Unformatted text preview: Debt Instruments and Markets Professor Carpenter Risk-Neutral
True probabilities Reading
Veronesi, Chapter 9
Tuckman, Chapter 9 Risk-Neutral Probabilities 1 Debt Instruments and Markets Professor Carpenter No Arbitrage Derivative Pricing
Last lecture, we priced a derivative by
constructing a replicating portfolio from the
– We started with a derivative with a payoff at
time 0.5. The payoff depended on the time 0.5
price of the zero maturing at time 1.
– We modeled the random future price of the
zero and the future payoff of the derivative.
– We constructed a portfolio of 0.5-year and 1year zeroes with the same payoff of the
derivative by solving simultaneous equations.
– We then set the price of the derivative equal to
the value of the replicating portfolio. General Bond Derivative
• Any security whose time 0.5 payoff is a function of
the time 0.5 price of the zero maturing at time 1 can
be priced by no arbitrage.
• Suppose its payoff is Ku in the up state, and Kd in the
0.5-year zero 0.973047 1-year zero 0.947649 General portfolio 0.973047 N0.5
+ 0.947649 N1 General derivative ? Time 0.5
1N0.5 + 0.97229N1 Ku
1N0.5 + 0.976086N1 Kd Risk-Neutral Probabilities 2 Debt Instruments and Markets Professor Carpenter Replicating and Pricing the
1) Determine the replicating portfolio by
solving the equations
1N0.5 + 0.97229N1 = Ku
1N0.5 + 0.96086N1 = Kd
for the unknown N's. (The two possible K's
2) Price the replicating portfolio as
0.973047N0.5 + 0.947649N1
This is the no arbitrage price of the derivative. Risk-Neutral Probabilities
• Finance: The no arbitrage price of the
derivative is its replication cost.
• We know that’s some function of the prices
and payoffs of the basic underlying assets.
• Math: We can use a mathematical device,
risk-neutral probabilities, to compute that
replication cost more directly.
• That’s useful when we only need to know
the price of the replicating portfolio, but not
the holdings. Risk-Neutral Probabilities 3 Debt Instruments and Markets Professor Carpenter Start with the Prices and Payoffs
of the Underlying Assets
• In our example, the derivative payoffs were functions
of the time 0.5 price the zero maturing at time 1.
• So the underlying asset is the zero maturing at time 1
and the riskless asset is the zero maturing at time 0.5.
• The prices and payoffs are, in general terms:
Time 0 Time 0.5 1
0.5 1 d0.5 d d1
0.5 1 d €
€ Find the “Probabilities” that “Risk-Neutrally”
Price the Underlying Risky Asset
• Find the “probabilities” of the up and down states, p
and 1-p, that make the price of the underlying asset
equal to its “expected” future payoff, discounted back
at the riskless rate.
• I.e., find the p that solves “Risk-Neutral Pricing
Price = discounted “expected” future payoff
for the underlying risky asset.
• In our example, this is the zero maturing at time 1, so
d1 = d0.5 [ p × 0.5 d1u + (1− p) × 0.5 d1d ]
− 0.5 d1d
0.5 d1 − 0.5 d1 € Risk-Neutral Probabilities 4 Debt Instruments and Markets Professor Carpenter Example of p
In our example,
0.972290 − 0.976086
1− p = 0.424 € Result
• The same p prices all the derivatives of the
• I.e., if a derivative has payoffs Ku in the up state
and Kd in the down state, its replication cost turns
out to be equal to Derivative price = d0.5 [ p × K u + (1− p) × K d ] € using the same p that made this risk-neutral pricing
equation (RNPE) hold for the underlying asset.
• That is, a single p makes the RNPE
price = discounted “expected” future payoff
hold for the underlying and all its derivatives. Risk-Neutral Probabilities 5 Debt Instruments and Markets Professor Carpenter Examples of Risk-Neutral Pricing
With the risk-neutral probabilities, the price of an
asset is its expected payoff multiplied by the
riskless zero price, i.e., discounted at the riskless
call option: (0.576 × 0 + 0.424 ×1.086) × 0.9730 = 0.448
or, 0.576 × 0 + 0.424 ×1.086
1.0277 Class Problem:
Price the put option with payoffs Ku=2.71 and Kd=0
using the risk-neutral probabilities. Examples of Risk-Neutral Pricing...
(0.576 × 0.9723 + 0.424 × 0.9761) × 0.9730 = 0.9476 or, € 0.576 × 0.9723 + 0.424 × 0.9761
1.0277 0.5-year zero (riskless asset):
(0.576 ×1+ 0.424 ×1) × 0.9730 = 0.9730 or, 0.576 ×1+ 0.424 ×1
1.0277 € Risk-Neutral Probabilities 6 Debt Instruments and Markets Professor Carpenter Why does the p that makes the RNP Equation hold for the
underlying also make the RNPE work for all its derivatives?
1) By construction, p makes
price of underlying risky asset = discount factor x
[p x underlying’s up payoff + (1-p) x underlying’s down payoff].
2) It’s also always true for any p that
price of riskless asset = discount factor
= discount factor x [p x 1 + (1-p) x 1]
so the p that works for the underlying also works for the riskless asset,
because any p does.
3) Therefore, this p also works for any portfolio of these two assets. I.e., for
any portfolio with holdings N0.5 and N1:
N0.5 x price of riskless asset + N1 x price of underlying =
[p x (N0.5 x 1 + N1 x underlying’s up payoff ) +
(1-p) x (N0.5 x 1 + N1 x underlying’s down payoff ) ]
portfolio price = disc.factor x
[p x portfolio’s up payoff + (1-p) x portfolio’s down payoff]
Since every derivative of the underlying is one of these portfolios, the
RNPE, using the same p, holds for all of them too. Class Problem
• Suppose the time 0 price of the zero maturing
at time 1 is slightly lower:
0.947007 Time 0.5
0.976086 • What would be the risk-neutral probabilities
p and 1-p of the up and down states? Risk-Neutral Probabilities 7 Debt Instruments and Markets Professor Carpenter Class Problem
1) Price the call using these new RN probs:
1.086 2) Calculate the replication cost of the call the old way
and verify that it matches the price above. Hint: the
payoffs are unchanged so the replicating portfolio is
still N0.5 = -278.163 and N1 = 286.091. Expected Returns with RN Probs
• Note that we can rearrange the risk-neutral pricing
equation, price = discounted “expected” payoff, as
V = d0.5 [ p × K u + (1− p) × K d ], or
p × K u + (1− p) × K d
1+ r0.5 /2
p × K u + (1− p) × K d
= 1+ r0.5 /2
V • I.e., “expected” return = the riskless rate.
€ (Here return is un-annualized. )
• Thus, with the risk-neutral probabilities, all assets have
the same expected return--equal to the riskless rate.
• This is why we call them "risk-neutral" probabilities. Risk-Neutral Probabilities 8 Debt Instruments and Markets Professor Carpenter True Probabilities
The risk-neutral probabilities are not the same
as the true probabilities of the future states.
Notice that pricing contingent claims did not
involve the true probabilities of the up or down
state actually occurring.
Let's suppose that the true probabilities are 0.5
chance the up state occurs and 0.5 chance the
down state occurs.
What could we do with this information?
For one, we could compute the true expected
returns of the different securities over the next
6 months. True Expected Returns
Recall that the unannualized return on an asset
over a given horizon is
For the 6-month zero the unannualized return
over the next 6 months is
−1 = 2.77%
0.973047 with certainty. This will be the return
regardless of which state occurs. That's why
this asset is riskless for this horizon.
Of course, the annualized semi-annually compounded ROR is
5.54%, the quoted zero rate. Risk-Neutral Probabilities 9 Debt Instruments and Markets Professor Carpenter True Expected Returns...
The return on the 1-year zero over the next
6 months will be either
−1 = 2.60% with probability 0.5, or
−1 = 3.00% with probability 0.5.
0.947649 € The expected return on the 1-year zero over
the next 6 months is 2.80%.
Notice that it is higher than the return of
2.77% on the riskless asset. True Expected Returns...
Why might the longer zero have a higher
– Investors have short-term horizons, and dislike
the price risk of the longer zero.
– Investors require a premium to hold securities
that covary positively with long bonds (bullish
securities) because government bonds are in
positive net supply.
Sometimes the reverse could be true.
In general, assets with different risk
characteristics have different expected returns.
Their expected returns also depend on how their
payoffs covary with other assets. Risk-Neutral Probabilities 10 Debt Instruments and Markets Professor Carpenter True Expected Returns...
What is the expected rate of return on the
call over the next 6 months?
The possible returns are:
−1 = −100% with probability 0.5, or
−1 = 142% with probability 0.5.
0.448 € The expected return on the call is 21%. True Expected Returns...
What is the expected rate of return on the put over
the next 6 months?
The possible returns are:
−1 = 78% with probability 0.5, or
−1 = −100% with probability 0.5.
1.52 € Risk-Neutral Probabilities The expected return on the put is -11%.
The put is bearish--it insures (hedges) the risk of
By no arbitrage, if bullish assets have positive risk
premia, bearish assets must have negative risk
Intuitively, investors must pay up for this insurance. 11 Debt Instruments and Markets Professor Carpenter Risk-Neutral Expected Returns
Using the risk-neutral probabilities to compute
expected (unannualized) returns sets all expected
returns equal to the riskless rate.
Asset 0.5-Year Zero
("prob"=0.424) 1/0.9730 - 1
1/0.9730 - 1
0.97229/0.947649 - 1 0.976086/0.947649 - 1
0/0.448 - 1
2.7103/1.519 - 1
= 78.42% 1.0859/0.448 - 1
0/1.519 - 1
= -100% "Expected"
2.77% Why Does the p that Works for the Underlying
Asset Also Work for All Its Derivatives?
1) The expected return on a portfolio is the average of the
expected returns of the individual assets.
2) The risk-neutral probabilities are constructed to make the
expected return on the underlying risky asset equal to the
riskless asset return.
3) So under the risk-neutral probabilities, the expected return
on every portfolio of the underlying and riskless assets is
also that same riskless return.
4) Every derivative of the underlying can be viewed as a
portfolio of the underlying asset and the riskless asset.
5) So the derivative’s expected return must also equal the
riskless return under the risk-neutral probabilities.
6) So the derivative’s price must equal its expected payoff,
using the risk-neutral probabilities, discounted back at the
riskless rate. Risk-Neutral Probabilities 12 ...
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- Spring '15
- Probability theory, Professor Carpenter