Questions tagged [stochastic-discount]

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2
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0answers
30 views

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 ...
1
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1answer
49 views

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{...
5
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1answer
237 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 ...
10
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1answer
675 views

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?
16
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5answers
1k views

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 methods are better suited than risk-neutral models for financial predictions. ...
1
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1answer
201 views

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}+[...
2
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1answer
785 views

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?
1
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1answer
386 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 $...
0
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1answer
108 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}{...
3
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3answers
278 views

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 ...
4
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2answers
483 views

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 ...
0
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0answers
84 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 ...
11
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3answers
5k views

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 ...
0
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0answers
97 views

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.
1
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1answer
449 views

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 ...
4
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1answer
386 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 ...
4
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1answer
131 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 ...
2
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1answer
129 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}...
2
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1answer
160 views

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 ...
13
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1answer
3k 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 ...
33
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6answers
7k 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 ...