Changes of numéraire, changes of probability measure and option pricing

1995 ◽  
Vol 32 (2) ◽  
pp. 443-458 ◽  
Author(s):  
Hélyette Geman ◽  
Nicole El Karoui ◽  
Jean-Charles Rochet
Keyword(s):  
Probability Measure ◽  
Option Pricing ◽  
Interest Rates ◽  
Complete Markets ◽  
Stochastic Interest Rates ◽  
Stochastic Interest ◽  
Hedging Portfolio ◽  
Pricing Problems ◽  
Key Theorem ◽  
Efficient Selection

The use of the risk-neutral probability measure has proved to be very powerful for computing the prices of contingent claims in the context of complete markets, or the prices of redundant securities when the assumption of complete markets is relaxed. We show here that many other probability measures can be defined in the same way to solve different asset-pricing problems, in particular option pricing. Moreover, these probability measure changes are in fact associated with numéraire changes, this feature, besides providing a financial interpretation, permits efficient selection of the numéraire appropriate for the pricing of a given contingent claim and also permits exhibition of the hedging portfolio, which is in many respects more important than the valuation itself.The key theorem of general numéraire change is illustrated by many examples, among which the extension to a stochastic interest rates framework of the Margrabe formula, Geske formula, etc.

Changes of numéraire, changes of probability measure and option pricing

1995 ◽  
Vol 32 (02) ◽  
pp. 443-458 ◽  
Author(s):  
Hélyette Geman ◽  
Nicole El Karoui ◽  
Jean-Charles Rochet
Keyword(s):  
Probability Measure ◽  
Option Pricing ◽  
Interest Rates ◽  
Complete Markets ◽  
Stochastic Interest Rates ◽  
Stochastic Interest ◽  
Hedging Portfolio ◽  
Pricing Problems ◽  
Key Theorem ◽  
Efficient Selection

The use of the risk-neutral probability measure has proved to be very powerful for computing the prices of contingent claims in the context of complete markets, or the prices of redundant securities when the assumption of complete markets is relaxed. We show here that many other probability measures can be defined in the same way to solve different asset-pricing problems, in particular option pricing. Moreover, these probability measure changes are in fact associated with numéraire changes, this feature, besides providing a financial interpretation, permits efficient selection of the numéraire appropriate for the pricing of a given contingent claim and also permits exhibition of the hedging portfolio, which is in many respects more important than the valuation itself. The key theorem of general numéraire change is illustrated by many examples, among which the extension to a stochastic interest rates framework of the Margrabe formula, Geske formula, etc.

scholarly journals Pricing Spread Options with Stochastic Interest Rates

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Yunguo Jin ◽  
Shouming Zhong
Keyword(s):  
Option Pricing ◽  
Interest Rates ◽  
Stock Returns ◽  
Grid Method ◽  
Pricing Models ◽  
Stochastic Interest Rates ◽  
Spread Options ◽  
Asset Pricing Models ◽  
Stochastic Interest ◽  
Spread Option

Although spread options have been extensively studied in the literature, few papers deal with the problem of pricing spread options with stochastic interest rates. This study presents three novel spread option pricing models that permit the interest rates to be random. The paper not only presents a good approach to formulate spread option pricing models with stochastic interest rates but also offers a new test bed to understand the dynamics of option pricing with interest rates in a variety of asset pricing models. We discuss the merits of the models and techniques presented by us in some asset pricing models. Finally, we use regular grid method to the calculation of the formula when underlying stock returns are continuous and a mixture of both the regular grid method and a Monte Carlo method to the one when underlying stock returns are discontinuous, and sensitivity analyses are presented.

EFFICIENT DYNAMIC HEDGING FOR LARGE VARIABLE ANNUITY PORTFOLIOS WITH MULTIPLE UNDERLYING ASSETS

2020 ◽  
Vol 50 (3) ◽  
pp. 913-957
Author(s):  
X. Sheldon Lin ◽  
Shuai Yang
Keyword(s):  
Interest Rates ◽  
Computing Time ◽  
Insurance Companies ◽  
Insurance Company ◽  
Simulation Approach ◽  
Stochastic Interest Rates ◽  
Dynamic Hedging ◽  
Market Risks ◽  
Stochastic Interest ◽  
Hedging Portfolio

AbstractA variable annuity (VA) is an equity-linked annuity that provides investment guarantees to its policyholder and its contributions are normally invested in multiple underlying assets (e.g., mutual funds), which exposes VA liability to significant market risks. Hedging the market risks is therefore crucial in risk managing a VA portfolio as the VA guarantees are long-dated liabilities that may span decades. In order to hedge the VA liability, the issuing insurance company would need to construct a hedging portfolio consisting of the underlying assets whose positions are often determined by the liability Greeks such as partial dollar Deltas. Usually, these quantities are calculated via nested simulation approach. For insurance companies that manage large VA portfolios (e.g., 100k+ policies), calculating those quantities is extremely time-consuming or even prohibitive due to the complexity of the guarantee payoffs and the stochastic-on-stochastic nature of the nested simulation algorithm. In this paper, we extend the surrogate model-assisted nest simulation approach in Lin and Yang [(2020) Insurance: Mathematics and Economics, 91, 85–103] to efficiently calculate the total VA liability and the partial dollar Deltas for large VA portfolios with multiple underlying assets. In our proposed algorithm, the nested simulation is run using small sets of selected representative policies and representative outer loops. As a result, the computing time is substantially reduced. The computational advantage of the proposed algorithm and the importance of dynamic hedging are further illustrated through a profit and loss (P&L) analysis for a large synthetic VA portfolio. Moreover, the robustness of the performance of the proposed algorithm is tested with multiple simulation runs. Numerical results show that the proposed algorithm is able to accurately approximate different quantities of interest and the performance is robust with respect to different sets of parameter inputs. Finally, we show how our approach could be extended to potentially incorporate stochastic interest rates and estimate other Greeks such as Rho.

Sign in / Sign up

Export Citation Format

Share Document