1 Warm-up & Motivation Exercise 1.1 Do problems 1.4 and 1.8 of [Shreve, 2005a].
Exercise 1.2 Let us extend the binomial tree model to include a second rate. How do Theorem 1.2.2 in (Shreve, 2005a], the formulas for 13 and -4 and the wealth equation (1.2.14) change, if the underlying stock pays a relative dividend of rf per time period? This means that an investor buying the stock at time n at price Sn will hold a portfolio worth Sn4.1(1 rf) at time n + 1. Typically, in practice, such situations are common: Think of S being the exchange rate of a currency pair such as GBP-USD. Then the foreign currency GBP (British Pound) pays an interest r1. With Gold being the underlying, the "interest rate of gold" rf is called the lease rate. For stock performance indices with dividends reinvested, one can think of r1 as a payment coming from the dividends of the individual stocks.
Exercise 1.3 Implement an N-step binomial tree model that allows to compute the value and delta of European style call and put options. Compute the values and deltas of put and call options for the following sample contract data: strike K = 1.4500, maturity T = 1 year. Use N = 120 time steps, so a single time step will be L‘t = N. Let the market data be given by So = 1.5000, r = 5% p.a., r1 = 4% p.a., volatility o- = 10%. For the implementation use the formulas
1
which is slightly different from those presented in class. The reason is that we assume the rates r and r1 to be continuous. For the discounting per time step you need to use e—rAt instead of
It would be good if your pricing tool was set up as general as possible, as we will keep extending it in class and the homework problems.
Programming Guidelines.
1. Restrict choice of language to: Matlab, Excel/VBA, Java, C/C, R.
2. Submit the original source code. This means do not submit your code pasted into a .doc or .txt file. The extension should be .rn,.xls,.java,.c,.cpp etc.
3. Make sure your code compiles. If it does not we will not grade this prob-lem.
Exercise 1.4. An up-and-out put is defined by the payout profile
XT =
max(0, K — ST), i f St
o,
Vt
where K denotes the strike of the option, St denotes the price of the underlying security at time t, and B denotes the barrier price. In other words, if St hits the barrier B at any point in time t during the lifetime of the option, the option dies immediately. Based on the example from Section 1.1 of the lecture, and assuming K = B = 1.05 and an interest rate r = 5%:
1. Devise a replication strategy for an up-and-out put on the security S.
2. Price the up-and-out put by expectation.
Exercise 1.5. Show that with Q defined by equation (1.10) it holds that
EQ(S3-) = S4(1 r)T•