Validity of assumptions in the Black-Scholes-Merton option Pricing Model.
- The Black-Scholes-Merton Model is based on assumptions that the price of financial option contract can be calculated by awareness of the price of the asset, strike price of the asset in the option contract, risk-free interest rate, and expiration of option and volatility of the price of asset.
The validity of these assumptions are assessed to meet satisfactory levels. The security markets offer no pressures; no transaction costs, taxes and restrictions placed on security trading. Additionally, the securities are divisible and a short time period is acceptable (Matei, 2005). The assets do not offer any form of additional payments in the period of lifetime option like dividends in common stock basis. From this, it is not rational for investors to practice an American option to the date of expiration, they can be used in assessing American options on non-dividend paying common stocks. The prices charged on the asset does not have lognormal distribution and changes according to Brownian motion process with progressive sample paths. While opportunities on riskless arbitrage are not offered.
Investors are able to borrow and lend at the same interest rate with no risk which is the same for a lifetime of the option. They opt for more on the function of underlying asset’s variance which is constant. Trading that is done in asset markets is done for a period of time.
Another assumption is the log-normal stock diffusion with constant volatility as the market players are aware of flawed. The equation of Black’s model with quoted prices and the use of volatilities for option strikes and maturities is known as volatility skew or smile and is valid in today’s market. Like the 1987 crash the volatilities of S&P500 created a smile pattern with the in or out of the money options were attributed to higher volatilities when compared to money options. On the other hand, the post-crash resemble the skew that showed a decline of volatilities with higher strikes (Matei, 2005). This is attributed to fear of declining market shifts.
The assumptions by Black-Scholes-Merton Model do not hold in a real world. If the dynamic hedging are not actualized, risks arise in the hedging payoffs of a call option. If the hedging portfolio is diversified (doing away with non-systematic risk) market risk-free return. Therefore, the payoffs of the call option are hedged and just market threats affects the hedging portfolio return.
There are varied implications that arise from this unrealistic assumptions. The violation of the assumptions related to continuous trading, flexible hedging and lognormal distribution of assets’ pricing exposes model to threats of tail processes that are termed to as highly questionable and have severe impacts. While the assumption on asset indivisibility does hold as the exchange-traded options are traded in blocks, it is hard to take a position. The implication is that the desire for options affect the rice of options contracts.
- The interest-Rate Cap is applied in these cases where one tries to protect themselves from against rising rates. In cases where the interest rate are rising, it is valid to secure funding costs while taking advantage of falling rates. It makes sure that one gets a borrowing cost bigger than the ‘capped’ level. If the rate is bigger than this, one is compensated. From this one is able to benefit from lower interest rates.
The chart above show the effect of a 4.5% 1 year Interest-Rate Cap. For every reset date. If the interest rate are below the cap rate, one simply pays the market rates as they are lower. While if the reset dates are bigger than the Interest-Rate Cap one does not pay more than the capped rate. In the reset date of 4.75% one receives compensation.
Data: St = 35, X = 35, Life of option = 1 year, Volatility = 20%, Rate of interest (Discrete compounding) 10%pa.
Through selling calls, we are able to hedge short position in the call option. Here we bring money against our debit. Here the stock trades down against us, the call we sold will lose value, leading to profit in the call position that assists to offset the loss in the stock.
For instance, if we sell a call at $35. The stock trades down to 28 at expiration date, our stock will lose $7 while a gain of $7 if at $42. From this, the short all will cover us to a $7 loss in stock. Showing that the stock would trade down to $28 with no loss. This hedging philosophy is perfect from the stock price of $35 to breakeven price of $28.
- The binomial option pricing model makes assumption about the underlying asset, also known as stock, taking one or two probable values every period. This assumption is quite unrealistic though resulting to accurate price options as applied by Wall Street practitioners to calculate prices of complex options (Matei, 2005). Another assumption is the absence of arbitrage opportunities which the model assumes that the current stock price could shift upward by u or down by d every period. This is probable since the market is complete. Trading in stock and bond results to payoffs that cover the two states.
- There are two determinants of the swap price of the forward LIBOR curve; they are the shape and level of the curve (Li, and Zhao, 2007). The curve is a curve that shows a varied number of yields or interest rates in varied lengths like 2 months or years for the same debt contract.
The curve displays the relation between the level of the interest rate (or borrowing price) and the time it gets to maturity also known as ‘term’ of the debt for a borrower in a precise currency. A good example is the US dollar interest rates that are issue on US Treasury securities for varied maturities that are keenly watched by traders.
The shape of the curve shows the cumulative priorities of the lenders in respect to some borrowers like the US Treasury or Japan. In most cases, the lenders are keen on a probable default (or increasing rates of inflation), hence they provide long-term loans for higher interest rates than they provide for shorter-term loans (Li, and Zhao, 2007). In most cases when lenders are looking for long term debt contracts more aggressive when compared to short-term debt contracts, the curve changes with interest rates being smaller for extended repayment period for the lenders to attract to long-term borrowing. The shape of the curve is affected by supply and demand like, if there is a huge demand for long bonds like pension funds to relate to their fixed liabilities to pensioners and limited bonds to acquire the demand.
The level and shape of the LIBOR forward curve display rich variations in the course of the period to maturity that spans to economic success like in the US history and crash of the technology bubble. The shape can be varied (short term interest rates bigger than long-term). A good example is in November 2004 where the curve for the UK bonds had been inverted. The yield for the 10 year maturity period was 4.68% though only 4.45% for the 30 year bond. The market’s projection of dropping interest rates led to these cases.
References
Li, H. and Zhao, F. (2007). Nonparametric Estimation of State-Price Densities Implicit in
Interest Rate Cap Prices. Retrieved on 21th February 2014 from: http://www.bauer.uh.edu/nlangberg/Finance%20Seminar/Zhao_Nonpara.pdf
Matei, E. (2005). “Black-Scholes-Merton approach – merits and shortcomings”. Retrieved on 21th February 2014 from: http://www.essex.ac.uk/economics/documents/eesj/au13/ec372matei.pdf
