MATH4210 Financial Mathematics ( ) Tutorial 6

Transcription

1 MATH4210 Financial Mathematics ( ) Tutorial 6 Enter the market with different strategies Strategies Involving a Single Option and a Stock Covered call Protective put Π(t) S(t) c(t) S(t) + p(t) Covered Call 1. Initial investment is needed to set up the portfolio 2. Protects a trader from sharp rise in the stock price 3. Not protected from the downside. Protective put 1. Initial investment is needed to set up the portfolio 2. Protects a trader from downside 3. Not protected from the upside. 4. Bullish move (Since S(t) + p(t) = c(t) + Ee rτ by put-call parity) 1

2 Reversing Just put a negative sign to the portfolio, bearish bullish or vice versa Spreads Involves taking a position in two or more options of the same type Strategies Involving a Single Option and a Stock Bull spread Bear spread Butterfly spread (E 1 < E 2 ) (E 1 < E 2 ) (E 1 < E 2 < E 3 ) with E 3 E 2 = E 2 E 1 ) Π(t) c(e 1, t) c(e 2, t) or p(e 1, t) p(e 2, t) c(e 2, t) c(e 1, t) or p(e 2, t) p(e 1, t) c(e 1, t) 2c(E 2, t) + c(e 3, t) or p(e 1, t) 2p(E 2, t) + p(e 3, t) Bull spread 1. Bull spread using calls: c(s(t), E 1, τ, r) > c(s(t), E 2, τ, r) when E 2 > E 1, initial investment is needed. 2. Bull spread using Puts: p(s(t), E 1, τ, r) < p(s(t), E 2, τ, r) when E 2 > E 1, initial cash inflow. 3. Risk is limited to the net premium paid. 4. Tailored risk profile: (a) narrow the spread with similar strike prices: less premium, restricting gains (b) widen the spread for aggressive trader: more premium, maximizing gains 5. Not optimal strategy if the underlying stock is expected to make big gain Bear spread 1. Reversing of the bull spread. 2. Intend to make profits from the decline of the underlying stock. 3. Limited both the profit and risk. 2

3 Butterfly spread Involves positions in options with three different strike prices 1. Non-directional options strategy 2. Maximum profit is attained when the underlying stock price is close to E 2 at expiration. 3. Requires a small investment initially (By No Arbitrage Assumption) Combinations (both call and put) Straddle Strip Strap Strangle (E 2 < E 1 ) Π(t) c(t) + p(t) c(t) + 2p(t) 2c(t) + p(t) c(e 1, t) + p(e 2, t) Constructing Piecewise Linear Payoff Functions Proposition 1. Suppose that European options expiring at time T are available with every single possible strike price. Then it is possible to combine them together to get any piecewise linear payoff function f(s(t )) that we prefer at time T. Using only call options If the piecewise linear payoff function f(s(t )) has the following property: then f(s(t )) can be formed by using only call options: f(s(t )) = 0 when S A for some A > 0, (1) f(s(t )) = a i max(s(t ) E i, 0), E 1 = A. i=1 3

4 Here a i s are the changes in slopes (forward). Note that the property (1) is a necessary condition when forming f(s(t )) by only call options. For instance, if f(0) 0, there is no way to form this payoff by using only call options (unless there are call options with negative strike prices). Using only put options If the piecewise linear payoff function f(s(t )) has the following property: f(s) = 0 when S > B for some B > 0, (2) then f(s(t )) can be formed by using only put options: n f(s(t )) = b i max(e i S, 0), E n = B. i=1 Here b i s are the changes in slopes (backward). Note that the property (2) is only a sufficient but not necessary condition when forming f(s(t )) by only put options. Using Butterfly spreads Note that any piecewise linear payoff function f(s(t )) can be formed by using butterfly spreads: f(s(t )) f(s(t )) = Π i (S), E 1 = A, (3) where is the minimum change allowed in the value of the stock price S(t) and 0, if S(T ) E i, S(T ) E Π i (S) = i +, if E i < S(T ) E i, S(T ) + E i +, if E i < S(T ) E i +, 0, if S(T ) > E i + i=1 is a butterfly spread of buying two call options with strike prices E i and E i + and selling two call options with strike price E i. Mathematically, it is also known as a hat function, which is centered at E i and zero outside [E i, E i + ] and has value at E i, i.e., Π i (E i ) = We let S(T ) = E j in (3). If i j, then Π(E j ) = 0. Hence the summation is reduced to only one term with i = j, and the RHS is f(e i ) Π i (E j ) = f(e j) Π j (E j ) = f(e j) = f(e j ). Example 1 European call and put options with expiry date T and strike prices E i, i = 1,..., 6 are available. Given E i+2 E i+1 = E i+1 E i > 0 for i = 1, 2, 3, 4. Construct a portfolio which has a payoff at expiry as in the following figure by 1. using only call options; 2. using only put options; 3. using put options with strike prices E i, i =, 1, 2, 3 and call options with strike prices E i, i = 3, 4, 5, 6. Answer: 1. The strategy is to sell one call with E 1, buy two calls with E 2, sell one call with E 4, sell one call with E 5, and buy one call with E 6. Let c i (t) denotes the price of European call options with strike prices E i for i = 1,..., 6. Then the portfolio we have constructed has the payoff c 1 (T ) + 2c 2 (T ) c 4 (T ) c 5 (T ) + c 6 (T ) = 0, if S(T ) E 1, S(T ) + E 1, if E 1 < S(T ) E 2, S(T ) E 3, if E 2 < S(T ) E 4, E 2 E 1, if E 4 < S(T ) E 5, S(T ) + E 6, if E 5 < S(T ) E 6, 0, if E 6 < S(T ), 4

5 2. Let p i (t) denote the prices of European put options with strike prices E i for i = 1,..., 6. By the put-call parity, we derive c 1 (t) + 2c 2 (t) c 4 (t) c 5 (t) + c 6 (t) = (p 1 (t) + S(t) E 1 e rτ )+ 2(p 2 (t) + S(t) E 2 e rτ ) (p 4 (t) + S(t) E 4 e rτ ) (p 5 (t) + S(t) E 5 e rτ )+ (p 6 (t) + S(t) E 6 e rτ ) = p 1 (t) + 2p 2 (t) p 4 (t) p 5 (t) + p 6 (t) 3. Simply apply put-call parity on c 1 (t) and c 2 (t), we derive p 1 (t) + 2p 2 (t) p 3 (t) + c 3 (t) c 4 (t) c 5 (t) + c 6 (t) Example 2 Let c E (t) denote a European call option at time t with strike price E and maturity T. Suppose at time t, a trader invested in a bull spread, which is a long c E1 (t) and a short c E2 (t) with E 1 < E 2. Let Π E1,E 2 (t) denotes such as bull spread at time t. 1. Show that Π E1,E 2 (t) 0 for all t T. 2. At time t, the following European call options are available c 20 (t), c 30 (t), c 40 (t), c 50 (t), c 60 (t) and c 70 (t). Construct a portfolio with only the given European call options such that it has a payoff at maturity as follows: 3. At time t, the following bull spreads are available: Π 20,30 (t), Π 30,40 (t), Π 40,50 (t), Π 50,60 (t) and Π 60,70 (t) Construct a portfolio with only the given bull spreads such that it has a payoff at maturity as shown above. 5

6 Answer: 1. Note that Suppose there exists a time t T such that Then at maturity, it becomes Π E1,E 2 (t) = c E1 (t) c E2 (t), t T. Π E1,E 2 (t ) = c E1 (t ) c E2 (t ) < 0. Π E1,E 2 (T ) = max(s(t ) E 1, 0) max(s(t ) E 2, 0) 0, if S(T ) E 1, = S(T ) E 1, if E 1 < S(T ) E 2, E 2 E 1, if S(T ) > E 2 Hence Π E1,E 2 (T ) Π E1,E 2 (t ) > 0, and an arbitrage opportunity exists. Hence we have Π E1,E 2 (t) 0 for all t T. 2. From the changes of slopes from left to right at the nodal points, we find out that the portfolio is c 20 (t) 2c 30 (t) + 2c 50 (t) c 70 (t). The corresponding action is to buy one call option with strike price 20, sell two call options with strike price 30, buy two call options with strike price 50, and sell one call option with strike price From (2), the portfolio is equivalent to [c 20 (t) c 30 (t)] [c 30 (t) c 40 (t)] [c 40 (t) c 50 (t)] + [c 50 (t) c 60 (t)] + [c 60 (t) c 70 (t)], which is Π 20,30 (t) Π 30,40 (t) Π 40,50 (t) + Π 50,60 (t) + Π 60,70 (t). 6