derivatives - session 02 and 03

7/28/2019 Derivatives - Session 02 and 03

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DERIVATIVESForwards & FuturesKAUSHIK DESARKAR

GOA INSTITUTE OF MANAGEMENT



DERIVATIVES

• FORWARDS

• PRICING & VALUING

• INVESTMENT ASSETS Vs. CONSUMPTION ASSETS



DERIVATIVES

• If you have Rs. 1 million and invest at 5% pa compounded annually. Then

• After 1 year you have Rs. 1.05 Million• After 5 years you have Rs.1.276..Million

• If the frequency of compounding is increased to 2 and then 4 we have…



DERIVATIVES

• Now if you were presented with the opportunity of continuous compounding,

that is 360 days a year, then you results would be

• 1 year = Rs.1.05126 Million• 5 year = Rs.1.284 Million

• The simplest way to complete the calculation is use the Exponential Formula• So in the above , set R = 5%, n= 1 and 5

• 1 year = 1 million X Exp (0.05 X 1) = Rs. 1.05127 Million• 5 years = 1 million X Exp (0.05 X 5) = Rs. 1.284 Million



DERIVATIVES

• There is a relationship between the Continuous Compounding Rate and the

Discrete Compounding Rate .

• That relationship is as follows

•

Rc = m X Ln(1+ R/m)

• Where • Ln = the natural logarithm (base e)• Rc is the Continuous Rate • m = Frequency of compounding • R = Discrete Rate



DERIVATIVES

• Since Futures and Forwards are closely related, we start with Forward Pricing to gaininsight into No Arbitrage(NA) pricing mechanism.

• The concept of No Arbitrage is premised on creating an equivalent portfolio that hasthe same payoff as the derivative contract. No arbitrage is enforced when both arepriced the same.

• Let us start with a simple case

• You want to trade 100,000 shares of SAIL 30 days from today. There is no dividenddate in these 30 days.

• Spot = 92.00 and 30 ‐day Forward = 94.00.• The risk free rate is 6.5%.

• Do you agree to the above ??



DERIVATIVES

• The easiest way to check the offer is

• Simply discount the Forward Price of (Rs.94.00) by the Risk Free Rate (6.5%) andcompare it with the Spot Rate (Rs.92.00).

• PV(Forward Contract) = Rs. 94 * exp( ‐0.065 * 30/365) = Rs.94.5035• Spot Price = Rs.92.00 • Difference of Rs.2.5035 per share ‐ Arbitrage ????

• And if the Forward Price is Rs. 92.25 ??

• This provides us with the first basic (NA) pricing mechanism:• the Present Value of the Forward Price (Future Price) = Spot Price OR …• The Future Value of the Spot Price = Forward(Future) Price



DERIVATIVES

• Hence, the Forward Price on any Investment Asset with Zero income/yield during thecontract’s life is simply

• F = S * exp(‐rf *T)

• Where F is today’s (T/365) year Forward Price• S is today’s spot price• Rf = risk free rate (usually Libor/Mibor rates)

• Usually Investment Assets will provide known income during there life so we can gothrough 2 cases

• Income/Dividend is known• The Yield is Known



DERIVATIVES

• Today is 14 th July 2013. When we know the value of the income – for instance weknow that the ex ‐div date for Company XYZ is 24 th July 2013 and I want a ShortForward Contract on 50,000 shs maturing on 31 st July 2013. Assume Rf = 6.5% pa.

• The simplest way to go about it is as follows• Spot Price = Rs.425.75•

Dividend per shs = Rs.12.00• PV(Dividend) = 12 * exp( ‐0.065 * 10/365) = Rs.11.94• The buyer(long) will not get the benefit of the dividend – he will only get the capital

gain benefit of holding the stock on 31 st July 2012.•

Deduct the PV(Dividend) from the Spot Price = 425.75 – 11.94 = Rs.413.81• Forward Price (NA) = 413.81 * exp(0.065 * 17/365) = Rs.415.065

• Formula : F = (S – I)*exp(rf * T) where I = PV(Known income)



DERIVATIVES

• If the underlying is known to provide a Known Yield, the formula is

• Assuming the known yield is q • F = S * exp(T * (rf ‐q))

•

The above is very widely used in the Forex Market for both• Pricing of Forwards• Reverse Engineering and its application for trading

•

Let us see 2 examples related to the above…



DERIVATIVES

• You are faced with the following situation on Friday 13 th (Superstitious !!)• INR‐$• Spot : 54.90• 30‐day INR Rf = 6.5% & 30‐Day $ Rf = 1.5%

• The 30‐day Forward Rate = 54.90 * Exp[(0.065 ‐0.015) X 30/365] = 55.126• The Formula is Spot X Exp[(rf – q) X T]• Where rf is the domestic risk free rate (the domestic spot value)

• If the Forward Rate is quoted at 56.50, is there a trading (aka Arbitrage) opportunity?

• (Work the numerical and compare)



DERIVATIVES

• We have understood the pricing part very well – valid for new contracts but how do we value Forward Contracts that were entered into previously ??

• Build a methodology for valuing of Contracts because• You should know the value of any position in any instrument you hold • The method of valuation provides the ground work for Risk Management

• Let us again use the non ‐dividend/income paying situation• You have a contract entered into sometime back with a Forward Price K. Today’s

Forward Price for the same maturity date is F. There is exactly T years to maturity andthe risk free rate is rf .

• Value of Long Forward = f = PV(F – K) = S – PV(K)• Where PV(X) = X * exp(‐rf * T)• What is the value of the Short Position ??



DERIVATIVES

• Using the previous discussed methodology, the value of Forward Contracts oninstruments known to pay an income or yield are the following

• f = S(0) – PV(Income) – PV(K)• f = S(0) * exp(‐q * T) – PV(K)

• Refer to Chap 5 of JCH for detailed exposition on valuing of Forward Contracts.

• The reason why we don’t value Future Contracts is because• THEY ARE TRADED IN THE MARKET HENCE THEIR VALAUTION IS COMPARED AND

SETTLED IN THE MARKET EVERYDAY. !!!



DERIVATIVES• We resign from the Investment World and hit the Commodities Universe.

•

The Pricing of Forwards (and Futures) is a bit different from that of Investment Assetsbecause Commodities have uses in Production, Manufacturing etc.

• Hence they have a CONVENIENCE factor. This gives rise to the famous CONVENIENCEYIELD.

• If you heading a Investment Bank – you can trade on Tata Motors using the underlyingStock, Futures Contract and Options.

• If you are heading the company Tata Motors – you cannot use Steel Futures tomanufacture cars. Cars are manufactured using the Raw Material itself.

• The benefit from holding Steel Inventory is a convenience factor rather than holdingderivatives on the commodity itself.



DERIVATIVES• This Convenience Yield is responsible for Forward/Future prices being below Spot

Prices (usually). This is known as BACKWARDATION.

• Analysts interpret the Convenience Yield as a measure/indicator of shortages andimportance of the underlying commodity.

•

On the basis of our statistical analysis… The results of our hypothesis tests support the theory and empirical validation stated by Weymar (1966) which says that convenience yields are not only related to inventory level at the same time but also to future inventory levels. As inventory levels show strong autocorrelation, we could show a similar relationship by using static lifetimes of stocks. …The convenience yield can therefore be seen as an indicator for future criticality of a commodity.

• FIM Research Center, Institute of Materials Resource Management, University of Augsburg Universitätsstrasse 12 86159 Augsburg, Germany



• Convenience Yield of US Oil Stock (Red = Oil Stk Cover, Blue = CY)



DERIVATIVES• Usually Commodities have Storage Costs associated with them hence when using NA

pricing we can state the following relationship

• If • r = Risk Free Rate• u = Storage Cost per unit as a constant proportion of the spot price• y = Convenience Yield

• Forward Price = Spot Price * Exp ((r + u – y)*T)

• It is very common for (r+u) to be referred to as the Cost of Carry (C). In the case of an investment asset, the cost of carry is computed as (r ‐q) where q = yield from theunderlying.

•

• Further Reading – JCH Chap 5 on Consumption Commodities.



DERIVATIVES

Forward No ‐Arbitrage (Strike) Price Value of Contract

No income underlying F = Spot x exp(rf x T) f = Spot – K * exp( ‐rf x T)

Known income during tenure

of contract

F = (Spot – PV of Income) x

exp(rf x T)

f = (Spot ‐ PV of Income) – K *

exp( ‐rf x T)

Known yield on the underlying : q% pa (continuous)

F = Spot x exp[(rf ‐q) x T] f = Spot*exp( ‐q*T) – K * exp( ‐rf x T)

On commodity F = Spot x exp[(rf+u ‐y) x T] Try to derive !!

Rf = risk free rate, T = time to maturity (in yrs), u = storage cost (continuous compounding), y = convenience yield



DERIVATIVES• In the opening session we had mentioned that a derivative contract

represents 3 markets

• The Spot market for the underlying• The Debt(money) market – provides the interest rates• The Derivatives market

• Try to break the Long and Short Forward Contracts on Non ‐dividend payingunderlying into appropriate Spot and Debt market instruments.

•

In short, financially engineer a Long Forward and a Short Forward using theSpot and the Debt market. See the result !!

derivatives - session 02 and 03

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