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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 ...
2
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1answer
89 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 ...
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46 views

How do I find the Sharpe Ratio?

Suppose I'm given two assets, x0, x1 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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1answer
56 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 ...
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17 views

discounted price economic meaning

Could you please explain why we discount the prices using bank account or some numeraire, what is its economic meaning. Specifically The movement of the security prices relative to each other ...
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1answer
146 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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1answer
76 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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1answer
46 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 ...
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1answer
92 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 ...
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1answer
691 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 ...