Closed form spread option valuation

Abstract

This paper considers the valuation of a spread call when asset prices are log-normal. The implicit strategy of the Kirk formula is to exercise if the price of the long asset exceeds a given power function of the price of the short asset. We derive a formula for the spread call value, conditional on following this feasible, but non-optimal, exercise strategy. Numerical investigations indicate that the lower bound produced by our formula is extremely accurate. The precision is much greater than the Kirk formula. Moreover, optimizing with respect to the strategy parameters (which corresponds to the Carmona–Durrleman procedure) yields only a marginal improvement of accuracy (if any).

Keywords:

• Finance
• Financial applications
• Financial derivatives
• Financial engineering
• Financial mathematics
• Financial modelling
• Financial options

JEL Classification::

• C6
• C63
• D8
• D81
• G1
• G12
• G13

Acknowledgements

We thank the Editor and anonymous referees for helpful comments and suggestions.

Notes

†By the put–call parity, the Kirk approximation of a put on the price spread S ₁ − S ₂ with strike K ≥ 0 and time to exercise T is p _K  = c _K  − e^−rT (F ₁ − F ₂ − K).

‡φ ∈ [0, π] and θ* ∈ [π, 2π] translate into sin φ ≥ 0 and cos θ* ≤ 0. To motivate this, observe from equations (9) and (11) that an increase in z ₁ will increase the pay-off from asset 1, and push the call more in-the-money (less out-of-the-money).

§There is a typo in equation (20) of Carmona and Durleman (2003a) as well as in equation (6.3) of Carmona and Durrleman (2003b). The trigonometric function entering the second term should read cos and not sin.

¶By the put–call parity, the Carmona–Durrleman approximation of a put on the price spread S ₁ − S ₂ with strike K ≥ 0 and time to exercise T is p _CD = c _CD − e^−rT (F ₁ − F ₂ − K).

†Rewrite equation (4) as

where the pay-offs C(T) and c(T) are defined in equations (3) and (14), and c is our spread option formula in equations (15–21). Clearly, the two pay-offs C(T) and c(T) are highly correlated. Consequently, the simulation error from evaluating the expectation on the RHS is much smaller than the simulation error from evaluating the expectation on the LHS.

†When K = 0, both formulas degenerate to the Margrabe exchange option formula, which represents the true value in this case. Hence, the pricing errors are zero in these cases.

†It can be shown that

One could of course simulate the delta directly from this expression. In order to increase the precision, however, we use the Kirk exercise strategy as a control variate and apply the result

(see appendix C) to obtain equation (24). We obtain equation (25) in a similar fashion using a result from appendix D.