Risk Management

Solution Preview

---

Please see response attached (also below). I hope this helps and take care.

RESPONSE;

1. Black-Scholes options pricing model

Assumptions of the Black and Scholes Model:

1) The stock pays no dividends during the option's life
Most companies pay dividends to their shareholders, so this might seem a serious limitation to the model considering the observation that higher dividend yields elicit lower call premiums. A common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price.

2) European exercise terms are used
Foreign currency: European exercise terms dictate that the option can only be exercised on the expiration date. American exercise term allow the option to be exercised at any time during the life of the option, making American options more valuable due to their greater flexibility. This limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same.

3) Markets are efficient
This assumption suggests that people cannot consistently predict the direction of the market or an individual stock. The market operates continuously with share prices following a continuous Itô process. To understand what a continuous Itô process is, you must first know that a Markov process is "one where the observation in time period t depends only on the preceding observation." An Itô process is simply a Markov process in continuous time. If you were to draw a continuous process you would do so without picking the pen up from the piece of paper.

4) No commissions are charged
Usually market participants do have to pay a commission to buy or sell options. Even floor traders pay some kind of fee, but it is usually very small. The fees that Individual investor's pay is more substantial and can often distort the output of the model.

5) Interest rates remain constant and known
The Black and Scholes model uses the risk-free rate to represent this constant and known rate. In reality there is no such thing as the risk-free rate, but the discount rate on U.S. Government Treasury Bills with 30 days left until maturity is usually used to represent it. During periods of rapidly changing interest rates, these 30-day rates are often subject to change, thereby violating one of the assumptions of the model.
6) Returns are lognormally distributed
This assumption suggests, returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options. http://bradley.bradley.edu/~arr/bsm/pg04.html

Measures of Risk associated with The Black and Scholes Model:

Delta: Delta is a measure of the sensitivity the calculated option value has to small changes in the share price.

Gamma: Gamma is a measure of the calculated delta's sensitivity to small changes in share price.

Theta: Theta measures the calculated option value's sensitivity to small changes in time till maturity.

Vega: Vega measures the calculated option value's sensitivity to small changes in volatility.
Rho:

(See graphs at http://bradley.bradley.edu/~arr/bsm/pg04b.html)

Black-Scholes in practice

The use of the Black-Scholes formula is pervasive in the markets.

Example 1:

In fact the model has become such an integral part of market conventions that it is common practice for the implied volatility rather than the price of an instrument to be quoted. (All the parameters in the model other than the volatility - that is the time to maturity, the strike, the risk-free rate, and the current underlying price - are unequivocally observable. This means there is one-to-one relationship between the option price and the volatility.) Traders prefer to think in terms of volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on.
However, the Black-Scholes model cannot be modelling the real world exactly. If the Black-Scholes model held, then the implied volatility of an option on a particular stock would be constant, even as the strike and maturity varied, and roughly equal to the historic volatility. In practice, the volatility surface (the two-dimensional graph of implied volatility against strike and maturity) is not flat. In fact, in a typical market, the graph of strike against implied volatility for a fixed maturity is typically smile-shaped (see volatility smile). That is, at-the-money (the option for which the underlying price and ...