Question

(Derivatives & Risk Management - Black-Sholes Option Pricing Model)

1. Discuss what is represented by the first and second terms of the B-S model. This should include the individual components of each term, what it represents, how it relates to the other terms, and how the two terms jointly reflect the equilibrium value of a call option.

2. Why is the variable ????? important in the B-S model? How does ????? relate to the expected range of values for the underlying (in both a qualitative and quantitative sense)

3. The instantaneous volatility of the B-S model: historical vs. implied

4. Explain how the delta (i.e., the N(d1) variable) can be used to estimate a hedge ratio and create a hedge position as part of a replicating portfolio. (Note that this refers to the B-S option pricing model.)

Extra:

-The replicating portfolio should be approximately delta neutral (i.e., the delta should be close to zero), but the value of the position changes as the price of the underlying moves away from the strike price. Explain why this is so, and discuss whether this has implications for options pricing models.

Answer 1

The Black-Scholes Model (also called Black-Scholes-Merton) is a widely used model for option pricing. It is used to calculate the theoretical value of European style options using current stock prices, expected dividends, the option's strike price, expected interest rates, time to expiration and expected volatility.

The Black-Scholes model makes certain assumptions:

• The option is European and can only be exercised at expiration date.
• Dividends are not paid out during the life of the option.
• Markets are efficient hence there is no arbitrage opportunity (that is, market movements cannot be predicted).
• It is possible to borrow and lend any amount, even fractional, of cash at the riskless rate.
• It is possible to buy and sell any amount, even fractional, of the stock (this includes short selling).
• There are no transaction costs of fees in buying the European option (that is, frictionless market).
• The risk-free rate and volatility of the underlying are known and constant.
• The returns on the underlying are normally distributed.

The model is essentially divided into two parts:

1. The first part, SN(d1), multiplies the price by the change in the call premium in relation to a change in the underlying price. This part of the formula shows the expected benefit of purchasing the underlying outright.
2. The second part, N(d2)Ke^(-rt), provides the current value of paying the exercise price upon expiration (since, the Black-Scholes model applies to European options that can be exercised only on expiration day). The value of the option is calculated by taking the difference between the two parts, as shown in the equation.

Interpretation

Rewrite the Black-Scholes formula as

• If the call option is exercised at the maturity date then the holder gets the stock worth S(T) but has to pay the strike price K. But this exchange takes place only if the call finishes in the money. Thus S(0) e^(rT) N(d1) represents the future value of the underlying asset conditional on the end stock value S(T) being greater than the strike price K.
• The second term in the brackets KN(d2) is the value of the known payment K times the probability that the strike price will be paid N(d2). The terms inside the brackets are discounted by the risk-free rate r to bring the payments into present value terms. Thus the evaluation inside the brackets is made using the risk-neutral or martingale probabilities. The term N(d2) represents the probability that the call finishes in the money where d2 is also evaluated using the risk-free rate.

In an options trade, both sides of the transaction make a bet on the volatility of the underlying security. While there are several methods for measuring volatility, options traders generally work with two metrics: Historical volatility measures past trading ranges of underlying securities and indexes, while implied volatility gauges expectations for future volatility, which are expressed in options premiums. The combination of these metrics has a direct influence on options' prices, specifically the component of premiums referred to as time value, which often fluctuates with the degree of volatility. Periods when these measurements indicate high volatility tend to benefit options sellers, while low volatility readings benefit buyers.

The shortcomings of the Black-Scholes method have led some to place more importance on historical volatility as opposed to implied volatility. However, using historical volatility also has some drawbacks. Volatility shifts as markets go through different regimes. Thus, historical volatility may not be an accurate measure of future volatility.

Historical Volatility : Historical volatility is the realized volatility of the underlying asset over a previous time period. It is determined by measuring the standard deviation of the underlying asset from the mean during that time period. The standard deviation is a statistical measure of the variability of price changes from the mean price change.

• Also referred to as statistical volatility, historical volatility gauges the fluctuations of underlying securities by measuring price changes over predetermined periods of time.
• This calculation may be based on intraday changes but most often measures movements based on the change from one closing price to the next. Depending on the intended duration of the options trade, historical volatility can be measured in increments ranging from 10 to 180 trading days.
• By comparing the percentage changes over longer periods of time, investors can gain insights on relative values for the intended time frames of their options trades. For example, if the average historical volatility is 25% over 180 days and the reading for the preceding 10 days is 45%, a stock is trading with higher-than-normal volatility
• Because historical volatility measures past metrics, options traders tend to combine the data with implied volatility, which takes forward-looking readings on options premiums at the time of the trade.

Implied Volatility : the implied volatility determined by the Black-Scholes method, as it is based on the actual volatility of the underlying asset.

• By gauging significant imbalances in supply and demand, implied volatility represents the expected fluctuations of an underlying stock or index over a specific time frame.
• Options premiums are directly correlated with these expectations, rising in price when either excess demand or supply is evident and declining in periods of equilibrium.
• The level of supply and demand, which drives implied volatility metrics, can be affected by a variety of factors ranging from market-wide events to news related directly to a single company. For example, if several Wall Street analysts make forecasts three days prior to a quarterly earnings report that a company is going to soundly beat expected earnings, implied volatility and options premiums could increase substantially in the few days preceding the report. Once the earnings are reported, implied volatility is likely to decline in the absence of a subsequent event to drive demand and volatility.