Practice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set?

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1 Derivatives (3 credits) Professor Michel Robe Practice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set? To help students with the material, eight practice sets with solutions shall be handed out. These sets contain mostly problems of my own design as well as a few carefully chosen, workedout end-of-chapter problems from Hull. None of these Practice Sets will be graded: the number of "points" for a question solely indicates its difficulty in terms of the number of minutes needed to provide an answer. Students are strongly encouraged to try hard to solve the practice sets and to use office hours to discuss any problems they may have doing so. The best self-test for a student of her/his command of the material is whether s/he can handle the questions of the relevant practice sets. The questions on the mid-term and final exams will cover the material covered in class. Their format, in particular, shall in large part reflect questions such as the numerical exercises solved in class and/or the questions in the practice sets. 1

2 Question 1 We are in A European call option on Netscape stock with strike price $50 matures in 1 year. Following developments in the Microsoft trial, Netscape s stock (then trading at $40) is expected to appreciate at a rate of 20% per annum. The standard deviation of that return is assessed at 30% per annum. The annualized, continuously compounded risk-free rate is 5%. (a) If you decide to carry out a binomial option pricing analysis by subdividing the 1-year time interval into two 6-month intervals, what is the risk-neutral probability of Netscape s stock going up every 6 months? (b) What would the value of Netscape s stock be after a year, if the stock value rose each period? (c) Using risk-neutral valuation, what is the value of the option today? (Hint) Do you need all of the information? (d) What would the price of this European option be, in the event that the market expects Netscape to pay out a $2 dividend in early December (we are in late May)? Question 2 (Bonus-type question) (a) Using data from Question 1, can you build a risk-free portfolio at t=0 without having to rebalance the portfolio at t=1/2? (Hint: At each node of the tree, describe the value of your risk-free portfolio: what is the hedge ratio ( ) at that node. Does the value of change over the life of the contract?) (b) If the investor finds himself in the down state at t=1/2, can he achieve the risk-free rate in any other way than by holding the risk-free asset from t=1/2 until t=t=1? Why, or why not? Question 3 Suppose that the Philadelphia Stock Exchange decides to list an American stock option with the same characteristics as the option in Question 1. (a) Would such a decision make sense if Netscape were not expected to pay out dividends or repurchase shares in the foreseeable future? Argue formally, and illustrate -- using the data from Question 1 -- any general principles you have used to answer the question. (b) What would the price of this American option be, in the event that the market expects Netscape to pay out a dividend of $5 (rather than $2) in December (we are in May)? 2

3 Derivatives (3 credits) Professor Michel Robe Practice Set #7: Solutions Question 1 We are in A European call option on Netscape stock with strike price $50 matures in 1 year. Following developments in the Microsoft trial, Netscape s stock (then trading at $40) is expected to appreciate at a rate of 20% per annum. The standard deviation of that return is assessed at 30% per annum. The annualized, continuously compounded risk-free rate is 5%. (a) If you decide to carry out a binomial option pricing analysis by subdividing the 1-year time interval into two 6-month intervals, what is the risk-neutral probability of Netscape s stock going up every six months? The risk-neutral probability of an up state, p, is given by: p = (e rt - d)/(u - d), where u = e t and d = e - t where t = 0.5 is the interval of time (one half of one year in this case) between two nodes in the tree. Therefore, since u = e x = and d = e - x = 0.809, we must have: p = (e 0.05 x 1/2 - d) / (u - d) = ( ) / ( ) = (b) What would the value of Netscape s stock be after a year, if the stock value rose each period? Suu = $40 x = $ (c) Using risk-neutral valuation, what is the value of the option today? (Hint) Do you need all of the information? As discussed in class, the rate of return on the underlying asset as well as the attitudes of investors towards risk are irrelevant in the risk-neutral valuation method (at least as long as the market is weak-form efficient). The following lattice can readily be established with bold and italic figures denoting, respectively, the stock and option prices in various periods and states of the world: 3

4 t=0 t=1/2 t= The payoffs at maturity (i.e., at t=t=1) are all of the form: Max[S T -X, 0], whereas the option values at earlier nodes are all of the type: e 0.05x1/2 (0.506 times option s up value in the next period times option s down value) For example, in the up node in period t=1/2, the call value is: f u = c u = e 0.05x1/2 (0.506 times $ times $0) = $5.50. (d) What would the price of this European call option be, in the event that the market expects Netscape to pay out a $2 dividend in early December (we are in late May)? From the dates given (early December vs. late May), we see that the dividend payout (or ex date) follows immediately the first tree node. Given a stock price drop of $2 just after time t=1/2 and assuming the stock s volatility remains the same ex-dividend, a recombining tree can be built from the ex-dividend price, S t = S t PV t (D). Since PV t=0 (D) = $1.95 and PV t=1 (D) = $2: S =47.03 S =58.13 S =38.05 S =38.05 S =30.78 S =24.91 S= S= S= S= S= S=

5 Question 2 (a) Using data from Question 1, can you build a risk-free portfolio at t=0 without having to rebalance the portfolio at t=1/2? (Hint: At each node of the tree, describe the value of your risk-free portfolio: what is the hedge ratio ( ) at that node. Does the value of change over the life of the contract?) In a nutshell, the answer is no. Formally, we argued in class that, because there are only two securities and there are three states of the world in t=t=1 (Suu, Sud=Sdu or Sdd), the markets are not complete but that they can be complete if, and only if, the risk-free portfolio is rebalanced in period 2 -- i.e., if the investor sells or buys some shares at t=1/2 (depending on whether the stock price went up or down from its original value at t=0, S). Put differently, in order to keep earning the risk-free rate by holding a combination of a certain number ( ) of shares and a short position in a single call option, the investor shall have to change at time t=1/2, depending on whether the state of the world at t=1/2 is the up state (Su) or the down state (Sd). To illustrate this, we shall build two trees: one with the value of the hedge ratio (D) at each node in t=0 and t=1/2, ƒ u ƒ d Su Sd and another tree with the value of the risk-free portfolio created using that particular hedge ratio for the realized value of S at the previous node: Su ƒ u or Sd ƒ d (i) The hedge-ratio tree ( values are in bold) shows the portfolio rebalancing in t=1/2: Whereas the investor buys 0.32 shares at t=0, he increases his holdings to 0.53 shares in the up state, and reduces them to nothing in the down state. 5

6 (ii) As a result, the investor can always guarantee a given risk-free pattern (in bold, are the values of the portfolio at the various points in time) (from Su) or (from Sd) Notice that, if the investor merely wants to invest from t=0 to t=1/2, he can readily earn the risk-free rate by buying 0.32 shares and shorting one call option. By doing so, that investor will have for sure $10.41 in t=1/2, an annualized return of 5% on the initial cost of that portfolio ($10.15). If the investor wants to earn the risk-free rate over the entire interval of time, then rebalancing is necessary. That is, in period 2, he will have to: (i) Get rid of all risky securities (shares and option) in t=1/2 in the down state. (ii) Buy an additional 0.21 shares at the price $49.45 at t=1/2 in the up state. What about the total rate of return until t=t=1? Well, there are 2 possibilities: (iii)if the share price went down from t=0 to t=1/2, then the investor liquidates all risky assets at t=1/2 for $10.41 which, reinvested for 6 months at the 5% risk-free rate, will yield $10.67 in both of the states that can be reached at t=t=1 from the Su node, i.e., in the states Sdd or Sud. (iv) if the share price went up from t=0 to t=1/2, then the investor borrows $10.41 at t=1/2 to buy 0.21 additional shares. In both states that can be reached at t=t=1 from the Su node (i.e., in the states Sdd or Sud), this strategy (net of the future value of the additional t=1/2 investment) will yield $ $10.41x e 0.05x1/2 = $ (b) If the investor finds himself in the down state at t=1/2, can he achieve the risk-free rate in any other way than by holding the risk-free asset from t=1/2 until t=t=1? Why, or why not? Starting from the downstate at t=1/2, the only way for the investor to earn the risk-free rate henceforth is to invest in the risk-free asset. Intuitively, this is because the option payoffs are 0 in both the Sdu and Sdd states in t=t=1 (i.e., the option can play no hedging role from that point onwards). 6

7 Question 3 Suppose that the Philadelphia Stock Exchange decides to list an American stock option with the same characteristics as the option in Question 1. (a) Would such a decision make sense if Netscape were not expected to pay out dividends or repurchase shares in the foreseeable future? Argue formally, and illustrate -- using the data from Question 1 -- any general principles you have used to answer the question. Nope. It is never optimal to exercise early an American option on a non-dividend-paying stock. In this particular case, the payoffs from early exercise at t=1/2 would be 0 in both states of the world and, hence, it is trivially not optimal not to exercise early. Thus, offering an American option on Netscape stock would make little sense. (b) What would the price of this American option be, in the event that the market expects Netscape to pay out a $5 dividend in early December (we are in late May)? This question differs from Q1(d) in two ways: (i) the option is American not European so early exercise is something we need to consider; (ii) the dividend is very large, so the method we used in Q1(d) to construct a recombining tree may not be appropriate in that the growth rate after the payout date may not be realistically the same as before the payout date (that is, when a company pays out $5 out of a $50 stock, it s a really big change in the company s life). Using the same methodology as in Q1(d) and as in Section 19.3 of Hull (7 th edition; 17.3 in the 6 th edition), we could (but arguably should not) construct a recombining tree with S. Specifically, from the dates given (early December vs. late May), we see that the dividend payout date (the ex-date ) follows immediately the earliest time when the option can be exercised. Given a stock price drop of $5 just after time t=1/2 and assuming the stock keeps appreciating at the same rate thereafter, a recombining tree can be built for the ex-dividend stock price, S t = S t PV t (D): S =43.43 S =53.70 S =35.13 S =35.16 S =28.41 S =22.99 Adding back PV t (D) at each note t when the dividend D has not yet been paid, we get: 7

8 S= S= S= S= S= S=22.99 By exercising early, the investor would therefore gain nothing: the option is still completely out of the money at t=1/2, regardless of the state of nature. The problem with this approach is that it assumes that the volatility remains the same after the dividend payment. In this context, one may consider assuming a greater percentage volatility (perhaps 10% greater volatility) from t=1 to t=2 than from t=0 to t=1. As discussed in class, Section 19.5 (Hull 7 th ; 17.5 in Hull 6 th ), all that would be needed is to re-estimate the tree parameters (u, d and p) for the second half of the tree, using this alternative volatility estimate. 8

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