because if the "going" annual interest rate is 6.25%, then £100 now will be worth £106.25 in a years time. The promise to £100 in a years time is only worth 100/106.25 = 0.9412 of a that of the £100 in hand now. (Note that the calculation does not the possibility of default take into consideration) The factor 0.9412 is the discount factor. The reader is referred to any introductory book on derivatives pricing for further details.
Discount factors can be implied from the price of various instruments. In particular, short term rates, FRAs and swaps. By combining the information from these instruments it is possible to construct a discount curve stretching to 20 years.
The cost of out-of-the-money options are more expensive than expected if their values are calculated using BS and volatilities and rates consistent with at-the-money options.
Why?
A good question.
BS assumes the price of the underlying at maturity is a log-normal distributed random variable, say X. I.e. X is normally distributed where X=ln(S) and the random variable S is the final price of the underlying.
Studies show that the distribution of prices at some point in the future has fatter tails and is more peaked than a normal distribution. The result is that out-of-the-money option prices are more expensive than predicted by a simple BS model. Other studies point out that markets, particularly equity markets tend to undergo sudden jumps. Models of jump diffusion are capable to producing smiles. All of these answers however pre-suppose an element of objectiveness that option prices do not really have.
Prices are driven by the market. Fund managers in particular are risk-averse and are blamed for driving up the price of out-of-the-money options which they purchase to protect themselves against losses from large market movements. The prices determined by the market determine the risk-neutral probabilities.
An analogy is useful. Bookies are able to quote odds for each animal. However bookies have no real (objective) method of determining the probability of any animal winning, rather the odds reflect the popularity of animals in the market, I.e. the "market view" of the likelihood of the animal winning. The probabilities implied by the odds are a "risk-neutral" probability measure - the odds are determined by the bookie ensuring that he does not risk any money on the outcome of the race.
The situation with options is similar. The volatility smile at any maturity determines the risk-neutral probability distribution at that maturity. The well known result is that if C_{K} is the value of a call with strike K, then the probability distribution is
The proof is relatively trivial.
, so and .
Exercise: Check the above by differentiating BS formula.
The formula above requires that the volatilty smile is twice continuously differentiable. Unfortunately the market only provides a limited sample of call price C_{i} with strike K_{i}, and all other values must be interpolated. That provides some constraints on how we must do that interpolation. Linear interpolation is out because it is not differential at K_{i}. The (probably) simplest interpolation method that can be made to work is natural cubic spline, which we implement. Other implementations are available in the literature, and these will be implement in the future. Cubic splines are not ideal - a change in the position of one point changes the shape of the whole curve, and that probably does not make business sense. If the demand for a option with a particular strike increases, why should the prices of options at the other end of the smile be effected.
Once the probability distribution p is known, it is possible to price more exotic options.
A snapshot of an early version of the volatility surface application is shown below.
Copyright © Bowmain Ltd, 2004