Questions tagged [stochastic-discount]

Anything to do with the Stochastic Discount Factor (SDF).

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Dealing with the ru term in an ADI Finite Difference Scheme

I'm trying to code up the algorithm from this paper. The paper presents an ADI algorithm for pricing options in the Heston-Hull-White model. The starting point is the Heston-Hull-White PDE, given ...
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Relationship between an Investor's utility function and Stochastic Discount Factor (SDF)

In real world, it is difficult to arrive at a single price for a risky asset since pricing of a risky asset would depend on the level of risk aversion of the investor. The following equation gives the ...
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Question on derivation step in portfolio replication under different borrowing and lending rates

I'm currently trying to understand the derivation of a pricing PDE on a european claim that considers stock lending fees: https://cs.uwaterloo.ca/~paforsyt/hjb.pdf In Appendix A.2, the author talks ...
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1 answer
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Hedge error - Willmot and Ahmad

I'm currently reading the paper: Willmot and Ahmad: Which free lunch would you like today, Sir? Delta Heding, volatility arbitrage. In case 1: They delta hedge with the actual volatility, by going ...
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2 votes
0 answers
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linear stochastic discount factor

I have heard some people say something like the following with regards to APT: Let returns be given by the factor model $r_t = B_tf_t + e$ with $E(f_t) = \lambda_t$ Assume that factors are ...
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Is market price of risk always negative?

I might have a gap in understanding, so clarifying: Basic pricing equation $E(R) = - cov(m, R)$ where $R$ = excess return and $m$ = stochastic discount factor (I think this is continuous case, in ...
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What is the link between the SDF in the Black-Scholes-Merton model and the exponential process in Girsanov's theorem?

Question I have been toying around to get some understanding of what the stochastic discount factor look likes in Black-Scholes-Merton and how it relates to the exponential process in Girsanov's ...
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1 vote
1 answer
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Covariance, stochastic discount factor (SDF) and risk aversion

John Cochrane states, that if the covariance between the stochastic discount factor and the payoff is zero - then risk aversion should have no impact on the pricing. I do not fully understand why this ...
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10 votes
3 answers
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SDF as an affine transformation of the tangency portfolio

I'm studying this paper. In the formulation of the theoretical setup they state: Our goal is to explain the differences in the cross-section of returns $R$ for individual stocks. Let $R_{t+1, i}$ ...
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2 votes
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61 views

Average individual consumption growth vs average aggregate consumption growth

Consumption growth is an essential thing in most asset pricing models and usually the Euler equation defines the return of an asset as a covariance between consumption frowth and the cash-flows of ...
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Stochastic discount factor for factor research

Often, after presenting a new factor technique, the paper calculates an SDF by doing $\Sigma ^{-1}\mu_F$ i.e. mean variance optimization on the factors. What is the significance of doing this ?
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State price deflator in the Vasicek model

I am trying to implement a simple bond pricing model using state price deflators in a Vasicek model. I am simulating paths of the processes $$\mathrm{d}r^{P} =\kappa^{P}(\theta^P - r^P(t))\mathrm{d}t ...
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1 vote
1 answer
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Confused About Ex-Ante vs. Ex-Post Pricing Representation

This is going to be a really simple question, but I am confused by it. The basic pricing formula is $p_t=E^p_t(m_{t+1}X_{t+1})$, where $p$ is the physical measure. We can also say that $R_{t+1}=\frac{...
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5 votes
1 answer
426 views

Behavioral SDF: modelling sentiment risk premium

With reference to Behavioral Asset Pricing models, I know that the discount factor (or required rate of return) is equal to: Discount rate = Risk-free rate + Fundamental risk premium + Sentiment ...
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12 votes
1 answer
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What is the difference between risk neutral probabilities and stochastic discount factor?

My question is regarding the difference between risk neutral probabilities and stochastic discount factor? I am confused as to how are they related?
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19 votes
5 answers
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Why quants think that the risk-neutral measure should not be used for financial forecasting?

In posts regarding the $\mathbb{P}$ vs $\mathbb{Q}$ debate (see 1, 2, 3 or 4), most answers conclude that historical-based forecast are better suited than risk-neutral models for financial predictions....
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1 vote
1 answer
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Stochastic Discount Factor of CIR bond pricing model

The CIR model states $dr=\kappa(\theta-r)dt+\sigma dW$ and the corresponding bond pricing equation can be derived from the general equilibrium approach. The equation is: $\frac{1}{2}\sigma^2rP_{rr}+[...
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What is the difference between stochastic discount factor and stochastic discount factor process?

What is the difference between stochastic discount factor and stochastic discount factor process and how are they both related?
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1 vote
1 answer
728 views

PDE for Pricing Interest Rate Derivatives

Suppose that interest rate $r(t)$ follows some short-rate models, say Vasicek, so that$dr = a(b-r) dt + \sigma dZ$, with constants $a,b,\sigma$. It is well known that the price of zero-coupon bond $...
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0 votes
1 answer
196 views

stochastic discount factor transformation

I have $$\frac{dM_t}{M_t}=-\frac{\mu}{\sigma} dW_t + \gamma_t dB_t, \tag{1}$$ where $B_t$ and $W_t$ are two independent Brownian Motions, which was further presented as $$ M_t=\exp \left( -\frac{\mu}{...
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3 votes
3 answers
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The portfolio whose return is the stochastic discount factor

I am trying to construct a portfolio whose return is $a + bm_{t+1}$ where $a$ and $b$ are some constants for a certain investor. $m_{t+1}$ is the stochastic discount factor at time $t+1$. I am ...
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5 votes
2 answers
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Real world monte-carlo (P-measure)

Consider the 2 following approaches to pricing a security: Monte-carlo ($\mathbb{Q}$-measure) $\begin{equation} C = \frac{1}{N} \sum_{i=1}^{n} e^{-rT} max(S_i(t) - K, 0) \end{equation}$ Monte-carlo ...
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0 votes
0 answers
105 views

Benchmarking option pricing under stochastic interest rates

I priced a long-term option (10 or 20 years) using two different models: one assumes constant interest rates, the other assumes stochastic interest rates. Is there a way (e.g. a benchmark) to ...
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12 votes
3 answers
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Black-Scholes under stochastic interest rates

I'm trying to implement the Black-Scholes formula to price a call option under stochastic interest rates. Following the book of McLeish (2005), the formula is given by (assuming interest rates are ...
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How do I find the Sharpe Ratio?

Suppose I'm given two assets, $x_0$, $x_1$ and the stochastic discount factor m. How do I find $m_p$, then use it to compute Sharpe($R_p$)? Any help is greatly appreciated.
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1 answer
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Proving there exists no arbitrage opportunities given 3 states and 2 assets

Assume there are 3 states of the world: w1, w2, and w3. Assume there are two assets: a risk-free asset returning Rf in each state, and a risky asset with Return R1 in state w1, R2 in state w2, and R3 ...
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4 votes
1 answer
517 views

Data Selection for Empirical Pricing Kernel Estimation (Stochastic Discount Factor)

I want to estimate an empirical pricing kernel for an index. Hence, I need to estimate a physical and risk neutral density. For estimating the physical density, only the index data in an observed time ...
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4 votes
1 answer
154 views

Discounted risky asset stochastic process problem

$S_t$ is the random variable representing the risky asset price at time $t$. M_t is the riskless asset. They are governed by the equations $\frac{dS_t}{dt}=\mu dt + \sigma dZ_t$ and $dM_t = rM_t ...
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2 votes
1 answer
283 views

Hansen-Jagannathan bounds derivation: last step is not clear

Pennachi's "Asset Pricing" chapter 4 derives: $$ \frac{E[R_{i}-R_{f}]}{\sigma_{R_{i}}}=-\rho_{m_{01},R_{i}}\frac{\sigma_{m_{01}}}{E[m_{01}]} $$ Then, he states that the fact that $-1\leq \rho_{m_{01}...
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2 votes
1 answer
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Discounting based on instrument type

Suppose we have an asset $A$, and we have modelled the cashflows for this asset to be $\{C_{1},\ldots C_{k}\}$ which occur at time $\{T_{1},\ldots T_{k}\}$. Now the present value of the asset can be ...
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13 votes
1 answer
5k views

Intuitive explanation of the Hansen-Jagannathan bound

The Hansen-Jagannathan bound states that the maximum Sharpe ratio of a portfolio can't exceed the ratio of the standard deviation of a stochastic discount factor to its mean. I more or less understand ...
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35 votes
6 answers
12k views

How to estimate real-world probabilities

In the world of finance, Risk-neutral pricing allow us to estimate the fair value of derivatives using the risk free rate as the expected return of the underlyings. However, the behavior of ...
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