QF101 Solutions of Week 12 Tutorial Questions Term /2018

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1 QF0 Solutions of Week 2 Tutorial Questions Term 207/208 Answer. of Problem The main idea is that when buying selling the base currency, buy sell at the ASK BID price. The other less obvious idea is that when buying the quote currency, you buy at /BID price, and when you sell the quote currency, you sell at /ASK price. In the case of EUR/USD, euro is the base currency, since we are selling it, we obtain mil.2373 dollars. In the case of USD/JPY, the greenback is the base currency, and so we obtain mil yens. In the case of GBP/JPY, Japanese yen is the quote currency. By selling the quote currency, you obtain mil /3.62 pounds. In the case of GBP/USD, the Sterling is the base currency, and you get mil / dollars. Finallly, for EUR/USD, the greenback is the quote currency, and the amount of euros obtained is e mil / /.2437 = e 997, The P&L is e 997,73.84 e,000, = e 2, But this is the academic answer. The answer in practice real-world is to take each trade one by one.. e,000, = $,23, $,23, = 3,845, , 845, = 865, , = $,2, $, 2, = e 997,73.84 It just happens that the rounding up and rounding down are offsetting each other and this answer is identical to the academic answer. Answer. of Problem 2 A Set up the payoff replication strategy: 26x + e 0.02 y = 00 24x + e 0.02 y = 0 By solving these linear equations, we obtain x = 45 shares, and the notional amount and the long positive or short negative direction of the bond is y = e = $, Borrow $,048.8 issue a bond and buy 45 shares at time 0. c Christopher Ting Quantitative Finance Group Page of 6

2 QF0 Solutions of Week 2 Tutorial Questions Term 207/208 B The cost of replication at time 0 is the present value or the price of this derivative $, = $76.9. C Let p be the risk-neutral probability for the stock to go up. $00p + $0 p = $76.9e 0.02 The solution is p = 24.75% for the stock to go down. Another method that yields the same result is $26p + $24 p = $25e These two methods are based on the first principle of QF. Answer. of Problem 3 A To find the level of the term structure, set T = 0. We need to prove that e T/ lim T 0 T =. Apply L Hôpital s rule, as it has the 0/0 indeterminate form: e T/ lim T 0 T Therefore, the two terms in the parenthesis lim T 0 Hence, the level of the yield curve is r. e T/ T = lim e T/ T 0 = e T/ = = 0. B The proof is a straightforward differentiation of Y T with respect to T and a re-arrangement of terms. [ Y T = β l β s e T/ T + ] T 2 e T/. C In considering the limit T 0, we need to focus on two terms in Y T : e T/ T e T/ T 2, c Christopher Ting Quantitative Finance Group Page 2 of 6

3 QF0 Solutions of Week 2 Tutorial Questions Term 207/208 which can be written as the 0/0 indeterminate form when T 0: ft := T e T/ e T/ L Hôpital s rule is applicable, and we have T 2 e T/ T lim ft = lim e T/ e T/ T 0 T 0 2T = lim T 0 e T/ 2 = 2. Therefore, lim Y T = β s T 0 = β l β s 2 2. D It is clear that as T, e T/ 0, and /T 0. Hence lim T Y T = β l. This result allows us to interpret the parameter β l as the gradient of the long end part of the yield curve. Answer. of Problem 4 The annual zero rates z i are %,.2%,.3%, and.4% for i =, 2, 3, 4. Each i is a half year, i.e. i = 2 is a year. The discount factors are DF = DF 2 = DF 3 = DF 4 = = = = = A Let n = 2. we have Therefore the annualized swap rate is.994%. B We need to solve for Hence K = DF 2 DF + DF 2 = %. c 2 DF + c 2 DF 2 + c 2 DF 3 + c 2 DF 4 + DF 4 =. c 2 = DF 4 4 i= DF i Substituting in the values for the discount factors, we obtain c =.398% as the answer for the par rate corresponding to the tenor of 2 years. c Christopher Ting Quantitative Finance Group Page 3 of 6

4 QF0 Solutions of Week 2 Tutorial Questions Term 207/208 Answer. of Problem 5 Take the data from Slide 28 of the lesson on Model-Free VIX. Given that the relevant annual risk-free rate is 0.5%, S t = 95.57, and T t = 3/365. A The call struck at 90 s midpoint is c t 90 = 2 $ $8.30 = $6.0. From the put-call parity, p t 90 = $6.0 + $90e /365 $95.57 = $0.49. Likewise From the put-call parity, we obtain c t 92.5 = 2 $.70 + $6.0 = $3.90 p t 92.5 = $ $92.5e /365 $95.57 = $0.79. Note that p t 92.5 > p t 90, which is the monotonicity property of the price curve for a put. B The midpoint of the ITM put option struck at 00 is p t 00 = 2 $ $6.50 = $4.50 Applying put-call parity, c t 00 = $ $95.57 $00e /365 = $0.. Likewise, the midpoint of the ITM put option struck at 02.5 is p t 02.5 = 2 $ $9.0 = $6.90 Applying the put-call parity, c t 02.5 = $ $95.57 $02.5e /365 = $0.0. Note that c t 02.5 < c t 00, which is the monotonicity property of the price curve for a call. c Christopher Ting Quantitative Finance Group Page 4 of 6

5 QF0 Solutions of Week 2 Tutorial Questions Term 207/208 Answer. of Problem 6 The static replication of a payoff function fs at maturity is fs T = fλ + f λs T λ + λ 0 f KK S T + dk + λ f KS T K + dk. A Note that in this context, λ is a known quantity at time 0. Therefore, the first term fλ is a sure cash flow and thus no risk. So its PV at time 0 is to discount it by e r0t where r 0 is the risk-free rate. The second term is the payoff of a long forward position. The number of forward contracts is f λ. Each contract can be replicated by a long position in a call, and a short position in a put. Both options have the same strike price of λ. The integrand in the third term is the payoff of a portfolio of put options at expiration T. Each put option at time 0 has the price of p 0 K. The number of contracts is f K for p 0 K struck at K. The integrand in the last term is the payoff of a portfolio of call options at expiration T. Its price PV at time 0 is c 0 K. The number of contracts is f K for c 0 K struck at K. Static replication of these European-style contracts means that no further transaction is needed once the discount bond and the option positions are established. All these replications follow the first principle of QF. B The payoff function is fs T = S0 2 ST 2, with S T being the variable and S 0 a constant since it is observable at time 0. Now, f S T = 2 S0 2 S T, and f S T = 2 S0 2. We let λ = F 0, the forward price, which is S 0 e r 0T. Given r 0 = % and S 0 = $6, F 0 = 0 e r 0T = $6e 0.0 = $6.06. Therefore, the price of the discount bond is, given T = year, e r 0T F0 S 0 2 = e r 0T e 2r 0T = e r 0T = e r 0 = $e 0.0 = $.0 Next, the number of forward contract is f F 0 = 2 S0 2 F 0 = 2er 0T = 2.0 =.0 S Theoretically, by put-call parity, c 0 F 0 p 0 F 0 = 0 if the strike price is exactly equal to the forward price. If that is the case, the forward contract s PV is zero. However, it is more likely that F 0 does c Christopher Ting Quantitative Finance Group Page 5 of 6

6 QF0 Solutions of Week 2 Tutorial Questions Term 207/208 not equal to any strike price of an option chain. We choose the nearest c 0 6 p 0 6 to replicate a contract of forward. Hence the cost is.0 c0 6 p We need to discretize the integral as a Riemann sum. The strike interval is K = 0.25, and 2 S 2 0 F0 0 p 0 K dk 2 S 2 0 I p 0 K i K, i= where K = 4 and K I = 6. Therefore, the PV involving put options is, since K = 0.25, Likewise, the PV involving call options is p0 4 + p p p S 2 0 F 0 c 0 K dk 2 S 2 0 J c 0 K i K, i=i where K I = 6 and K J = 8. Accordingly, the PV is c0 6 + c c c The minimum price total PV is, for one contract of obtaining the payoff ST PV = c0 6 p p0 4 + p p p c c c The bank will add a fee to this minimum PV. S 0 2, Remark: In practice, since there is no strike corresponding to F 0 = 6.06, the quant need to choose between K = 6 or K = Here K = 6 is chosen. In practice, quants will consider the prices of p c 0 6 versus p c They will choose the one that has a higher premium. Also, for the forward contract, they will look at c 0 6 p 0 6 versus c p Again, they will choose the one that has a higher value. The main consideration is that the replicated payoff must not be less than the payoff fs T. c Christopher Ting Quantitative Finance Group Page 6 of 6