FRM 2006-Delta Normal VAR

发布时间:2010-01-19 共1页

Today we published a 1 hour 20 minute movie on delta normal VAR and Monte Carlo simulation methods.These movies focus on two Jorian chapters that are *critical* toNovember’s FRM exam. We hope to help especially if you are confused bycovariance and correlation matrices (see, we read our mail). Plus,we’ve got a new set of Quantitative questions. It’s summer and you should have already started your preparation. You’ve seen the Piratessequel, now view a movie with real production value. This one is adramatic nail-biter. It gets so dicey we have to open Excel (twice) tocope with matrix math. Can Geometric Brownian Motion (GBM) be appliedto interest rate dynamics (hint: can a mean-reverting series really be a random walk?). Sorry, we won’t give away the ending…
SwapsThereare several swap instruments: equity swaps (you pay me LIBOR and inexchange I’ll pay you the S&P 500 return), commodity swaps (pay methe floating price of oil and I’ll pay you a fixed price), currencyswaps (pay me interest in German marks and I will pay you interest inAmerican dollars). The largest category is interest rates swaps and a"plain vanilla" interest rate swap is where the company pays a fixedrate (fixed rate payer) in exchange for receiving a floating rate (floating-rate receiver).

Of course, the counterparty to this swap is the opposite position. The counterparty is a floating-rate payer, fixed-rate receiver. A few things to keep:
  • Most of them do not require an exchange principal at inception (that’s why it’s called notional amount). The notable exception is currency swaps: in general, currency swaps exchange principal at inception and at maturity.
  • Atinception, the value of the swap is typically zero (why would acounterparty enter into a negative value?). Subsequent to inception, asthe truth is revealed on the floating side, the swap becomes netpositive or negative. That’s why the pricing problems must besubsequent to inception.
  • For the interest rate swap,remember it breaks down into a floating-rate bond on one side and afixed-rate bond on the other side. Most of the examples in Hull arevariations on the simple idea that the Value [swap] = Present Value[fixed-rate bond] - Present Value [floating-rate bond].
  • The key simplification in the pricing of the plain-vanilla interest rate bond is this: the value of the floating-rate bond is exactly its notional immediately after a coupon (interest) payment.
Forexample, assume a semi-annual interest rate swap on a notional of $100.The fixed-rate payer will pay 6%. Assume the LIBOR curve is nicelysmooth. At 3, 6, 9 and 12 months, the LIBOR rates respectively are4.5%, 5%, 5.5% and 6% (further, the rate curve isn’t moving over time;that’s a convenience so we can assume the same rates three months ago).Finally, assume we are only 15 months away from maturity (convenient sowe only have three cash flows). The value of the swap is the differencebetween the two (implicit) bonds. We price the fixed bond as usual; itsprice is the present value of the three cash flows to be payed/received($3, $3, and $103):

Theimplicit floating-rate bond isn’t too much trouble after all, if weremember that its price is $100 (the principal) immediately after itpays the next coupon! Why? At that point, the coupon is paying LIBORand rates are LIBOR, so you’d pay $100 for such a bond because bydefinition it earns market rate. So will only need to present valuethis implicit bond at the next coupon date; i.e., in three months itwill pay 1/2 (semi-annual) of 5% (the six month rate) of the $100notional. In three months, the future value of this bond is $2.5 plusthe principal:

Thevalue of this swap is therefore $101.40 - $101.35 = $0.05. That’spositive five cents for one counterparty and negative five cents forthe other.
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Section V Notes ReadyPosted by David Harper on 15th July 2006
Yesterday we posted the Section V Study Notes (Investments), a question set on Investments, and a new movie tutorial, Introduction to Value at Risk(VAR). We hope you agree this is the best way to learn about VAR. Thisweek’s movie is introductory: We review one-period VAR, absolute versusrelative VAR, and n-period VAR.
Interest rate parity (IRP) Theinterest rate parity (IRP) formula is a just a flavor of thecost-of-carry model that we reviewed in the last two posts. Thecost-of-carry model says the forward rate is a function of thecompounded spot rate. The difference is that, instead of an underlyingphysical commodity (e.g., corn, oil futures), we are dealing withforeign currency. So the forward exchange rate is a function of the spot exchange rate:

Inthe IRP, the spot exchange rate is simply the result of (continuously)compounding the difference between the domestic riskless rate and theforeign country riskless rate (r - rf). What if they happened to beequal? Then exp(0) = 1 and the forward exchange rate would equal thespot exchange rate. Why? Because if the country rates are equal, you’llend up at the same place regardless of whether you hold home currencyor covert immediately to foreign currency. The IRP, as a flavor of thecost-of-carry model, depends on the "no arbitrage" assumption: you needto be roughly indifferent to holding domestic or foreign currency.
Nowassume a 1.2 spot exchange rate, a domestic riskless rate of 5% and aforeign riskless rate is 2.75%. For a three month period (t=0.25), theIRP says the forward exchange rate must be 1.207:

Theforward rate must be higher. If it were not, you would always hold thedomestic currency and an arbitrage opportunity would (temporarily)exist.
Normal BackwardationKeep in mind that contango is not when the forward rate is greater than the spot rate. Contango is when the forward rate exceeds the expected spot rate:

Itis not obvious why the forward rate would be different from theexpected (future) spot rate. Contango is not what we expect; after all,why should we expect speculators to pay more for a futures contractthan its expected spot price. Normal backwardation, however, isreasonable when we consider that speculators (buyers of the foward contract) expect a profit.If speculators expect a profit, then they will pay something less thanthe expected (future) spot price. Therefore, normal backwardation is areasonable phenomenon:

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