Option Pricing Equity Portfolio Management

Question: Calculate the option price using Blacks formula. We are in June 27, 2011. A sugar producer has 1, 120, 000 pounds of sugar to sell in October. He wants to hedge the risk of a price fall between September and December. He hesitates between selling October futures and buying the put option described in question 2. We assume the margin requirement is 10% of notional (it is calculated once and for all at the inception of the position). The position is cleared on a monthly basis and the maintenance margin is the same as initial margin requirement (hence, a margin call is triggered every time the account balance falls below the initial level). Calculate the dynamics of the margin account balance and the margin calls between June and October 2011. Explain the possible problem for the sugar producer. Compare the final selling prices for the sugar producer in October 2011 if he does not hedge, if he hedges with futures contracts and if he hedges with a put option (we assume the final spot price in October is 24.84 cents per pound). Imagine again we are in June 27, 2011 but we dont know the final outcome in October. Calculate the minimal selling price secured when the producer buys the put. Plot the sugar selling price in terms of final Oct price (assumed unknown) within the 3 alternative strategies. Comment briefly on the pros and cons of the three strategies. Answer: According to (Collins and Fabozzi 1999, p.64), there are several assumptions that the Black Formula is based on. These include; The Black-Scholes model assumes European Options that can only be carried out at expiration. During the lifespan of an option, payments of dividends does not take place The Black-Scholes model makes the assumption that the markets are efficient and as such cannot be predicted by people easily. It uses the application of the random walk. The model also makes the assumption that there are no charges and any other transaction costs when purchasing and trading options and that there exists no barrier whatsoever to tradeoff. It assumes lognormal distribution that the underlying stock is normally distributed. The Black-Scholes model also makes the hypothesis that the interest rates are do no change and are known. It applies the risk-free rate to the known constant. It is also assumes that the volatility of the underlying stock which can be described as the measure of the expectations of near-future stock movements is known. The realism of this the assumptions in the real-life commodity markets. While some of the assumptions taken by the Black-Scholes model are reasonable and applicable in the real world scenarios like the lognormal distribution, others such as the risk free rates and known and constant interest rates do not apply in the real world markets. This is because rates are very likely to be affected by the volatility. Calculations N is the C.M.F of the standard normal distribution T-t is the time to maturity St is the spot value of the stock K is the strike worth of the underlying assets r is the rate not attached to any risk, usually in continues compounding terms is the volatility Replacing the figures on the formula with the values provided. =K e-r(T-t) St +C (St t) Risk free rate= 0.02 Spot price= 25.91 Strike price= 24.84 Time= 90 days Volatility=0.25 =24.84e-0.02(90)-25.91 +C (St t) C (St t) = N (d1) St -N (d2) K e r(T-t) Put price = 0.7458 3. Hedging for a price fall between September and December Time=6 X 30 = 180 days Put price =24.84e-0.02(180)-25.91 +C (St t) Put price = 0.0350 Possible problem: The new put price is too low for the sugar producer. While hedging for a price fall between September and December, the value of the put option is 0.0350. The final value of the put option by the end of October is highest if the sugar producer does not hedge. While hedging itself is the act of buying and retaining shares so as to reduce the risk on the portfolio, the put option price is highest at the end of November if the sugar producer does not hedge. The minimal price secured when the producer buys the put option is 24.0942. The advantages of hedging will include the fact they a hedged portfolio has lower risk than one that is not hedged. References Collins, B. and Fabozzi, F. (1999). Derivatives and equity portfolio management. 1st ed. New Hope, Pa.: Frank J. Fabozzi Associates, p.Page 64.