巴黎银行-2013-C9_solution

For the third period starting on 15th November 2012 and ending on 21st December 2012, To find the length of these periods in years, it’s convenient to use the function YEARFRAC in Excel, but remember to use the actual/actual day count basis. The stock price and call values as well as the hedge ratios at each node of the tree are presented in Table 3. First, g the value of the underlying asset S is computed at each node starting from time 0 (28 of August, 2012) where S = 31.12. For the first period, we have S(u) = S * u1 = 31.12 * 1.08 = 33.61; the value of S(u,d) = S(u) * d2 = 33.61 * 0.9 = 30.25, and so on. The call values on the tree are solved backward, i.e., starting from maturity when the value of the Call is C = Max[S – K, 0]. For example, the value of C(u,d,u) is Max[S(u,d,u) – K, 0] = Max[33.88 – 30, 0] = 3.88. Next, let’s take for example the calculation steps of C(u,d): we need to compute the hedging ratio Δ(u,d). Following technical document 1, ( ) , the value of C(u,d) is then ( )
For the first period starting on 28th August 2012 and ending on 30th September 2012, For the second period starting on 30th September 2012 and ending on 15th Novembng the hedge ratios and call values the same way for all nodes until you reach the first one at 28 August, 2012.
Table 3 S(u,u,u) C(u,u,u) S(u,u) Δ(u,u) C(u,u) S(u) Δ(u) C(u) 33.61 0.76 4.69 S(u,d) Δ(u,d) C(u,d) S Δ (hedge C (Call value) ratio) 31.12 0.6532 3.1463 S(d,u) Δ(d,u) C(d,u) S(d) Δ(d) C(d) 28.63 0.48 1.44 S(d,d) Δ(d,d) C(d,d) 25.77 0 0.00 S(d,d,d) C(d,d,d) 22.6753 0 31.49 0.6976 2.73 S(d,u,d) C(d,u,d) S(d,d,u) C(d,d,u) 27.71 0 28.86 0 30.25 0.53 2.009 S(u,d,d) C(u,d,d) S(d,u,u) C(d,u,u) 26.62 0 35.27 5.27 36.97 1 7.12 S(u,u,d) C(u,u,d) S(u,d,u) C(u,d,u) 32.53 2.5341 33.88 3.88 41.41 11.41
Question 2: The correct answer is presented in table 2: Table 2 28th of August : Δ (hedge ratio) 0.6532 C (Call value) 3.1463 15th of November C(u,d) 2.0090 Δ(d,u) 0.6976 21st of December S(d,d,d) 22.6753 C(u,u,d) 2.5341 Explanation: On the 28th of August, the price of the underlying asset is 31.12 million Aces. Following technical document 1 (binomial trees), we must construct the binomial tree of this Call over 3 periods (Table 3). Each period has its own up “u” and down “d” coefficients, and the length of periods must be in fractions of years. 28-Aug Period (in years) S (Start-up price) Risk-free rate 30-Sep 15-Nov 21-Dec 0.09016393 0.12568306 0.098361 31.12 K (Exercise price) 30 0.05 Period u (up coef) d (down coef) 1 1.08 0.92 2 1.1 0.9 3 1.12 0.88


Since the option (Call or Put) is an insurance against changes in the price of the underlying asset: it fixes the price at which the asset may be bought (Call) or sold (Put), to buy at a higher strike reduces the insurance (Call value), and to sell at a higher strike increases the insurance (Put value). On the other hand, a higher current value of the underlying assetreduces the probability of “the asset price to decrease lesser than the strike price”, so the Call value will be higher, and vice-versa. The more time is left till the maturity of the option, the higher is the likelihood that the option moves into the money (that exercising it will generate a profit) or further into the money. Thus, the option value is higher the longer is the time to maturity. Similarly, the higher is the asset’s volatility, the more likely it is that the option moves in the direction of becoming profitable or more profitable for the holder, as the price may make larger jumps in a shorter time period.
Case 9 - Solution / expected answers
Note: For extreme precision, all intermediate computations have been carried out with exact values. Figures of intermediate results in this solution may be rounded for simplicity of notation only. Final results are the only ones to be rounded according to the rules of the game. Part 1: Question 1: a. The correct answer is: A Call option gives the holder the right but not the obligation to buy the underlying asset. A Put option gives the holder the right but not the obligation to sell the underlying asset. A European option may be exercised only at the expiration date; while an American option may be exercised any time before the expiration date. These answers simply rephrase the definitions of call, put, European and American options as stated in technical document 1.