Gaussian Change of Measure in a Kirk Spread Option Derivation

Summary

The document presents a question about a derivation used in Bjerksund and Stensland’s treatment of spread options. Two correlated lognormal futures are expressed using standard normal shocks, and the target expectation includes the first asset multiplied by an indicator for a threshold involving the second. The question is how the asset factor is moved outside the expectation while the indicator’s arguments are shifted.

The displayed transformation is an application of exponential tilting for jointly normal variables. Weighting by the first asset’s lognormal exponential changes the normal means: the first shock shifts by its own volatility, and the correlated second shock shifts by the correlation times that volatility. This rewrites the weighted expectation as the first futures price times a probability under the shifted normal distribution. The excerpt states the transformation but does not give a full derivation, define every parameter’s role, or discuss broader assumptions behind the spread option approximation.

Key ideas

• Exponential weighting by a lognormal variable changes the distribution of its underlying normal shock.
• With correlated shocks, tilting the first shock also shifts the mean of the second shock.
• The shifted indicator expectation can be expressed as the first futures price times a probability.
• The derivation relies on jointly normal shocks and the specified lognormal representation.

Tags

# Implicit Kirk Strategy proof in Bjerksund and Stensland (2006)


# Implicit Kirk Strategy proof in Bjerksund and Stensland (2006)












I'm reading this paper "Closed form spread option valuation" by Bjerksund and Stensland (2006). You can access the paper here: https://openaccess.nhh.no/nhh-xmlui/bitstream/handle/11250/164107/2006.pdf

In Appendix C, they derive a reformulation of the Kirk strategy but I don't know how they get to the last line shown below.

Consider:

$$ X_1 = F_1exp({-1 \over 2}v_1^2 + v_1\varepsilon_1) \\X_2 = F_2exp({-1 \over 2}v_2^2 + v_2\varepsilon_2) $$ The two error terms are standard normal random variables with correlation $\rho$

Since the two variables follow a log normal distribution, we have: $$ {aX_2^b \over E[X_2^b]} = a*exp\{{-1 \over 2}b^2v_2^2 + bv_2\varepsilon_2\} $$ In page 14 of the paper, they show that:

$$ E[X_1I(X_1 \ge {aX_2^b \over E[X_2^b]})] \quad (1) \\= E[F_1exp\{{-1 \over 2}v_1^2 + v_1\varepsilon_1\} I(F_1exp\{{-1 \over 2}v_1^2 + v_1\varepsilon_1\} \ge a*exp\{{-1 \over 2}b^2v_2^2 + bv_2\varepsilon_2\})] \quad (2) \\ = F_1E[I(F_1exp\{{-1 \over 2}v_1^2 + v_1(\varepsilon_1 + v_1)\} \ge a*exp\{{-1 \over 2}b^2v_2^2 + bv_2(\varepsilon_2 + \rho v_1)\})] \quad (3) $$

I understand the first two lines (from (1) to (2)) just fine but I don't know how they arrive at the indicator function in the last expression (3).

Thank you!

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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