代做金融作业 OPTION PRICING MODELS

OPTION PRICING MODELS: THE BLACK-SCHOLES-MERTON MODEL

代做金融作业 A stock is selling for $100 with a volatility of 40 percent. Consider a call option on the stock with an exercise price of 100 and an

1. 
   

   Consider the right-hand side of the Black-Scholes-Merton formula as consisting of the sum of two terms. Explain what each of those terms represents.  代做金融作业

   
   

The formula has two terms:

S₀N(d₁), is the present value of the expected value of the stock price, conditional on the option expiring in-the-money discounted at the risk-free rate.

The second term, , is the present value of the expected payout of the exercise price at the expiration. N(d₂) is the probability of exercise. All of this is based on the condition of risk neutrality, meaning that the probability distribution is based on a risk-free expected return on the stock.

2. Answer the following questions as they relate to volatilities.


3. Can implied volatilities be expected to vary for options on the same stock with the same exercise prices but different expirations?

Implied volatilities can indeed vary for options on the same stock with the same exercise price and different expirations. The volatility is supposed to be the volatility of the stock over the life of the option so it can indeed vary with a different time to expiration. The difference in volatilities is called the term structure of implied volatility.

1. 
   

   Can implied volatilities be expected to vary for options on the same stock with the same expiration but different exercise prices?

   
   

In theory implied volatilities should not vary for options on the same stock with the same expiration and different exercise prices. The volatility is referring to the volatility of the stock over a given period of time. There cannot be but one such volatility. In practice, however, there are multiple volatilities, which is often referred to as the volatility smile or skew.

1. Why and how are implied volatilities used to quote options prices?

Often option prices are quoted using the implied volatility. A given option price could be quoted by stating its implied volatility, while another option on the same stock could be quoted by stating a different implied volatility. It is assumed that the actual option price is then found by plugging into a model such as Black-Scholes-Merton. The purpose of quoting option prices this way is because it is much easier to see which are the more expensive options.

1. 
   

   Suppose that you subscribe to a service that gives you estimates of the theoretically correct volatilities of stocks. You note that the implied volatility of a particular option is substantially higher than the theoretical volatility. What action should you take?

   
   

If you accept the "theoretically correct" standard deviation as the true volatility, then the market price of the option is implying a higher volatility of the stock than is reasonable. That is, the implied volatility obtained by setting the Black-Scholes-Merton price equal to the market price is higher than it should be. This means that the market price is too high. You would consider selling the overpriced option, perhaps in conjunction with the purchase of the stock.

3. Explain each of the following concepts as they relate to call options.


4. Delta

The delta is the change in the call price for a given change in the stock price. Strictly speaking the delta applies only when the stock price changes by a very small amount. The delta also gives the hedge ratio, which tells how many shares of stock must be held (or sold short) to hedge a given short (or long) position in calls. The delta is between zero and one and, as expiration approaches, converges to one if the option is in-the-money and zero if the option is out-of-the-money.

1. 
   

   Gamma  代做金融作业

   
   

Gamma is the change in delta for a given (again very small) change in the stock price. The gamma measures the risk involved in not adjusting the hedge ratio to equal the delta. The gamma will be large when the option is at-the-money and nearly zero when the option is deep in- or out-of-the-money.

1. Rho

Rho measures the change in the option price when the risk-free rate changes. The relationship is nearly linear and is fairly weak. In other words, the call price is not very sensitive to the risk-free rate.

1. 
   

   Vega

   
   

Vega (also called kappa and lambda) measures the change in the option price for a change in the volatility. The relationship is nearly linear when the option is at-the-money. The option price is very sensitive to the volatility.

1. Theta

Theta is the relationship between the option price and the time to expiration. For European calls, the theta is negative meaning that the option price will fall as expiration approaches.

4. What factors contribute to the difficulty of making a delta hedge be truly risk-free?

The primary reason is the inability to trade at no cost over a very small time interval. Delta hedging assumes only very small stock price moves. If the stock price makes a sufficiently large move, the delta changes too quickly. To manage this risk, one must hedge the gamma. These types of large moves are more prone to occur over longer time intervals. Thus, if one cannot trade frequently, large stock price moves can occur between trades, and this will make it harder to maintain the delta neutral position. In addition transaction costs make it difficult to delta hedge perfectly. Transaction costs discourage frequent trading and make it impossible to actually earn the risk-free rate.  代做金融作业

5. Estimate the historical volatility of the stock for use in the Black-Scholes-Merton model Following is the sequence of daily prices on the stock for the preceding month of June:. (Ignore dividends on the stock)

A stock is selling for $100 with a volatility of 40 percent. Consider a call option on the stock with an exercise price of 100 and an expiration of one year. The risk-free rate is 4.5 percent. Let the call be selling for its Black-Scholes-Merton value. You construct a delta-hedged position involving the sale of 10,000 calls and the purchase of an appropriate number of shares. You can buy and sell shares and calls only in whole numbers. At the end of the next day, the stock is at $99. You then adjust your position accordingly to maintain the delta hedge. The following day the stock closes at $102.

In all cases use the spreadsheet BSMbin8e.xls to price the call.

Plugging into the Black-Scholes-Merton model

S₀ = 100, X = 100, r_c = 0.045, T = 1, s = .40, we obtain C = 17.80 and a delta of 0.6227.

Sell 10,000 calls and buy 6,227 shares. The value of the portfolio is now

V = 6,227(100) – 10,000(17.80) = 444,700.

1. 
   

   Compare the amount of money you end up with to the amount you would have had if you had

   
   

invested the money in a risk-free bond. Explain why the target was or was not achieved.

At the end of Day 1, S₁ = 99, T = 364/365 = 0.9973. Plugging into the Black-Scholes-Merton model, we obtain C = 17.1559. Our portfolio is now worth

V₁ = 6,227(99) – 10,000(17.1559) = 444,914.

The new delta from the Black-Scholes-Merton model would be 0.6129. So we need 6,129 shares. We sell 6,227 – 6,129 = 98, raising 98(99) = 9,702, which we invest in risk-free bonds.

Now at the end of Day 2, S₂ = 102, T = 363/365 = 0.9945. Plugging into the Black-Scholes-Merton model we obtain 19.0108. Our portfolio is now worth

V₂ = 6,129(102) – 10,000(19.0108) + 9,702e^0.045(1/365) = 444,753.

The amount we should have had is the initial amount invested plus two days interest:

444,700e^0.045(2/365) = 444,810.

The target was not achieved because a delta hedge works well only for very small changes in the stock price. The discrepancy between the amount we obtained and the target is due to the gamma risk, which reflects large stock price moves.

1. Now add another option, one on the same stock with an exercise price of 105 and the

same expiration. Reconstruct the problem by delta and gamma hedging. Explain why the target was or was not achieved.

Going back to Day 0, on the original option, we would obtain a gamma of 0.0095. Now we add a new option. Its Black-Scholes-Merton value is 15.6929, its delta is 0.5756 and its gamma is 0.0098. Remember that we have 10,000 of the first option. The number of the second is

10,000(0.0095/0.0098) = 9,694.

The number of shares of stock we need is

10,000(0.6227 – (0.0095/0.0098).5756) = 647.

So now our portfolio value is

V₁ = 647(100) – 10,000(17.80) + 9,694(15.6929) = 38,827.

Now on Day 1, the stock is at 99. From the Black-Scholes-Merton formula, the first call price is 17.1559, its delta is 0.6129 and its gamma is 0.0097. The second call is at 15.0958, its delta is 0.5654 and its gamma is 0.0100. The value of the portfolio is  代做金融作业

V₂ = 647(99) – 10,000(17.1559) + 9,694(15.0958) = 38,833.

Now we revise the number of the second call to

10,000(0.0097/0.0100) = 9,700.

We revise the number of shares to

10,000(0.6129 – (0.0097/.0100).5654) = 645.

So we need to buy 9,700 – 9,694 = 6 more of the second call and sell 647 – 645 = 2 shares. This will generate

2(99) – 6(15.0958) = 107, which we invest in bonds.

Now on Day 2, the stock is at 102. The Black-Scholes-Merton value is 19.0108 for the first call and 16.8092 for the second. The portfolio is now worth

V₂ = 644(102) – 10,000(19.0108) + 9,700(16.8092) + 107e^0.045(1/365) = 38,838.

The amount we should have is 38,827 plus two days interest:

38,827e^0.045(2/365) = 38,837.

We see that we have nearly a perfect hedge. The target was achieved because gamma hedging is able to eliminate risk caused by the larger price movement better than delta hedging alone.

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