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Derivative Market Futures Forwards Options What is in today’s lecture? Introduction to Derivative Forward and Futures Financial Derivatives Various aspects of forwards Pricing of forward contracts Options Forward Prices and Spot Prices How forward price are determined? In other words, how to price future contracts? See example on the next slide and draw conclusion for yourself 1. 2 3. 4. Forward Prices and Spot Prices Suppose that the spot price of gold is $1000 per ounce and the risk-free interest rate for investments lasting one year is 5% per annum. What is a reasonable value for the one-year forward price of gold? • Suppose first that the one-year forward price is $1300 per ounce. A trader can immediately take the following actions: • 1. Borrow $1000 at 5% for one year. • 2. Buy one ounce of gold. • 3. Enter into a short forward contract to sell the gold for $1300 in one year • The trader earns a riskless profit of? • The trader pays a total price for one ounce of gold = 1000 + (1000x5% interest) = $1050 The trader sell the gold for Rs. 1300 His riskless profit is = 1300-1050 = $250 The example shows that $1300 was too high a forward price Pricing aPrices forward Forward andcontract Spot Prices Forward price = So + CC Where So is the spot/current/cash price today CC is the cost of carry ( in the previous example CC is the financing cost / interest paid for buying gold) F = $1000 + (1000x.05) = 1000(1+.05) = $1050 The case of continuous compounding Like in time value of money concept, when continuous compounding is the rT assumption, the interest rate formula ie becomes: Where e = 2.71828 Forward price for a non-dividend paying asset is F Soe rT Example Consider a four-month forward contract to buy a zerocoupon bond that will mature one year from today. The current price of the bond is Rs.930. (This means that the bond will have eight months to go when the forward contract matures.) Assume that the rate of interest (continuously compounded) is 6% per annum. T = 4/12 = .333 r = 0.06, and So = 930. The forward price, F Soe rT 930 e . 06 *. 333 948 . 79 For dividend or interest paying securities Since the forward contract holder does not receive dividend/interest on the underlying asset, but the present price So reflects the future income from the asset, the present value of dividends/interest should be deducted from So while calculating Forward price F ( S o l )e rT l is the present value of future dividends/ interest Example Consider a 10-month forward contract on Nishat Mills Ltd (NML) stock with a price of Rs.50. Assume that the risk-free rate of interest (continuously compounded) is 8% per annum for all maturities. Also assume that dividends of is 0.75 per share are expected after three months, six months, and nine months. The present value of the dividends 0 . 75 e 0 . 08 * 3 / 12 0 . 75 e 0 . 08 * 6 / 12 0 . 75 e 0 . 08 * 9 / 12 2 . 162 Example continued.. The forward price of the contract is F ( S o l )e rT F ( 50 2 . 162 ) e 0 . 08 *10 / 12 Rs . 51 . 14 Assets with storage costs Storage costs can be regarded as negative income If U is the present value of all the storage costs that will be incurred during the life of a forward contract, then the forward price is given by: F ( S o U )e rT Example Consider a one-year futures contract on gold. Suppose that it costs $2 per ounce per year to store gold, with the payment being made at the end of the year. Assume that the spot price is $450 and the riskfree rate is 7% per annum for all maturities. This corresponds to r = 0.07, and S0 = 450, T=1 0 . 07 *1 U 2e 1 . 865 Example continued.. F ( S o U )e rT F ( 450 1 . 865 ) e 0 . 07 *1 484 . 63 Forward price with known yield If an asset pays income that can be expressed as a present yield on per annum continuous basis, the forward price of the asset can be determined as, F Soe ( r q )T q = average yield per annum the asset underlying the forward contract Consider a six month forward contract on an asset that is expected to provide a yield of 3.96% p.a. with continuous compounding. Then price of a six month contract is???? Valuing the forward Contracts The value of a forward contract at the time it is first entered into is zero. At a later stage, it may prove to have a positive or negative value. we suppose K is the delivery price for a contract that was negotiated some time ago, the delivery date is T years from today, and r is the T-year risk-free interest rate. The variable Fo is the forward price that would be applicable if we negotiated the contract today. Valuing the forward Contracts f:Value of forward contract today At the beginning of the life of the forward contract, the delivery price, K, is set equal to the forward price, Fo, and the value of the contract, f, is 0 Here K remains constant but Fo keeps changing due to changing So therefore the value of a forward contract becomes either positive or negative. Valuing the forward Contracts Value of a long forward contract is, Value of a short forward contract is, Example A long forward contract on a non-dividend-paying stock was entered into some time ago. It currently has 6 months to maturity. The risk-free rate of interest (with continuous compounding) is 10% per annum, the stock price is $25, and the delivery price is $24. In this case,. So= 25, r = 0.10, T = 0.5, and K = 24. From the 6-month forward price, Fo, is given by the value of the forward contract is Pricing an option Pricing an option means “finding the amount that you pay to have the right/option” Or pricing an option is finding the “fair” value of the agreement “fair” means the value that precludes any arbitrage opportunity Let’s use an example to find the value of call option at expiration Intrinsic Values If a call is in the money, it will have positive intrinsic value. Intrinsic value is equal to the difference between current price and exercise price Example: FFC current price is Rs.65, a call option on FFC is Rs.60, the option is in the money IV = 65-60 = 5 Pricing call option at expiration Suppose you own a call option on one share of PTCL stock with a strike price of Rs. 30 Suppose that at expiration one share of PTCL trades at Rs. 35 What should be the minimum price of this call now? Price of call option Minimum price should be 35 – 30 = 5 OR Market price – Exercise Price Or S – E which is equal to Intrinsic value In other words, price of call option can never be lower than intrinsic value The Black-Scholes Model for valuing options Black-Schole model uses the five variables to value non-dividend paying call option • • • • • S = Current stock price E = Exercise price of call r = Annual risk-free rate of return, continually compounded δ2 =Variance (per year) of the continuous return on the stock t = Time (in years) to expiration date Determinants of option prices 1. Exercise Price An increase in the exercise price reduces the value of the call In the previous example, if exercise price was 32 instead of 30, the value of the call option will be 35 – 32 = 3 instead of 5 Determinants of option prices Expiration date : the higher the maturity, the higher will be the premium of both call and put option The reason is that the option holder has more time to exercise the option The extra time increases the probability of profit for option holder and loss for option writer Determinants of option prices Stock Price: Other things being equal, the higher the stock price, the more valuable the call option will be. In our earlier example, if current market price of PTCL share was Rs. 40, then option value would have been 40 – 30 = 10 in stead of 5 If S>E -------high profit if call option is exercised-------high premium for call options If S>E------- loss if put option is exercised----low premium for put options Determinants of option prices The Variability of the Underlying Asset prices The greater the variability of the underlying asset, the more valuable both the call and put option will be Greater variability can subject the option writer to higher losses, this is why option writer will need higher premium for giving such option Determinants of option prices • • • • • The Interest Rate Call prices are also a function of the level of interest rates. Buyers of calls do not pay the exercise price until they exercise the option. The ability to delay payment is more valuable when interest rates are high and less valuable when interest rates are low. Thus, the value of a call is positively related to interest rates. Difference between futures and option contracts Initial investment on the contract are different ◦ margin requirements in forward contracts ◦ Premium paid on the option contracts In future contracts money is needed on daily basis for marking to market the contract whereas in options money is only needed when the option is exercised Profits from the future contracts are linear whereas profits from the option contracts are non linear