Intro-Maths-Fin-1Financial Derivatives (A Brief Introduction )Background This chapter deals with the two basic building blocks of financial derivatives:1. Options2. Forwards and futures.We briefly introduce the third class of derivative: swap. We see how a complex swap can be decomposed into a number of forwards and options.Definitions Derivatives securities are financial contracts that ‘derive’ their value from cash market instruments such as stocks, bonds, currencies and commodities.At the time of the maturity of the derivative contract, denoted by T , the price F(T) of the derivative asset is completely determined by the market price of the underlying asset (S T ).For instance, the value at maturity (T ) of a long position in a call option of strike (K) written on an asset (S T ) is:Max [S T −K ;0]Also, the value of time T of a long position in a forward contract of forward price (F) written on an underlying asset worth (S(T) at time T is given by:S (T )−FTypes of derivatives We group derivatives into three general headings:1. Futures, Forwards, Repos, Reverse Repos and Flexible Repos (Basic building blocks )2. Options and3. SwapsMany of these instruments will be discussed in other parts of the syllabus for the QF Exam.The underlying asset: We let (S t ) represent the price of the relevant cash instrument, which we call the underlying asset . The five main groups of underlying asset : We list five main groups of underlying assets:1.Stocks (These are claims on “real” returns)2.Currencies3.Interest rates: Interest rate in not an asset, so we are referring to the direction of interest rates. The assets are Treasuries, bonds.4.Indexes (S&P 500) andmodities: they are not financial assets either, they are goods in kind. There is another method for classifying the underlying asset:1.The cash and carry markets and2.The price discovery marketsLet us discuss these two marketsThis new classification is important to us.In the cash and carry market, one can borrow at risk-free rates, buy and store the product,and insure it until the expiration date of any derivative contract.Pure cash and carry market have one property: Information about demand and suppliesof the underlying instrument should not influence the spread between cash andfutures (forward) prices.In the Pure cash and carry market, any relevant information concerning future supplies anddemands of the underlying instrument is expected to make the cash price and the future price change by the same amount (This is not so, in the price discovery market).In the price discovery market, it is physically impossible to buy the underlying instrumentfor cash and store it until some future expiration date. That strategy (of borrowing, buying and storing) is no longer applicable.In the price discovery market, any information about the future supply and demand of theunderlying commodity cannot influence the corresponding cash price.Expiration DateAt the expiration of the forward/futures contract, we expect:At expiration,tℎe Futures price=F(T)=Tℎe spot price of tℎe underlying asset=S T But, during the life of the futures contract (t<T), the futures price may well be different to theForward1 and FuturesForwards and Futures are linear instruments (while options are nonlinear instruments).“Options are non-linear instruments because the derivative of the payoff function changes sign around the strike price. In fact, the payoff of an option is a convex function of the underlying asset”Definition of a forward contractA long forward contract is an obligation to buy an underlying asset at a specified forward price (or the strike price F) on a known date (the maturity T). The contract requires no initial premium (it costs nothing to enter into the contract). At expiration, the holder of the forward contract (the long position) purchases the asset at the forward price (agreed upon at contract inception)2.The long position in a forward contract makes money when the price at expiration of the underlying asset exceeds the agreed-upon forward price. Thus:Payoff of long forward position=S(T)−F Where:S(T)=Tℎe spot price of tℎe underlying asset at tℎe expiration of tℎe contract(T) F=Tℎe forward price (agreed upon at contract inception)The short position profits only when prices fall, and thus:Payoff of sℎort forward position=F−S(T)=−Payoff of long position Graphically:1 In Chapter 5 of Fixed Income Securities (FIS-5), we discuss forward contracts as they apply to fixed income instruments: The underlying asset will be (i) a ZCB and then (ii) a coupon-paying bond. 2From a risk management perspective, the long position (buyer of the forward contract) isThe short position in the forward contract is graphically represented below3:Note the following points:Though the initial value of the forward contract is zero, the contract surely has valueduring its life. At any time (t) less than maturity (T), the contract is worth (S t−PV t(F)),where (PV t(F)) is the present value of the forward price calculated at time (t) and (S t)is the underlying asset spot price as of time t.4If at expiration (T), the spot price of the underlying asset matches the agreed uponfuture price (F), then there is no profit to be made under the forward contract.The forward contract is a zero-sum game in a sense that the gain to one party is the lossto the other party.The short position in the forward contract is the party who wrote the contract and soldto the long position. The short position has an obligation/liability at maturity towards3 The payoff clearly exhibits an unlimited loss when the underlying asset price (S T) exceeds the forward price (F).the long position only if (S(T)>F). However, if (S(T)<F) as can be seen in the payoff’sdiagram, the short position in the forward contract will benefit from the contract.The slope of the line is 1, and the forward contract is referred to as a ‘linear contract’.Futures and forwards are similar instruments. The major differences between them can be stated briefly as follows:1.Futures are traded in organized exchanges and forwards are custom-made and traded over the counter.2.Futures exchanges are cleared through exchanges clearing houses and there is a mechanism designed to reduce default risk. Forwards are not cleared and there is default risk.3.Futures contracts are market to market. Every day, the contract is settled and a new contract is entered.4.The security underlying the futures contract is standardized (the type of security is clearly specified and the timing and method of delivery as well). Forward contracts can be customized.5.Because of the mark-to-market system in the futures contracts, they have less credit risk than forward contracts (same point as point 2).Repos, Reverse Repos, and Flexible Repos5In a Repo (Repurchase agreement), one party sells securities to another party in returnfor cash, with an agreement to repurchase the securities (or equivalent) at a pre-agreed price (the repurchase price) and pre-agreed time (the maturity date).The long position in a repo (the buyer) acts as a lender of cash, and the short position (theseller) as a borrower.For the party selling the security and agreeing to repurchase it in the future, the transactionis a repo. For the party on the other end, the transaction is a reverse repo.The securities are used as collateral in this transaction.Profit from a repoAssume that a trader is entering into a repo transaction with a Repo Dealer as follows:P t=Tℎe price of tℎe collateral asset at time t5At time t, the trader exchanges this underlying asset in return for cash received from the Repo Dealer. The Repo Dealer pays (P t )6Amount borrowed from tℎe Repo Dealer =P t.At maturity of the repo (time T ), the trader must repurchase the underlying asset from the Repo Dealer. This underlying asset worth is now worth (P T ). But under the repo transaction, both parties would have agreed upon a repurchase price (reflecting the interest earned on the repo transaction). Let:X T =Tℎe agreed upon repurcℎase price of tℎe collateralWe have:X T =P t ×[1+Repo rate ]At time T, the trader repurchases the security at price (X T ) and sells it back in the market for (P T ).Uses of repos Repos are used to raise short term capital and are classified as money market instruments as a consequence.Categories of repos There are three broad categories of repos:1) Overnight repos: A one day maturity repo transaction. 2) Term repos: This is a repo with a specified maturity. 3) Open repo: This repo has no end date.A flexible repo is a repo with a flexible withdrawal schedule. Therefore, the party holding the collateral can sell it in parts before or at the maturity of the repo. There are two types of flexible repos:1) Secured : The municipality/customer receives collateral in the form of Treasury bonds,GNMA bonds, agency MBS/CMO.2) Unsecured : The customer does not receive collateral. The deal commands a higher spread.Differences between flexible repos and traditional repos: They are four major differences with a traditional repo:1) Convexity due to cash withdrawals, 2) Formal written auction like trade, 3) Enhanced documentation,4) Counterparties are usually municipal bond issuers.Options6Forwards and futures obligate the holder to deliver or accept the delivery of the underlyinginstrument at expiration. Options, on the other hand give the owner the right (not the obligation) to purchase or sell an asset.Call option Consider an investor who purchases a call option written on an underlying asset 7As such, the payoff of the long call option (the call option buyer) is: . The initial spot price of the asset is (S 0). The investor pays a premium (c ) to be able to take advantage of theflexibility offered in the option contract. The option matures at time (T ), when the underlying asset has a spot price of (S T ). The future spot price is unknown to all market participants when entering into the option initially (t=0). The option gives the investor the right (but not the obligation) to purchase the asset at time T , for a pre-agreed price of K (or X, or Strike), called the strike price.Long call option maturity payoff =�S T−K if S T >K 0 if S T ≤KTo put words into the mathematics, we say this:If at maturity of the option (T ), the underlying asset (S T ) is worth less than the strike (K ), the option buyer will not exercise his/her option. The instrument ends worthless.If at maturity of the option (T ), the underlying asset (S T ) is worth moreLong call option maturity payoff =Max (S T −K ;0)than the strike (K ), the option buyer will exercise his/her option, and the payoff is the excess of the asset’s value over the strike (S T −K ). Thus, we also write:The payoff of the short position (the call option writer) is the opposite of the long position as follows:Short call option maturity payoff =�−(S T −K ) if S T >K 0 if S T ≤KThe premium (c)8Profit for tℎe long position =(Maturity Payoff of tℎe long position )−FV (Premium ) must be adjusted from the payoff in order to get the net profit for each position: AndProfit for tℎe sℎort position =(Maturity Payoff of tℎe short position )+FV (Premium ) Or:Profit for tℎe sℎort position =FV (Premium )−(Maturity Payoff of tℎe long position )7 We assume that the underlying asset pays no dividends during the life of the option.8 This premium represents the compensation the seller accepts for the right granted to the buyer.option .Graphically:Put option Consider an investor who purchases a put option written on an underlying asset. The initial spot price of the asset is (S 0) at contract inception. The investor pays a premium (p) for the put option. The option matures at time (T ), when the underlying asset has a spot price (S T ). The future spot price (S T ) is unknown to both parties at the inception of the put contract. At maturity of thecontract, the long position will decide on whether to exercise his/her option to sell the asset for the strike price (K ):If at maturity of the option (T ), the underlying asset (S T ) is worth less than the strike (K ), the put option buyer (the long position in the put option) has the ability to sell at price (K ), an asset that is worth less than (K ). He/she will exercise his/her option. The payoff from exercising this option is clearly equaled to (K −S T ).If at maturity of the option (T ), the underlying asset (S T ) is worth more than the strike (K ), the put option buyer (the long position in the put option) has the ability to sell at price (K ), an asset that is worth moreLong put option maturity payoff =� K −S T if S T <K 0 if S T ≥Kthan (K ). He/she will not exercise his/her option. Thus, the instrument ends worthless. Thus, the maturity payoff of the long put option is:Or also:Max (K −S T ;0)The payoff of the short position (the option writer) is the mirror image as follows:Short put option maturity payoff =�−(K −S T ) if S T <K 0 if S T ≥KNote : From the perspective of the buyer, it is interesting to buy the put option when you think that the underlying asset ‘might’ decrease below the strike price. You can also purchase the put option ifoption + asset) is comparable to owning an insurance contract9 on an asset (subject to damage in value).Graphically the payoff/profit of a call option (long/short) are plotted below: Also, the payoff/profit of a put option (long/short) are plotted below:Let us consider other reasons why would a trader may consider buying options:Reasons why traders may want to calculate the arbitrage-free price of a call option1.New contract: before the option is first issued, a trader may want to know the price that the option will trade at.2.Mispricing: A trader may want to calculate the arbitrage free value of an option to determine if the option is mispriced in the market.9 In fact, in the financial economics literature, the portfolio containing the stock and the put option written on the stock is called a protective put. Likewise, the writer (seller) of the call might want to protect himself from a huge increase in the stock index, as such, he/she would buy the underlying7.1).Profit of call and put options 10The profit at maturity (T ) of the derivative to the longProfit (Long position )=Payoff of the Long Position (T )−FV (Premium )position is calculated as: WhereFV (Premium )=Tℎe Future Value of tℎe Premium paid at inception by tℎe long positionAlso, the profit at maturity (T ) of the derivative to the shortProfit (Short position )=FV (Premium )−Liability of the Short Position (T )position is calculated as:WhereLiability of tℎe Sℎort Position (T )=Payoff of tℎe Long Position (T )We build the cash flows table as follows:As already explained:The long position in a stock is purchasing the stock price today (a negative cash flow) to receive the proceeds from the sale of the stock at maturity (a positive cash flow).The short position in a stock receives money from selling the stock today (a positive cash flow) and has to redeem the stock at maturity (negative cash flow). The long stock and the short stock are in a zero-sum game.The long position in a ZCB of redemption value K has to pay for the ZCB today. The price of which is the PV of the redemption amount. Because this is a purchase, it is a negative cash flow. At maturity though, the long position receives the redemption value of the bond K, a positive cash flow.The long position in any derivative has to pay a premium for entering into the derivative transaction at time t=0 (negative cash flow). At maturity, this long position is entitled to a payoff.The short position in any derivative receives the premium from the long position (positive cash flow) and at maturity; the short position is responsible for paying off the payoff arising to the long position. Because this payoff is a liability, it is a negative cash flow.10 Though not directly discussed in Neftci-1, the concept here is discussed in FIS-6 and frankly theSwapsA swap is an agreement between two counterparties for selling and purchasing cash flowsinvolving various currencies, interest rates and a number of other financial assets.The counterparties borrow in sectors where they have an advantage and then exchange theinterest payments.In a simple IRS, at the end, both counterparties secure a lower rates and the swap dealerwill earn a fee.The simple IRS exampleThis contract allows parties to exchange payments between two differently indexed legs, starting from a future time instant (Tα). At every time (T i), the fixed leg pays out the amount:Notional×τi×KWhere: τi=Tℎe year fraction between T i−1and T iK=Tℎe fixed rate of interest for tℎe fixed payments of tℎe IRS swapNotional=Tℎe notional amount of tℎe swap Whereas the floating leg pays out the amount:Notional×τi×L(T i−1,T i)Where: L(T i−1,T i)=Tℎe floating rate tℎat resets at tℎe previous time (T i−1) and is used tℎe payment at time (T i) Graphically, we have:A counter-party in this plan vanilla swap may be able to close out the transaction by payingthe net present value (NPV) of future swap payments.Pricing swaps11One method for pricing swaps and swaptions is to decompose them into forwards andoptions.The forwards can then be priced separately, and the corresponding value of the swap can bedetermined from these numbers.Two examples of swaps1)The simple IRS (Interest Rate Swap): Each counter-party borrows in the market (fixed rate and floating rate market) where it has an advantage and they both exchange the payments.2)The Cancelable Swaps: In this swap, each party has the option to cancel the transaction before maturity and extinguish the obligation to pay the PV of future payments. They come in two flavors: Callable swaps and Puttable swaps.Some properties of Cancelable swapsPopular among institutions with an obligation in which they are to repay principal beforematurity. The embedded option on the swap can be exercised to honor such liabilities.They can be used as hedge instruments.They allow institutions to mitigate maturity mismatch between assets and liabilities due toprepayments options or early surrenders.Past examsSOA Spring 2015 QFIC Q11 on Repo (Must Read)SOA Spring 2013 APM Q3 (Must Read)PAK Practice questionsQuestion 1a)List the differences between forwards and futures:Question 2Assume that the S&P 500 index is at 100. For a single premium of $100, a life insurance company had sold the following type of products:Contract 1: promising to pay 100 at maturity of the contract in 5 years plus any excess of the S&P 500 over its initial value of 100.Contract 2: The Company promises to pay the excess of two quantities:90% of the initial premium accumulated at 2% per annum andThe proceeds from the investment of the initial premium into a fund that performs exactlyFor each contract, determine the following quantities:1.The maturity of the contract2.The type of derivative embedded in the liability and the underlying asset.3.Identify the payoff of the liability, and the strike amountQuestion 3a)Describe the bull call spread optionQuestion 4a)Show the payoff of a cap portfolio (short stock + long call): In mathematical terms andthe plot.b)What is the net profit of the cap portfolio?c)Show that the ca portfolio can be viewed as a long put position.SolutionQuestion 1: Differences between futures and forward contractsThere are important differences between these two contracts:Forward contracts are settled at expiration while futures contracts are settled daily. At the end of each trading day, the clearinghouse adjusts the margin accounts to reflect the daily gain/loss to each counterparty of the transaction. This is called mark-to-market.Forward contracts are settled at the agreed-upon forward price while futures contracts aresettled at the settlement price determined on the last trading date.In a forward contract, there are no cash flows until expiration whereas for a futurescontract, there are daily cash flows to reflect the gain and loss to each counterparty.Futures contracts are more liquid than forward contracts. Futures contracts can be offsetany day by entering into an opposite transaction.Because of the daily mark-to-market accounting, futures contracts have lower (if any) creditrisk than forward contracts.There are typically daily price limits in future markets. Such limits are market moves thattrigger a temporary halt in trading.Question 2: Life insurance index contract1.Contract 1 has a maturity of 5 years and contract 2 has a maturity of 10 years.2 and 3.Embedded derivative in Contract 1:A call option on the S&P 500 of maturity 5 and strike 100.Embedded derivative in contract 2: Payoff is: Max (0.9×100×(1.02)10;F 10) Where: F 10=tℎe terminal value of tℎe fund tℎat mimics tℎe S &P 500 The payoff is equal to: Max (109.7;F 10)=F 10+Max (0;109.7−F 10)=F 10+Payoff of a put option Thus, the embedded derivative is a put option on the underlying fund of strike 109.7 and maturity 10 years . Question 3: The Bull spread 12Bull Call Spread Option (Neftci practice problem 6 on page 11)A bull spread call is a strategy that involves purchasing call options at a specific strike pricewhile also selling the same number of calls of the same asset and expiration date but at a higher strike. A bull call spread is used when a moderate rise in the price of the underlying asset isexpected. The maximum profit in this strategy is the difference between the strike prices of the long and short options, less the net cost of options. Question 4: The cap portfolio The combination (short asset and long call option on the asset) is called a cap 13. Let us now look at the mathematics of the cap (c-S ). The payoff table is captured below:Note the following about the payoff of the cap position: Max (0; S (T )−K )−S (T )=�If S (T )<K or if (–S (T )>−K ) Tℎe payoff is –S (T ) If S (T )>K or if (–S (T )<−K ) Tℎe payoff is –K12 More of this type of strategies in QFIQ-120-19 section 4. 13 The insight here is that without the call option, the short position in the asset has an obligation/liability of amount (S(T)) at maturity time T . However, with the long call option added to exceed that strike level .Note:That is why in the table, the final payoff for the position is simply (Max(−S(T);−K)).It is also very important to realize that the expression (Max(−S(T);−K)) is not the same as(−Max(S(T);K))14.Graphically, the maturity payoff of (the sum of the short stock + the long call option) yields a liability (opposite payoff) that cannot exceed a certain cap (the opposite of the strike level) as can be depicted below:Where we clearly see that when (S(T)<K), the liability (opposite payoff) of the cap is (-S(T)) and when (S(T)>K), the liability (opposite payoff) of the cap is (−K).The net profit of the capAs explained earlier on also, the profit at maturity (T) of the derivative to the short position is always calculated as:Profit(Short position)=FV(Premium)+Liability of the Short Position(T) Where Liability of tℎe Sℎort Position(T)=−Payoff of tℎe Long Position(T) Thus, the net profit from the cap is calculated as:Max�−K;−S(T)�+FV[S(0)−c]Thus:14For instance, (Max(−3;−10)=−3) while (−Max(3;10)=−10). These are two differentProfit from the cap portfolio=Max�−K;−S(T)�+[S(0)−c]×(1+r)TThe cap can be viewed as a long putWe are claiming that the net profit of the cap is equal to the net profit of a long put. How so? Once again, we make use of the put call parity identity:c−S(0)=p−K×(1+r)−T Multiplying this line by the accumulation factor (−(1+r)T), we get:[S(0)−c]×(1+r)T=−p×(1+r)T+KBy substitution of the FV of the premium ([S(0)−c](1+r)T) into the net profit for the cap, we get: Profit from the cap portfolio=Max�−K,−S(T)�−p×(1+r)T+KBy allowing K to enter into the Max-term, we get:Profit from the cap portfolio=Max�K−K;K−S(T)�−p×(1+r)T Thus: Profit from the cap portfolio=Max�0;K−S(T)�−p×(1+r)T=Profit for a long put option。