Pricing Forwards And Futures

Market Value IndicesnnnnMV(0) 150(50) 40(100) 10(500) 16500MV(t) 150(50) 80(100) 30(500) 30500I(0) 100I(t) I(0)[MV(t)/MV(0)] 100[30500/16500] 184.85Peter RitchkenForwards and Futures Prices45Price Weighted IndexnnnV(0) [150 40 10]/3 66.667V(t) [150 80 30]/3 86.67I(t) [V(t)/V(0)]I(0) [86.67/66.667]100 130Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices4623

Pricing Futures on Price Weighted IndicesnnnStocks in the index are A and B.A pays a dividend of size dA at time tA.B pays a dividend of size dB at time tB.F (0 ) S(0 )erTAS(0 )erTB der (T tA der (T tBF (0 ) [SA(0 ) S[dAer (T tF (0 ) I (0 )ePeter RitchkenrTA)B( 0 )] e d FVBerT)AB ) r (T tB )( dividends])Forwards and Futures Prices47Stock Index Arbitrage with a ValueWeighted IndexPriceSharesDivA401m0.5Time toDiv.10 daysB352m0.512 daysC252m--Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices4824

Example:Stock Index Arbitrage with Price WeightedIndexnComposition of Index:nnnnnnA 40/ (40 35 25) 1/4B 7/16C 5/16I(0) 400Multiplier 500Each contract controls 400(500) 200,000.Peter RitchkenForwards and Futures Prices49Stock Index ArbitragenPortfolio Composition:nnn(1/4 )(200,000) 50,000 or 1,250 shares of A(7/16)(200,000) 87,500 or 2,500 shares of B(5/16)(200,000) 62,500 or 2,500 shares of CPeter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices5025

Stock Index mberShares1,2502,5002,500Dividend IncomeReceived andInvested at 10%Peter Ritchken628.43 1256.18-Forwards and Futures Prices51Stock Index ArbitrageAmount owed 200,000exp(0.10(30/365) 201,650.61n less dividends and interest. - 1,844.61.n Total . . 199,766n Theoretical Futures Price 199,766/500 399.53Note, in this example I(0) 400nPeter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices5226

Stock Index Futures Prices and IndexPricesnnIf Stock index futures prices are upward sloping as afunction of maturity, then market participants areforecasting that the market will increase ( ie thesentiment is bullish)Is this true?Peter RitchkenForwards and Futures Prices53Forward Contracts on FOREXnnPrice of 1 British Pound is S(0) dollars.Strategy 1:nnnInvest S(0) dollars at the risk free rate to obtainS(0)exp(rDT) dollars at date T.S(0) dollars grows to S(0)exp(rDT) dollarsPeter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices5427

Forward Contracts on FOREXnStrategy 2:nnnnSell exp(rFT) forward contracts on the pound.Each contract gives the obligation to exchange 1 pound forF(0) dollars.Buy 1British pound. Invest in riskless rate in Britain to obtainexp(rFT) pounds.Deliver the pounds for exp(rFT) F(0) dollars.S(0) grows to exp(rFT) F(0)Peter RitchkenForwards and Futures Prices55Forward Contracts on FOREXnnTo avoid arbitrage opportunities:S(0)exp(rDT) exp(rFT) F(0)orF(0) S(0)exp[(rF-rD)T]nForward prices relate to spot prices depending oninterest rate differentials.Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices5628

Pricing Futures Contracts on DividendPaying “Stocks”nnnnAssume interest rate is r and storage costs are afixed percent of the spot price. The underlying paysno dividends.F(0) S(0)exp[(r u)T]Now assume the underlying pays a continuousdividend yield, expressed as a percent of the price.ThenF(0) S(0)exp[(r u-d)T]Peter RitchkenForwards and Futures Prices57Pricing Futures Contracts on DividendPaying “Stocks”nnnF(0) S(0)exp[(r u-d)T]The spot price of the S&P500 index is 1000. Thedividend yield is 3% per year. Interest rates are 5%continuously compounded. Storage costs are 0%. Aone year futures contract should have a futures priceofThen F(0) S(0)e (r-d)T 1000 e0.02 1020.20Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices5829

Forward Prices of a PropertynYou can buy a property now for 100m. Alternatively,you can enter into a forward contract to purchase theproperty in two years time.nnThe property has a maintenance fee that is 1% of the price.The property has buildings that provide rental income that isestimated at 6% of the price per year.The fair forward price of the property is100e(r u-d)T 100e(0.05 0.01-0.06)2 100 million dollars.Peter RitchkenForwards and Futures Prices59Futures Prices of Storable Commodities.nCommodity forward contracts have two important features thatare not present when the underlying is a financial asset.nnnStorage CostsConvenience YieldsStorage CostsnnnWarehouse space, transportation costs, spoilage,insurance.We will represent these charges as a fraction of the marketprice of the commodity.If storage costs are 20%, this implies that the annualizedstorage costs are about 20% of the spot market price.Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices6030

Futures Prices of Storable Commodities.nThe convenience yieldnnnnnUnlike securities, commodities are usually consumed or used in aproduction process.Having a commodity on hand has value since it allows theproduction process to continue without disruptions.If the commodity is abundant then there is not much benefit fromhaving it in storage. However,If the commodity is scarce, then having the commodity in inventoryis very beneficial.One can view the potential benefit of having a commodity onhand as a yield, just like a continuous dividend yield. Weexpress the convenience yield as a percent of the price in anannualized formPeter RitchkenForwards and Futures Prices61Futures Prices of Storable Commodities.nA 2% convenience yield means that having the item readilyavailable comes at a cost that accrues at a rate of 2% ofthe current price of the commodity per year.Example:Spot price 100; Interest Rate 5%; Storage Cost 8%Convenience Yield 7%. Time to expiration is 1 year.F(0) S(0) e(r u-k)T 100 e(0.05 0.08 –0.07) 106.18Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices6231

Upper Bound for Commodity Forwards:The Cost of Carry ModelnIf forward price is very high:nnnnnSell the forward contractBuy the commodity using borrowed funds.Pay the storage fees with borrowed fundsDeliver the commodity for the forward price.Profit is:FO(0) – S(0)e(r u)TClearly, to avoid riskless arbitrage:F0(0) S(0)e(r u)TPeter RitchkenForwards and Futures Prices63Lower Bound for Commodity ForwardsnnnnIf the forward price was very low we would like toinitiate the reverse cost of carry. In particular, weneed to short sell the commodity.We must borrow the commodity from party, A, andthen sell it.This deprives A from a convenience yield. A is givingup having inventory around to meet unexpectedneeds! Clearly A needs to be compensated for this.This compensation is the convenience yield,represented by k% per annum as a fraction of thecommodity pricePeter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices6432

Lower Bound for Commodity ForwardsnnnnA does get to save the storage fees, represented byu% per annum as a fraction of the commodity price.So A requires a net compensation that accrues at therate of k-u%If we borrow one unit now, we have to compensateA by providing e(k-u)T units at date T.Equivalently, for every e-(k-u)T units that we taketoday we owe 1 unit at date T.Peter RitchkenForwards and Futures Prices65Lower Bound for Commodity ForwardsnnSo we go long 1 forward contract, and we borrow eunits, and sell them.At date T, we purchase 1 unit for the forward price,and return 1 unit to the borrower. The net profit is:(k-u)TProfit [e-(k-u)T]S(0)erT –FO(0)S(0)e(r u-k)T - FO(0)nProfit if S(0)e(r u-k)T FO(0)Hence to avoid riskless arbitrage:nFO(0) S(0)e(r u-k)TnPeter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices6633

Commodity Forward Prices.S(0)e (r u-k)T FO(0) S(0)e (r u)TnnnThe upper bound is easy to obtain. The lower boundis a problem, since the convenience yield is notknown.Forward prices on commodities can have such wideupper and lower bounds, that little can be said aboutthese prices from arbitrage arguments alone!We need a deeper model.Peter RitchkenForwards and Futures Prices67Bounds on Forward PricesForward PriceUpper Bound ( easy)Actual PricesLower Bound ( ?)MaturityPeter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices6834

Interpreting the Convenience YieldnnLet k be the convenience yield. You can view this likea dividend yield, but it is not observable!Then, we will write:F(0) S(0)exp[(r u-k)T]nnnWhen will k be large?When will k be small?Can k depend on the time horizon? ( ie. Different kvalues for different futures contracts.)Peter RitchkenForwards and Futures Prices69The Implied Convenience YieldnIf the convenience yield was k%, thenFO(0) S(0)en(r u-k)TIn practice the forward prices are given. So we can implyout the average convenience yield over the period [0,T].Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices7035

On April1st,ExampleS(0) 1.96, u 1%, r .24%21.28%15.46%13.96%Implied k 16.13%Peter RitchkenForwards and Futures Prices71Implied Convenience YieldsnnnnConsider maturity s and t contracts with s tFOt(0) FOs(0)e(r u-k)(t-s)Why is this true?We can imply out convenience yields over specificperiods.Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices7236

On AprilExample (Continued)S(0) 1.96, u 1%, r mplied k 16.13%ImpliedratesApril-May May –July July-Sept16.13%19.29%25.83%Peter RitchkenForwards and Futures Prices73Forward Prices, Spot Prices, and ForecastsnExpect future prices to be high:nnnnnnSellers will increase their inventories so as to capture thehigh prices.Buyers will want to buy more today.Buyers will demand more from sellers who prefer to sellless.These conflicting positions will drive spot prices upPrice expectations are an important part of the formation ofspot prices.With inventory, changes in future expectations haveimpact on current prices.Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices7437

Price Variations and Storage IssuesnAs storage costs increase, the connections betweenexpected future prices and current prices diminish.nnnnWith high storage costs the alternative of holding inventoriesfor future sale is less attractive than if storage costs arenegligible.As storage costs increase, expectations of higher futureprices will have less impact on current spot prices.As storage costs increase, the price variation over time inthe forecasts increase.As storage costs increase, actual price variations over timeincrease.Peter RitchkenForwards and Futures Prices75Example: ElectricitynnnElectricity is difficult to store. Expected priceforecasts vary by the hour.Actual prices fluctuate widely.Cost of carry model and reverse cost of carry modeldo not hold.Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices7638

Spot Forecasts Versus Forward PricesnWhat is the link between forward prices andexpected spot prices for non storables?Peter RitchkenForwards and Futures Prices77Futures Prices and Expected Future SpotPricesnnSuppose k is the expected return required byinvestors on an asset. (k r risk premium)We could invest Fexp(-rT) today to get S(T) at dateT.nnnnnHow would you do this?HenceFexp(-rT) E[S(T)]exp[-kT]No systematic risk?Positive systematic risk?Negative systematic risk?Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices7839

Futures Price For NonstorableCommoditiesnnnE[S(T)] 100F(0) 100Perhaps a speculator would enter a forward contractat 90. Expect to make a 10 profit.nnnWhat if F(0) 99The speculator expects to make 1, but risk .Futures prices may not be unbiased estimators offuture spot prices.Peter RitchkenForwards and Futures Prices79Normal BackwardationnIf all hedgers were SHORT, all speculators, LONG.nnnnHedgers will pay a premium to reduce riskSpeculators will require a premium to accept risk.F(0) E[S(T)]This market is called normal backwardation.Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices8040

A Contango MarketnnIf all hedgers were LONG, all speculators, SHORTHedgers will pay a premium to reduce risknnSpeculators will require a premium to accept risk.F(0) E[S(T)]nThis market is known as CONTANGO.nIf F(0) E[S(T)] E[F(T)] .nFutures prices are martingalesPeter RitchkenForwards and Futures Prices81Expected Spot Prices and Forward PricesnThe risk premium equals the difference in prices( expected spot less forward), as a percentage of theforward.PriceSpot ForecastsForward CurveMaturityPeter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices8241

Risk PremiumsProducers PerspectivennnnnThere is no theoretical reason for forward prices to act as anunbiased estimator of future expected prices.The risk premium may be positive or negative!The risk premium may start out big positive, say, and thendecline to zero at the expiration date.The risk premium exists due to risk aversion. Producers (sellers)are generally risk averse. They may have debt obligationsassociated with their capital stock. They need stable cash flows.So, they may be willing to accept forward prices that are lessthan expected spot prices.Peter RitchkenForwards and Futures Prices83Risk PremiumsConsumers PerspectivennnBuyers may also have a desire to avoid price risk.They may prefer to pay a premium for price security.This implies that forwards might be higher thanexpected prices and risk premiums would benegative.Relative desire for price security between thesegroups that will determine the sign of the riskpremium.Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices8442

SpeculatorsnnSpeculators are also present. They are concernedabout forward prices relative to expected futureprices.Does the strategy of buying forward and then sellingat spot prices increase the total risks of a welldiversified portfolio?nYes-speculators need a positive risk premium. Forward willbe below expected pricesPeter RitchkenForwards and Futures Prices85The Market Price of RisknThe Risk Premium can be decomposed into twoparts:nnnnVolatility ( or the amount of risk)The price per unit of risk.The price per unit of risk is called the market price ofrisk.Estimating the market price of electricity risk isdifficult.Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices8643

SummarynnnnnnForward prices and futures prices are close to eachother.Futures give cash flows right away, forward do not.This has hedging implications.For securities, the cost of carry model and thereverse cost of carry model gives us forward prices.Market imperfections lead to bounds on prices.For commodities storage costs and convenienceyields must be considered.The upper bound is easily computable, but the lowerbound depends on the convenience yield.Peter RitchkenForwards and Futures Prices87SummarynnnnFor commodity forwards and futures there is someflexibility as to where the forward prices can fall.That is, arbitrage alone, cannot identify thetheoretical price, and we need richer models.When storage costs are high, forward prices andindeed spot prices and their forecasts can be morevolatile. Cheap storage has the effects of smoothingprices over time.Implied convenience yields can be extracted fromfutures prices and can be informative.Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices8844

SummarynnnIn general commodity forward curves will reflectseasonality since the convenience yields will fluctuateaccording to aggregate inventory levels.Electricity is not readily storable, and forward priceshere represent a challenge.For non storables, forward prices are linked toexpected future spot prices. However, they are notunbiased estimators.Peter RitchkenForwards and Futures Prices89SummarynnFor electricity prices, we may expect spikes and other“weird” phenomenon. For storables, as futureexpectations change, a ripple effect is felt through tocurrent prices. Price changes should therefore bemore smooth.If storage costs increase, then price volatility shouldincrease.Peter RitchkenPricing Futures and Forwards by PeterRitchkenForwards and Futures Prices9045

Pricing Futures and Forwards by Peter Ritchken 2 Peter Ritchken Forwards and Futures Prices 3 Forward Curves n Forward Prices are linked to Current Spot prices. n The forward price for immediate delivery is the spot price. n Clearly, the forward price for delivery tomorrow should be close to todays spot price. n The forward price for delivery in a year may be further