Zvi Wiener

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Framework for pricing derivatives

• Zvi Wiener
• 02-588-3049
• http//pluto.mscc.huji.ac.il/mswiener/zvi.html

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The Market Price of Risk

• Observable underlying process, for example stock,
  interest rate, price of a commodity, etc.

Here dz is a Brownian motion. We assume that m(x,
t) and s(x, t). x is not necessarily an
investment asset!
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• Suppose that f1 and f2 are prices of two
  derivatives dependent only on x and t. For
  example options. We assume that prior to maturity
  f1 and f2 do not provide any cashflow.

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• Form an instantaneously riskless portfolio
  consisting of ?2f2 units of the first derivative
  and ?1f1 units of the second derivative.

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• An instantaneously riskless portfolio must earn a
  riskless interest rate.

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• This implies

or
? is called the market price of risk of x.
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• If x is a traded asset we must also have

But if x is not a financial asset this is not
true. For all financial assets depending on x and
time a similar relation must held.
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Volatility

• Note that ? can be positive or negative,
  depending on the correlation with x.
• ? is called the volatility of f.
• If sgt0 and f and x are positively correlated,
  then ?gt0, otherwise it is negative.

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Example 19.1

• Consider a derivative whose price is positively
  related to the price of oil. Suppose that it
  provides an expected annual return of 12, and
  has volatility of 20. Assume that r8, then
  the market price of risk of oil is

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Example 19.2

• Consider two securities positively dependent on
  the 90-day IR. Suppose that the first one has an
  expected return of 3 and volatility of 20
  (annual), and the second has volatility of 30,
  assume r6. What is the market price of
  interest rate risk? What is the expected return
  from the second security?

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Example 19.2

• The market price of IR risk is

The expected return from the second security
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Differential Equation
f is a function of x and t, we can get using
Itos lemma

• Finally leading to

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Differential Equation
Comparing to BMS equation we see that it is
similar to an asset providing a continuous
dividend yield qr-m?s. Using Feynman-Kac we can
say that the expected growth rate is r-q and then
discount the expected payoff at the risk-free
rate r.
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Risk-neutral approach

• True dynamics

Risk-neutral dynamics
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Example 19.3

• Price of copper is 80 cents/pound. Risk free
  r5. The expected growth rate in the price of
  copper is 2 and its volatility is 20. The
  market price of risk associated with copper is
  0.5. Assume that a contract is traded that
  allows the holder to receive 1,000 pounds of
  copper at no cost in 6 months. What is the price
  of the contract?

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Example 19.3

• m0.02, ?0.5, s0.2, r0.05
• the risk-neutral expected growth rate is

The expected (r-n) payoff from the contract is
Discounting for six months at 5 we get
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Derivatives dependent on several state variables

• State variables (risk factors)

Traded security
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Multidimensional Risk

• Here ?i is the market price of risk for xi.
• This relation is also derived in APT (arbitrage
  pricing theory), Ross 1976 JET.

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Pricing of derivatives

• To price a derivative in the case of several risk
  factors we should
• change the dynamics of xi to risk neutral
• derive the expected (r-n) discounted payoff

If r is deterministic
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Example 19.5

• Consider a security that pays off 100 at time T
  if stock A is above XA and stock B is above XB.
  Assume that stocks A and B are uncorrelated.
• The payoff is 100 QA QB, here QA are QB are r-n
  probabilities of stocks to be above strikes.

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Example 19.5
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Derivatives on Commodities

• The big problem is to estimate the market price
  of risk for non investment assets. One can use
  futures contracts for this.
• Assume that the commodity price follows (no mean
  reversion and constant volatility)

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Derivatives on Commodities

• The expected (r-n) future price of a commodity is
  its future price F(t).

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Example 19.6

• Futures prices
• August 99 62.20
• Oct 99 60.60
• Dec 99 62.70
• Feb 00 63.37
• Apr 00 64.42
• Jun 00 64.40
• The expected (r-n) growth rate between Oct and
  Dec 99 is ln(62.70/60.60)3.4, or 20.4 annually.

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Convenience Yield

• y - convenience yield
• u - storage costs, then then r-n growth rate is
• m - ? s r - y u

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Martingales and Measures

• A martingale is a zero drift stochastic process
• for example dx s dz
• an important property ExT x0,
• fair game.

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Martingales and Measures

• Real world

Risk-neutral world
In the risk-neutral world the market price of
risk is zero, while in the real world it is
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Martingales and Measures

• By making other assumptions we can define other
  worlds that are internally consistent. In a
  world with the market price of risk ? the drift
  (expected growth rate) ? must be

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Equivalent Martingale Measures

• Suppose that f and g are price processes of two
  traded securities dependent on a single source of
  uncertainty. Define xf/g.
• This is the relative price of f with respect to
  g.
• g is the numeraire.

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Equivalent Martingale Measures

• The equivalent martingale measure result states
  that when there are no arbitrage opportunities, x
  is a martingale for some choice of market price
  of risk.
• For a given numeraire g the same market price of
  risk works for all securities f and the market
  price of risk is equal to the volatility of g.

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Equivalent Martingale Measures
Using Itos lemma
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Equivalent Martingale Measures
A martingale
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Forward risk neutral wrt g
Since f/g is a martingale
(19.19) in Hull
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Money market as a numeraire

• Money market account dg rgdt
• zero volatility, so the market price of risk will
  be zero and we arrive at the standard r-n world.
  g01 and

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Zero-Coupon Bond as a Numeraire

• Define P(t,T) the price at time t of a
  zero-coupon bond maturing at T. Denote by ET the
  appropriate measure.
• gT P(T,T)1, g0 P(0,T) we get

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Zero-Coupon Bond as a Numeraire

• Define F as the forward price of f for a contract
  maturing at time T. Then

In a world that is forward risk neutral with
respect to P(t,T) the forward price is the
expected future spot price.
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Important Conclusion

• We can value any security that provides a payoff
  at time T by calculating its expected payoff in a
  world that is forward risk neutral with respect
  to a bond maturing at time T and discounting at
  the risk-free rate for maturity T.
• In this world it is correct to assume that the
  expected value of an asset equals its forward
  value.

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Interest Rates With a Numeraire

• Define R(t, T1, T2) as the forward interest rate
  as seen at time t for the period between T1 and
  T2 expressed with a compounding period T1- T2.
• The forward price of a zero coupon bond lasting
  between T1 and T2 is

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Interest Rates With a Numeraire
A forward interest rate implied for the
corresponding period is
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Interest Rates With a Numeraire
Setting
We get that R(t, T1, T2) is a martingale in a
world that is forward risk neutral with respect
to P(t,T2).
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Annuity Factor as a Numeraire

• Consider a swap starting at time Tn with payment
  dates Tn1, Tn2, , TN1. Principal 1. Denote
  the forward swap rate Sn,N(t). The value of the
  fixed side of the swap is

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Annuity Factor as a Numeraire

• The value of the floating side is

The first term is 1 received at the next payment
date and the second term corresponds to the
principal payment at the end. The swap rate can
be found as
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Annuity Factor as a Numeraire

• We can apply an equivalent martingale measure by
  setting P(t,Tn)-P(t,TN1) as f and An,N(t) as g.
  This leads to

For any security f we have
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Multiple Risk Factors
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Multiple Risk Factors
Equivalent world can be defined as
Where ?i are the market prices of risk
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Multiple Risk Factors

• Define a world that is forward risk neutral with
  respect to g as a world where ?i?g,i. It can be
  shown from Itos lemma, using the fact that dzi
  are uncorrelated, that the process followed by
  f/g in this world has zero drift.

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An Option to Exchange Assets

• Consider an option to exchange an asset worth U
  to an asset worth V. Assume that the correlation
  between assets is ? and they provide no income.
  Setting gU, fTmax(VT-UT,0) in 19.19 we get.

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An Option to Exchange Assets
The volatility of V/U is
This is a simple option.
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An Option to Exchange Assets
Assuming that the assets provide an income at
rates qU and qV.
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Change of Numeraire

• Dynamics of asset f with forward risk neutral
  measure wrt g and h we have

When changing numeraire from g to h we update
drifts by
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Change of Numeraire

• Set v be a function of traded securities. Define
  ?v,i as the i-th component of v volatility. The
  rate of growth of v responds to a change of
  numeraire in the same way. Define qh/g, then
  ?h,i- ?g,i is the i-th component of volatility of
  q. Thus the drift update of v is

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Quantos

• Quanto provides a payoff in currency X at time T.
  We assume that the payoff depends on the value
  of a variable V observed in currency Y at time T.
• F(t) - forward value of V in currency Y
• PX(t,T) - value (in X) of 1 unit of X paid at T
• PY(t,T) - value (in X) of 1 unit of Y paid at T
• G(T) forward exchange rate units of Y per X
• G(t) PX(t,T)/PY(t,T)

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Quantos

• In equation 19.19 set gPY(t,T), f - be a
  security that pays VT units of currency Y at time
  T.
• fTVT/ST, and gT1/ST
• f0 PY(t,T)EY(VT), no arbitrage means
• F(0)f0/PY(0,T), hence
• EY(VT)F(0)
• When we move from X world to Y world the expected
  growth rate increases by ??F?G

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Quantos

• This means that approximately

Since V(T)F(T) and EY(VT)F(0)
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Example 19.7

• Nikkei 15,000
• yen dividend yield 1
• one-year USD risk free rate 5
• one-year JPY risk free rate 2
• The forward price of Nikkei for a contract
  denominated in yen is
• 15,000e(0.02-0.01)115,150.75

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Example 19.7

• Suppose that the volatility of the one-year
  forward price of the index is 20, the volatility
  of the one-year forward yen/USD exchange rate is
  12. The correlation of one-year forward Nikkei
  with the one-year forward exchange rate ?0.3.
• The forward price of the Nikkei for a contract
  that provides a payoff in dollars is
• 15,150.75 e0.30.20.1215,260.2

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Siegels paradox

• Consider two currencies X and Y. Define S an
  exchange rate (the number of units of currency Y
  for a unit of X).
• The risk-neutral process for S is

By Itos lemma the process for 1/S is
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Siegels paradox

• The paradox is that the expected growth rate of
  1/S is not ry - rX, but has a correction term.
• If we change numeraire from currency X to
  currency Y the correction term is ??2-?2.
• The process in terms of currency Y becomes