1a) Options are European. Hence options can be exercised only on maturity and not before that. Under a long straddle strategy, an investor purchases a call as well as a put on the same asset with the same expiration date and strike price, generally at-the-money options. The main aim of a long straddle strategy is to ensure a potential profit irrespective of what direction the stock price moves in and to benefit out of market volatility. On the expiration date, if the stock price is more than the strike price, the investor will exercise the call option and will let the put option expire. Again, if the stock price is less than the strike price, the investor would exercise the put option and sell the asset at the strike price. But he would let the call option expire. (Kaeppel, 2016) Long Straddle: Buy 1 call option and 1 put option S = $22 T = 6 months = ½ year r = 5% p.a. = 0.05 = 0.025 in 6 months = 2.5% (continuously compounded) K3 = $22.50 Call Price = $1.75 Put Price = $1.75 The expected stock price on the date of the maturity of the options will be, S' = S * (erT); since interest is continuously compounded Or, S' = $22 * (e0.025*½) = $22.28 ∴ on the date of expiration, K3 > S' The investor adopts a long straddle strategy by buying the call option as well as the put option. Since the strike price is more than the stock price, he would not exercise the call option. He will exercise the put option and sell the stock at $22.50 instead of the ongoing market price of $22.28. The payoff from the long straddle is, Π = $ (22.50 – 22.28 – 1.75 – 1.75) = $(-3.28) Thus under these conditions, the investor acquires a net loss from the long straddle strategy. He would still exercise the put option to minimize his loss. 1b) Under a short strangle, a trader writes or sells a call option and a put option on the same asset with different strike prices; both options are slightly out-of-the-money and have the same expiration date. The short strangle strategy is a neutral strategy which offers a limited profit and unlimited risk. It is generally undertaken when the options trader anticipates little or no volatility in the stock market in the recent future. The maximum profit from a short strangle is acquired when the stock price on the date of maturity is somewhere between the strike prices of the two options. In this case, both options will expire and the trader benefits from acquiring the entire profit of the initial credit. (DraKoln, 2016) Short Strangle: Sell 1 call option and 1 put option. S = $22 T = 6 months = ½ year r = 5% p.a. = 0.05 = 0.025 in 6 months = 2.5% (continuously compounded) K3 = $22.50 (strike price of call) K2 = $20 (strike price of put) Call Price = $1.75 Put Price = $0.75 The expected stock price on the date of the maturity of the options will be, S' = S * (erT); since interest is continuously compounded Or, S' = $22 * (e0.025*½) = $22.28 On the date of expiration, K2 < S' < K3 Hence, the investor is making a profit out of the short straddle strategy. The net profit from the short strangle is, Π = $(1.75 + 0.75) = $2.50 (considering commissions = 0) 2a) The Black-Scholes model is used to determine the call price theoretically. It is based on fixed inputs: stock price, strike price, volatility, time to expiration, dividend yield and the short term interest rate (risk-free). The main drawback of this model is that it is not flexible enough to price options with non-standard features such as a compulsory exercise requirement and a price reset feature. Moreover, this model can calculate the option price at only the expiration date and hence cannot be used to accurately determine the prices of American options which can be exercised at a pre-mature date. (Stringham, 2015) In contrast, a binomial model segregates the maturity period into a number of distinct time intervals. At each time point, based on volatility and period of expiration the model anticipates two feasible moves for the stock price by certain amounts (up and down). This produces a binomial tree of possible stock prices representing all the possible paths that can be taken by the stock prices. Hence, it can be used to accurately determine prices of American options. (Neto, 2016) 2b) The binomial model divides the maturity period into a large number of time intervals. This produces a binomial tree of stock prices base on the time to expiration and market volatility. The anticipated movements in the stock price (both up and down) produce a binomial distribution that can be used to evaluate the stock price at any point in time. Binomial model is mathematically simple to use. It is very useful for evaluating American options which can be exercise at any pre-mature date. Binomial model is also very useful in pricing Bermudan options that can be exercised at different time points throughout the life of the option. (Sayal, 2012) 4) American call Stock Price (S) = $120 S.D = 20% D1 = $5 in 40 days D2 = $5 in 130 days Interest rate (r) = 5% p.a. Exercise Price under call option = $150 Maturity Period = 100 days r100 = 0.01370 (interest rate in 100 days) The stock price can either go up or down by 20% = 20% of $120 = $24 S' = $ (120 + 24) = $144 S'' = $ (120 – 24) = $96 The stock pays a dividend of $5 in 40 days and $5 in 130 days. ∴ on the date of maturity, that is, after 100 days, $5 dividend would be there for (100 – 40) = 60 days and $5 would be received after (130 – 100) = 30 days. r30 = 0.00427 (interest rate in 30 days) r60 = 0.00833 (interest rate in 60 days) ∴ the value of the total dividend received on the maturity date = $ [ 5 * (1 + 0.00833) ] + 5 / [ (1+0.00427) ] = $ (5.04165 + 4.97874) = $10.02039 The actual value of the stock on the date of the maturity is $ (144 + 10.02039) = $154.02039 (S') $ (96 + 10.02039) = $106.02039 (S'') The call option will not be exercised under S'' in which case the call price will be $0. The option will be exercised under S', that is, when the net value of the stock on the maturity date is $154.02039. ∴the call price will be, $ [ 154.02039 – 150 / (1 + 0.01370) ] = $6.0272 ≅ $6 (Karatzas, 1988) REFERENCES DraKoln, N 2016, Get A Strong Hold On Profit With Strangles, viewed 21 August 2016, < http://www.investopedia.com/articles/optioninvestor/08/strangle-strategy.asp>. Kaeppel, J 2016, Profit On Any Price Change With Long Straddles, viewed 21 August 2016, . Karatzas, I 1988, On the Pricing of American Options, viewed 22 August 2016, http://www.math.columbia.edu/~ik/Kar_AMO_88.pdf. Neto, A 2016, Breaking Down The Binomial Model To Value An Option, viewed 21 August 2016, . Sayal, V 2012, Binomial Option Pricing Model, viewed 21 August 2016, < http://www.simplilearn.com/binomial-option-pricing-model-article/all-resources>. Stringham, T 2015, Option Pricing: Black-Scholes v Binomial v Monte Carlo Simulation, viewed 21 August 2016,