RMSC 2001 Introduction to Risk Management

Transcription

1 RMSC 2001 Introduction to Risk Management Tutorial 6 (2011/12) 1 March 19, 2012 Outline: 1. Option Strategies 2. Option Pricing - Binomial Tree Approach 3. More about Option ==================================================== 1 Option Strategies Options Two basic types of options : Call option and Put option Option holder has NO obligation to exercise his/her right! Option holder needs to pay a premium when he/she holds the option. i.e. The initial cost is NOT ZERO. Terminal payoff option at maturity T: Long a call: max(s T K, 0) Long a put: max(k S T, 0) Short a call: max(s T K, 0) Short a put: max(k S T, 0) Example 1 Trish buys a 80-strike European call on Asset A and sells a 70-strike European call on Asset B. Coincidentally, it turns out that the spot price at expiration of both assets are $78. What is Trish s total payoff from the two options? (A) -$8 (B) -$6 (C) $6 (D) $8 (E) $10 Sol: long 80-strike call: max(78 80, 0) = 0 short 70-strike call: max(78 70, 0) = 8 total payoff: $0 + ( $8) = $8 Ans: (A) 1 All rights 2012 by Wang Weiyin 1

2 Option Strategies Q: What is your trading strategy if you expect the stock price will increase (bull) in the near future? Ans: 1) Buy the stock and sell it later 2) Enter into a long future contract 3) Buy a call option on that stock 4) Buy a call, but reduce the cost (even if it means your profit will also be lowered) The last one is the motivation for the option strategy called Bull Spread. Option Strategy Combination Market Expectation Profit Graph Bull Spread Short a call with K 2 (K 1 < K 2 ) stock price will increase Bear Spread Short a call with K 2 (K 1 > K 2 ) stock price will decrease Straddle Long a put with K 2 (K 1 = K 2 ) stock price will be volatile Strangle Long a put with K 2 (K 1 > K 2 ) stock price will be volatile (cheaper than straddle) Butterfly Short two calls with K 2 Long a call with K 3 (K 1 < K 2 < K 3 ) stock price will be stable 2

3 Example 2 (2011 Final Q2) Bank A is offering three European call options which mature in 6 months. Call 1: Strike price K=$95, call option price c=$7 Call 2: Strike price K=$100, call option price c=$5 Call 3: Strike price K=$105, call option price c=$3 (a) Describe how to create a butterfly option trading strategy from the above call options. (b) Draw a graph represents relationship between the payoff and stock price at the maturity. (c) Is there any arbitrage opportunity available? Construct one if there exists. Sol: (a) long one Call 3, long one Call 1, short two Call 2 (b) payoffs: long Call 1: max(s T 95, 0) long Call 3: max(s T 105, 0) short two Call 2: 2max(S T 100) 0, if S T [0, 95) S T 95, if S T [95, 100) total payoff = 105 S T, if S T [100, 105] 0, if S T [105, ) (c) Portfolio: hold the long the butterfly in part(a) initial price P 0 = = 0 final payoff P T 0 Arbitrage! 3

4 2 Option Pricing - Binomial Tree Approach Example 3 (2011 Final Q5) Consider the following Binomial tree model (a) There is a financial derivative which gives you the payoff max(a-28,0) at time 2, where A is the average value of the stock price (S) over time 0,1 and 2. What is the name of this financial derivative? (b) Let the risk-free interest rate for each period be 2%. Price this derivative. Sol: (a) Asian Option max( 98 28, 0) = 14 if S 3 3 t : max( 94 (b) final payoffs = 28, 0) = 10 if S 3 3 t : max(29 28, 0) = 1 if S t : max( 82 28, 0) = 0 if S 3 t :

5 At node B: 35X Y = X Y = 10 3 X = 1 3 Y = C B = = At node C: 30X Y = 1 31X Y = 0 X = 0.2 Y = C C = = At node A: 33X Y = X Y = X = Y = C A = = price = $ More about Option Put-Call Parity Proof: C E + Ka 1 (T ) = P E + S Consider the followting two portfolios: A: hold a call option and deposit Ka 1 (T ) into the bank B: hold a put option and one unit of the stock One maturity date: A max(s T K, 0) + K = K max(s T K, 0) + K = S T B max(k S T, 0) + S T = K max(k S T, 0) + S T = S T A T = B T, by no arbitrage argument, A 0 = B 0, the proof is completed. Bounds for European Options Proof: For European Calls, max(s 0 Ka 1 (T ), 0) < C E < S 0 max(ka 1 (T ) S 0, 0) < P E < Ka 1 (T ) (a) C E < S 0 we consider the following two portfolios: A: hold a Call Option B: hold a unit of asset 5

6 One maturity date: A max(s T K, 0) = 0 max(s T K, 0) = S T K B S T S T A T < B T, by no arbitrage argument, A 0 < B 0, the proof is completed. (b) S 0 Ka 1 (T ) < C E we consider the following two portfolios: A: hold a unit of asset, borrowka 1 (T ) from the bank B: hold a Call Option One maturity date: A S T K < 0 S T K B max(s T K, 0) = 0 max(s T K, 0) = S T K A T B T, by no arbitrage argument, A 0 < B 0, the proof is completed. (c) C E > 0 One maturity date: max(s T K, 0) = 0 max(s T K, 0) = S T K 0 C E > 0 Remark: Option premium must be a positive amount of cash since the right is solely given to the holder. Similar proof can be constructed for the European Put Options. 6