Financial Mathematics Problems
Financial Mathematics Problems MATH 3615
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University of Connecticut Math 3615 Financial Mathematics Problems Spring 2009 31909 Examples Module 8 1 Positive Polly buys one share of a non dividend paying stock at the ask price of 6025 and pays a commission of 060 a What is the accumulated value two years later of the total amount Polly paid to acquire the stock Assume that Polly could have invested money at a risk free interest rate of 5 an annual effective rate during those 2 years b Two years after the purchase date Polly sells the stock at the bid price of 6475 and pays a commission of 060 What is the value as of the sale date of her profit on this pair of transactions If there is a loss show the loss as a negative profit c Sketch a graph of the value of Polly s profit as of the sale date for any possible bid price for the stock That is make a graph of Profit fx where x is the stock s sale price before commission Label the graph to show the slope of the function and its x intercept not its y intercept The x intercept is the ending price at which Polly s profit would be exactly zero 2 Negative Nellie sells short one share of the same stock that Polly purchased and this sale occurs at the same time as Polly s purchase Nellie s short sale is executed at the bid price of 5975 and she pays a commission of 060 a What is the accumulated value two years later of the total amount Nellie paid or received by selling the stock short Was this amount paid or received by Nellie Assume that the risk free interest rate at which Nellie can invest money during the year is 5 an annual effective rate b Two years after the date of the short sale and at the same time that Polly sells her share Nellie covers her short sale by buying one share of the stock at the ask price of 6525 and paying a commission of 060 What is the value as of the sale date of Nellie s profit on this set of transactions If there is a loss show the loss as a negative profit c Sketch a graph of the value of Nellie s profit as of the sale date for any possible ask price for the stock Make a graph of Profit 2 f x where x is the stock s ask price as of the sale date Label the graph to show the slope of the function and its x intercept U3 Although Polly and Nellie were on opposite sides of the same transactions their combined profit is not zero Calculate the value of their combined profit as of the sale date and explain how it arose In other words describe each element that contributed to this combined non zero profit amount and show that the sum of these elements adjusted for the effect of interest exactly equals the combined profit for the two investors Note also that the x intercepts of the two graphs in 1 and 2 are not the same For each of the remaining two problems assume that the riskfree oneyear interest rate is 6 compounded continuously ie 5 006 4 Suppose that on March 18 2007 when the SampP 500 index is at 130989 you enter into a forward contract to sell the SampP 500 index on March 18 2009 Assume that the forward contract s delivery price is equal to the arbitrage free price determined as of March 18 2008 a Write the formula for your gain from this contract as of March 18 2009 expressed as a function of the value of the index on that date b Sketch a graph of your gain vs the value of the index as of March 18 2009 5 On March 18 2007 you purchased a one year call option on stock NCA with a strike price of 70 and also purchase a one year put option on the same stock with a strike price of 80 On March 18 2008 the stock price is 74 and you exercise both options a If your profit on the call is equal to your profit on the put and if you paid 5 to purchase the call how much did you pay to purchase the put b Sketch a graph showing as a function of the stock s price on the expiration date i the profit for the call option ii the profit for the put option iii the total profit for the two options combined Label your graph to show the key stock prices for these transactions and indicate which lines correspond to i ii and iii Hint It may be helpful to make a table of values for a b and c and plot the points on the graph University of Connecticut Math 3615 Financial Mathematics Problems Fall 2008 Summary Module 10 McDonald Chapter 3 Lombardi text Units 6amp7 OPTION STRATEGIES Note In the following discussion whenever two or more derivatives are combined it is assumed that all of the derivatives are based on the same underlying asset eg stock and that all have the same expiration date It is also assumed that the underlying asset does not pay dividends A floor is the combination of owning an asset and owning a put option on that asset This is also referred to as having insurance on the price of the asset The profit from buying both a stock and a put option on that stock is the w as the profit from buying a call option with the same strike price as the put option However the initial cost and the eventual payoff are different because the stock plus put option is more expensive than a call option 2 Nonetheless the pro ts from the two positions must be identical otherwise there would be an arbitrage opportunity3 A Qp consists of a purchased call option on a stock combined with a short position in the stock itself The call option provides insurance for the short position The combined short position and call option are equivalent in profit to a purchased put option4 Selling or writing an option is called covered writing if the seller owns the underlying asset in the case of a call or has shorted the asset in the case of a put Writing options without having a position in the underlying asset is called naked writing Writing a covered call owning a stock and selling a call option on it is equivalent in profit to writing a put option at the same strike price To match cash ows as well as profit the written put option would need to be combined with a purchased bond Writing a covered pu shortng a stock and selling a put option on it is equivalent in profit to writing a call option at the same strike price To match cash ows the written call option would need to be combined with a short bond position 1 The put option provides insurance against a decline in the asset s price below the option s strike price If the market price does decline below the strike price the investor will exercise the put option and sell the asset for the strike price so a decline below the strike price does not affect the investor A call option plus a riskfree bond of appropriate size would have cash flows as well as profit identical to the stockplusput combination The purchase price of the bond should equal the total cost of the stock plus put option less the cost of the call option In general adding a riskfree bond or a short bond position to any position in derivatives moves its payoff diagram up or down but leaves its profit diagram unchanged For example if the call option is overpriced relative to the put option a profitable arbitrage position could be created by purchasing the stock and a put option and writing a call option Note that the combination of call option and a short position in the stock is equivalent in cash flow and profit to a put option plus a short position on a riskfree bond ie a put option plus a loan N w s A synthetic forward can be created by purchasing a call option and writing a put option with the same strike price5 On the options expiration date either the spot price will be above the strike price and the investor will exercise the call option to buy the stock at the strike price or the spot price will be below the strike price and the written put will be exercised against the investor forcing himher to buy the stock at the strike price In either case the asset will be purchased at the strike price as would happen with a long forward If the strike price of a synthetic forward equals the no arbitrage forward price then the premium paid for the purchased call equals the premium received for the written put and there is no net cost for establishing the synthetic forward contract More generally CallKT 7 PutKT PVFOVT 7 K ie the premium for a call option at strike price K with duration T less the premium for a put option at strike price K with duration T equals the present value of the difference between the no arbitrage ie zero cost forward price with the same expiration date and the options strike price Note An offmarket synthetic forward contract is one for which the premium is not zero ie the price is not equal to the no arbitrage forward price so one party must pay the other to enter into the contract Equivalence of Different Positions The very important formula shown above can be rearranged to show the equivalence of the prices and payoffs and profits of a variety of different combinations of positions To analyze the components of the formula PVFOVT is the present value of the no arbitrage T year forward price it equals the current spot price of the stock6 and therefore represents a long position in the stock alternatively PVF0T represents a short position in the stock PVK or PVKT is the present value of an amount K payable in T years it represents a T year zero coupon bond with a maturity value of K And PVK represents a loan with a maturity value of K PutKT or CallKT is the premium for a T year put or call option with a strike price K it represents a purchased option and PutKT or CallKT represents a written option Here are some rearrangements of the formula along with descriptions of what they mean PVFOYT PutKT CallKT PVK Buying a stock plus a put option with strike price K is equivalent to buying a call option with strike price K and buying a bond PVFOYT CallKT PutKT PVK Writing a covered call is equivalent to writing a put option and buying a bond PVFOYT PutKT CallKT PVK Writing a covered put is equivalent to writing a call and taking out a loan 5 This is a long forward A short synthetic forward would combine a purchased put option and a written call option 6 This assumes that the asset pays no dividends during the term of the forward contract Spreads Bull spread Buy a call and also write a call with a higher strike price or to produce the same result buy a put at the lower strike price and sell a put at the higher strike price and buy a bond Bear spread Sell a call and also buy a call with a higher strike price or to produce the same result sell a put at the lower strike price and buy a put at the higher strike price and short a bond ie take out a loan Box spread Create a synthetic long forward at one price and a synthetic short forward at a lower price This creates no stock risk but the paid and received premiums amount to a loan The investor has agreed to make a net payment in the future by buying at the higher price and selling at the lower price so the financial effect is the same as a loan A ratio spread consists of m purchased calls at one strike price and n written calls at a different strike price A ratio spread can also be created with puts A ratio spread can be designed so that there is no net cost to acquire the position Note that because the purchased and written options do not balance m 2 n there is a net exposure at all prices above the lower strike price or below the higher strike price if puts are used A collar consists of a purchased put option and a written call option at a higher strike price Note that this is a short position whose value decreases as the stock price increases except that the collar s value does not change as a result of price movements between the two strike prices Adding a long position in the stock ie buying the underlying asset converts the collar to a bull spread A written collar consists of a written put and a purchased call with a higher strike price Note that this is a long position when a short position in the stock is added the written collar becomes a bear spread A zerocost collar can be created by selecting strike prices such that the premiums of the purchased and written options are equal and the net cost is zero Theoretically there are an infinite number of possible combinations of strike prices that will produce a zero cost collar If both options strike prices are equal to the no arbitration forward price then the cost of the collar is zero and the collar is identical to a forward contract a short forward in the case of a purchased collar or a long forward in the case of a written collar Speculating on Volatility A straddle consists of a put option and a call option with the same strike price There is guaranteed to be a positive payoff unless the stock price at expiration is identical to the strike price The market s assessment of the underlying asset s volatility determines the cost of a straddle Greater perceived volatility results in higher option premiums which means that greater price movement is required in order for the straddle buyer to recoup the cost of the options A strangle consists of a put option and a call option that are out of the money ie with strike prices below and above the current spot price respectively A strangle is less expensive than a straddle but there is no payoff unless the stock price moves enough so that one of the options is in the money A written straddle or strangle is a bet against volatility A butter y spread or insured written straddle is the combination of a written straddle and a purchased strangle This is an insured or limited bet against volatility Unlike a written straddle or strangle a butter y spread does not have unlimited liability in the case of large price movements However the cost of this insurance is that there are smaller gains from the written straddle for small price movements Note A butter y spread may be created by using only calls or using only puts or by a long or a short position in the underlying asset combined with both calls and puts A butter y spread created with only call options involves a purchased call at strike price K1 2 written call at strike price K2 and a purchased calls at strike price K3 where K2 is midway between K1 and K3 An asymmetric butter y spread created with call options involves m purchased calls at strike price K1 n written calls at strike price K2 and nm purchased calls at strike price K3 where K2 is not midway between K1 and K3 For an asymmetric butter y spread the following relationship must hold K3 K2 K3 K1 As with the symmetric butter y spread an investor can create an asymmetric butter y spread by using only calls using only puts or by using a long or short position in the stock combined with both puts and calls l1 l l