ACT370H1S - TEST 2 - MARCH 25, 2009 Write name and student number on each page. Write your solution for each question in the space provided. Do all calculations to at least 6 significant figures. The only aid allowed is a calculator. 1. A 3-period binomial tree is constructed. The continuously compounded risk-free rate < of interest per period is <, and you are given that / œ Þ per period. You are also given that? œ Þ and. œ Þ) for each period. The current stock price is 1000. For parts (a), (b) and (c) assume that there is no dividend. 3 (a) A European derivative that expires at the end of 3 periods pays 100 if the stock is at either node 9 or node 10 and pays 0 otherwise. Find the price of the derivative at time 0. 3 (b) It is possible to replicate the payoff at time 3 in part (a) using a combination (some short some long) of European put options expiring at time 3 with strike prices of 1000 and 600. Find the combination. (c) Consider a 3-period American put option with a strike price of 1000. The option can be exercised at any node at time 1, 2 or 3. 2 (i) Find the price of the option at time 0. 2 (ii) Find the replicating portfolio at time 0. 2 (iii) Indicate if and where it would be optimal to exercise the option before expiry. 3 (d) Suppose that in part (c) there is a dividend of 100 payable at time 2.5. A 3-period tree is constructed using the prepaid forward price of the stock at time 0 (which is Þ&<!!!!!/ œ *Þ! at time 0), and the pv of the dividend is added back to the value of the stock prior to time 2.5. Find the price of the put option with strike price 1000 and indicate if and where it would be optimal to exercise the option before expiry. BONUS: Continuing part (d) find the maximum value of the dividend at time 2.5 that would 5 result in it being optimal to exercise the option (at some node) at time 2.
< 2. A two-period binomial model has / œþ,?œþß.œþ). The stock price at time 0 is 100, so the tree of stock prices is Time 0 1 2 144 120 100 96 80 64 A non-standard European derivative expiring at time 2 is defined as follows. The payoff at time 2 is Ð\!Ñ, where \ is the average value of the three stock prices that occurred on the path taken through the tree, so each path will have its own average. 3 (a) Find the (no-arbitrage) value of the derivative at time 0. 2 (b) Find the replicating portfolio = of the derivative at time 0 and at node 120. 2 (c) Verify that the self-financincg strategy is valid from node 100 (starting node) to node 120.
3. The cost today to buy one US dollar is $1.30 in Canadian funds. The continuously compounded risk free rate in Canada is 2%. The volatility of the US dollar is 5 œþ!. The risk free rate of interest in the US is 0. The Black-Scholes option pricing formula is used to find the price of a 3-month (.25 year) European Call option to purchase a US dollar at a strike price of $1.25 in Canadian funds. 5 (a) Find the price of the option. (b) Find the following option Greeks `G ` G `G 3 (i) Delta œ (ii) Gamma œ (iii) Vega œ `W `W ` 5
ACT370H1 S - TEST 1 SOLUTIONS - MARCH 19, 2008 Þ Þ) 1.(a) ;œþ Þ) œþ(&. Derivative value at time 0 is $< $ / Ò$;Ð ;Ñ Ð ;Ñ ÓÐ!!Ñ œ Þ(%. (b) + units of put-1000 and, units of put '!! Ð!!! (')Ñ+ œ!! p + œ Þ%$! on put-1000 Ð!!! &Ñ+ Ð'!! &Ñ, œ!! p, œ Þ&$* (short on put-600). (c) The tree of stock prices is 1728 1440 Put=0 1200 Put=0 1152 1000 Put=11.98 960 Put=0 Put=53.62 800 Put=52.73 768 Put=200 640 Put=232 Put=360 512 Put=488 The risk-neutral probabilities are ;œþ(&ß ;œþ&. The expected present value of the put at node 640 is Þ ÒÐÞ(&ÑÐ$Ñ ÐÞ&ÑÐ%))ÑÓ œ '*Þ!*, the exercise value at that node is!!! '%! œ $'! p early exercise at node 640. The expected present value of the put at node 960 is Þ ÐÞ&ÑÐ$Ñ œ &Þ($, the exercise value at that node is!!! *'! œ %! p no early exercise at node 960. The expected present value of the put at node 800 is Þ ÒÐÞ(&ÑÐ&Þ($Ñ ÐÞ&ÑÐ$'!ÑÓ œ (Þ((, the exercise value at that node is!!! )!! œ!! p early exercise at node )!!. The expected present value of the put at node 1200 is Þ ÐÞ&ÑÐ&Þ($Ñ œ Þ*), the exercise value at that node is 0 p no early exercise at node 1200. The expected present value of the put at node 1000 is ÒÐÞ(&ÑÐÞ*)Ñ ÐÞ&ÑÐ!!ÑÓ œ &$Þ'. Þ
Þ*)!! The replicating portfolio at time 0 consists of!! )!! œ Þ%( shares of stock (short ÐÞÑÐ!!Ñ ÐÞ)ÑÐÞ*)Ñ.47 shares) and risk-free investing of œ &$Þ'(. Þ Þ Þ) (d) The tree of prepaid stock prices (with dividend added back) is 1591.84 1326.53 0 +95.35 = 1421.88 1105.44 Put=0 1061.22 +86.68 = 1192.12 0 921.20 Put=15.11 884.35 +78.80 736.96 +95.35 = 979.70 =1000 +86.68 = 823.64 Put=66.48 707.48 Put=50.38 Put=176.36 589.57 Put=292.52 +95.35 = 684.92 Put=319.51 471.66 Put=528.34 The risk-neutral probabilities are ;œþ(&ß ;œþ&. The expected present value of the put at node 589.57 is Þ ÒÐÞ(&ÑÐ*Þ&Ñ ÐÞ&ÑÐ&)Þ$%ÑÓ œ $*Þ&, the exercise value at that node is!!! ')%Þ* œ $&Þ!) p no early exercise. The expected present value of the put at node 884.35 is Þ ÐÞ&ÑÐ*Þ&Ñ œ ''Þ%), the exercise value at that node is!!! *(*Þ(! œ!þ$! p early exercise. The expected present value of the put at node 736.96 is Þ ÒÐÞ(&ÑÐ''Þ%)Ñ ÐÞ&ÑÐ$*Þ&ÑÓ œ (Þ*%, the exercise value at that node is!!! )$Þ'% œ ('Þ$' p early exercise. The expected present value of the put at node 1105.44 is ÐÞ&ÑÐ''Þ%)Ñ œ &Þ. No early exercise Þ The expected present value of the put at node 912.20 is Þ ÒÐÞ(&ÑÐ&ÞÑ ÐÞ&ÑÐ('Þ$'ÑÓ œ &!Þ$). No early exercise.
BONUS: With dividend H, the prepaid stock price at the bottom two nodes at time 3 are H H $ Ð!!! ÞÞ& ÑÐÞ)Ñ ÐÞÑ and Ð!!! ÞÞ& ÑÐÞ)Ñ. The expected present value of the option at the previous node would be (assuming that these are both!!!) H Þ ÐÒ!!! Ð!!! ÞÞ& ÑÐÞ)Ñ ÐÞÑÓÐÞ(&Ñ H $ Ò!!! Ð!!! ÞÞ& ÑÐÞ)Ñ ÓÐÞ&ÑÑ œ '* Þ&!%H. The stock price, including dividend, at that node is H H Ð!!! ÞÞ& ÑÐÞ)Ñ Þ Þ& œ '%! Þ%%*H, so early exercise has a payoff of!!! Ð'%! Þ%%*HÑ œ $'! Þ%%*H. Early exercise will be optimal if $'! Þ%%*H '* Þ&!%H, or equivalently, if H *&Þ&. 2. The payoffs at time 2 are 144 11.33 120 100 96 0 (upper and lower) 80 64 0 (a) Risk neutral probs are ;œþ(&ß ;œþ& The expected pv at time is Þ$$ÐÞ(&Ñ œ &Þ( Þ (b) The derivative value at node 120 is Þ ÒÞ$$ÐÞ(&ÑÓ œ (Þ(& and the derivative value at node 80 is 0, since the average will be 0 for the two lower paths. The replicating portfolio at time 0 consists of (Þ(&! ÐÞÑÐ!Ñ ÐÞ)ÑÐ(Þ(&Ñ! )! œ Þ*$ shares of stock and Þ Þ Þ) œ %Þ!% in risk free investing (borrowing 14.04). At node 120 the replicating portfolio consists of Þ$$! ÐÞÑÐ!Ñ ÐÞ)ÑÐÞ$$Ñ %% *' œ Þ$' shares of stock and Þ Þ Þ) œ!þ'! of investing (borrowing 20.60). (c) The self-financing strategy at node 120 is to borrow 5.16 and buy.043 shares of stock to have a total of.236 shares. The loan amount is %Þ!%ÐÞÑ &Þ' œ!þ'.
X 3.(a) G œ W/ $ <X FÐ. Ñ O/ FÐ. Ñ W œ Þ$! (today's value of US$), $ œ! (US interest rate is dividend rate on asset) <œþ!ß XœÞ&ß 5 œþß OœÞ& 68ÐÞ$ÎÞ&Ñ ÒÞ! ÐÞÑ ÓÐÞ&Ñ. œ È œ Þ%* ß. œ. 5 È X œ Þ$*. ÐÞÑ Þ& Þ!ÐÞ&Ñ. G œ Þ$ FÐÞ%*Ñ Þ&/ FÐÞ$*Ñ œ ÐÞ$ÑÐÞ'))Ñ Þ%$)ÐÞ'&Ñ œ Þ!)$ (b)? œ FÐ. Ñ œ FÐÞ%*Ñ œ Þ')) 9Ð. Ñ 9ÐÞ%*Ñ ÐÞ%*Ñ Î / Î È 1 W5ÈX ÐÞ$ÑÐÞÑÈÞ& ÐÞ$ÑÐÞÑÈÞ& > œ œ œ œ Þ( Vega œw9ð.ñè ÐÞ%*Ñ Î X œþ$ð/ Î È 1ÑÈÞ&œÞ$! (,0023 per.01 change in 5)