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Risk-Neutral Valuation of an Exponential Claim in a Brownian Market

Article Quant Q&A · Author: Buddy_
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Summary

The document presents a risk-neutral valuation question for a claim paying the exponential of the sum of two terminal, non-discounted state variables. It specifies a three-dimensional Brownian market through the dynamics of discounted assets, with deterministic bounded interest rates, and states that the market has already been found to be arbitrage-free. The central challenge is to use a risk-neutral measure consistently when the given dynamics are for discounted quantities but the payoff is written using non-discounted quantities.

No solution or answer is included, so the document does not identify a pricing measure, compute an expectation, or give a price. In particular, arbitrage freedom alone does not establish that the market is complete or that the risk-neutral price is unique. A full treatment would need to determine the admissible measure or measures and translate the discounted dynamics and payoff into a consistent valuation expression. The supplied setup makes this a useful exercise, but not a standalone pricing method.

Key ideas

  • The claim depends on two terminal state variables through an exponential payoff.
  • The stated dynamics describe discounted assets, while the payoff uses non-discounted values.
  • Risk-neutral valuation requires consistency between the measure, asset dynamics, and payoff.
  • Arbitrage freedom alone does not guarantee a unique risk-neutral price.
  • The document poses the problem but does not provide a solution.

Tags

Derivatives pricingStatisticsRisk management
Full text
# Risk-neutral price of $H=e^{X_T^1+X_T^3}$


# Risk-neutral price of $H=e^{X_T^1+X_T^3}$












Let $B=(B_t^1,B_t^2,B_t^3)$ a $\mathbb R^3$-valued Brownian motion. Let $r_t$ (risk free rate) be bounded and deterministic. Let consider the DISCOUNTED market $$d\overline X_t^1=\frac52dt+2dB_t^1-dB_t^2-dB_t^3$$ $$d\overline X_t^2=7dt+2dB_t^1+2dB_t^2-10dB_t^3$$ $$d\overline X_t^3=\frac72dt+4dB_t^1-3dB_t^2+dB_t^3$$ I have already found that the market is arbitrage free.

I would like to find the risk-neutral price of the following claim:$$H=e^{X_T^1+ X_T^3}$$ (note: here the $X_T^1,X_T^3$ are not discounted) but i'm stack. Any help please?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.