Question

Question 1 a. A stock price is currently $30. It is known that at the end of two months it will be either $33 or $27. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a two-month European put option with a strike price of $31? b. What is meant by the delta of a stock option? A stock price is currently $100. Over each of the next two three-month periods it is expected to go up by 8% or down by 7%. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a six-month European call option with a strike price of $952 I (3+2+5= 10 marks) C.

Answer 1

2a)

Nd₂ = Cumulative area unde the houmal Solving the sums using Black Scholes Model Formulai Ve = Po Na expl Ndz Where Ve Value of call option Po= Current Stock Price t = time remain RRE= Risk free rate of interest aining No, = Cumulative area under the normal distribution curve to di to maturity forection distribution curve to dz Strike (Exercise) price of the option Volatility (standard deviation) of exchange rate In the given problem we have the following values :- Po $73, x = $70, t= 6months = 0. 5 years 0=40%pa, Rap - 10% pa = 0.4 First we calculate d, & de :- d = ln / + [kers +(0.5)(6)]+ Natural logarithm d₂ = dort dl, = ln 6) + ((0.1) +(0.5)(0:47]0.5 (0.4) 10.5 = 0.0419 + [(0:1) +0.08]0.5 0.2828 0-0419 + 0.09 0.2828 0.1319 0.2828 = 0.4664 de ad -ort = 0.4664 -(0.41o.s = 0.4664 -0.2828 = 0.1836

There are several assumptions underlying the Black-Scholes model:

1) Constant volatility. The most significant assumption is that volatility, a measure of how much a stock can be expected to move in the near-term, is a constant over time. While volatility can be relatively constant in very short term, it is never constant in longer term. Some advanced option valuation models substitute Black-Schole's constant volatility with stochastic-process generated estimates.

2) Efficient markets. This assumption of the Black-Scholes model suggests that people cannot consistently predict the direction of the market or an individual stock. The Black-Scholes model assumes stocks move in a manner referred to as a random walk. Random walk means that at any given moment in time, the price of the underlying stock can go up or down with the same probability. The price of a stock in time t+1 is independent from the price in time t.

d, ~ 0.50 d, 20.20 Fox From Normal Distribution table we we get :- = 0.6915 (0.50 2 N60.26 = 0.5793 Vc = (73) (0.6915) - 70 x(0.5793) (0.1) (0.5) е = 50.4795 70 *10.5793). 1.0513 = 50.4795- 38.5722 = 11.9073

3) No dividends. Another assumption is that the underlying stock does not pay dividends during the option's life. In the real world, most companies pay dividends to their share holders. The basic Black-Scholes model was later adjusted for dividends, so there is a workaround for this. This assumption relates to the basic Black-Scholes formula. A common way of adjusting the Black-Scholes model for dividends is to subtract the discounted value of a future dividend from the stock price.

4) Interest rates constant and known. The same like with the volatility, interest rates are also assumed to be constant in the Black-Scholes model. The Black-Scholes model uses the risk-free rate to represent this constant and known rate. In the real world, there is no such thing as a risk-free rate, but it is possible to use the U.S. Government Treasury Bills 30-day rate since the U. S. government is deemed to be credible enough. However, these treasury rates can change in times of increased volatility.

5) Lognormally distributed returns. The Black-Scholes model assumes that returns on the underlying stock are normally distributed. This assumption is reasonable in the real world.

6) European-style options. The Black-Scholes model assumes European-style options which can only be exercised on the expiration date. American-style options can be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility.

7) No commissions and transaction costs. The Black-Scholes model assumes that there are no fees for buying and selling options and stocks and no barriers to trading.
8) Liquidity. The Black-Scholes model assumes that markets are perfectly liquid and it is possible to purchase or sell any amount of stock or options or their fractions at any given time.

3b) Advantages of future options over spot options:

1.) Futures are great for trading certain investments

Futures may not be the best way to trade stocks, for instance, but they are a great way to trade specific investments such as commodities, currencies, and indexes. Their standardized features and very high levels of leverage make them particularly useful for the risk-tolerant retail investor. The high leverage allows those investors to participate in markets to which they might not have had access otherwise.

2.) Fixed upfront trading costs

The margin requirements for major commodity and currency futures are well-known because they have been relatively unchanged for years. Margin requirements may be temporarily raised when an asset is particularly volatile, but in most cases, they are unchanged from one year to the next. This means a trader knows in advance how much has to be put up as initial margin. On the other hand, the option premium paid by an option buyer can vary significantly, depending on the volatility of the underlying asset and broad market. The more volatile the underlying or the broad market, the higher the premium paid by the option buyer.

3.) No time decay

This is a substantial advantage of futures over options. Options are wasting assets, which means their value declines over time—a phenomenon known as time decay. A number of factors influence the time decay of an option, one of the most important being time to expiration. An options trader has to pay attention to time decay because it can severely erode the profitability of an option position or turn a winning position into a losing one. Futures, on the other hand, do not have to contend with time decay.

4.) Liquidity

This is another major advantage of futures over options. Most futures markets are very deep and liquid, especially in the most commonly traded commodities, currencies, and indexes. This gives rise to narrow bid-ask spreads and reassures traders they can enter and exit positions when required. Options, on the other hand, may not always have sufficient liquidity, especially for options that are well away from the strike price or expire well into the future.

5.) Pricing is easier to understand

Futures pricing is intuitively easy to understand. Under the cost-of-carry pricing model, the futures price should be the same as the current spot price plus the cost of carrying (or storing) the underlying asset until the maturity of the futures contract. If the spot and futures prices are out of alignment, arbitrage activity would occur and rectify the imbalance. Option pricing, on the other hand, is generally based on the Black-Scholes Model, which uses a number of inputs and is notoriously difficult for the average investor to understand.

3c) Vega is the measurement of an option's price sensitivity to changes in the volatility of the underlying asset. Vega represents the amount that an option contract's price changes in reaction to a 1% change in the implied volatility of the underlying asset.