Esscher Transform: Pricing Maturity Guarantees

• This paper focuses on using the Esscher transform, to find the price of maturity guarantees for a unit-linked contract

Key Points

• Three stochastic processes for the interest rate underlying the unit fund are considered.
• The choice of central assumptions is more important than the choice of model.
• Keywords: Esscher transform, Maturity guarantees, Option pricing, Financial economics.

Introduction

• Developments in financial economics have sparked interest in the actuarial community, particularly in the life insurance industry

Nobel Prize

• Merton and Scholes were awarded the Nobel Prize in Economics in 1997.
• The method adopted by this year's laureates can be used to value guarantees and insurance contracts, viewing insurance companies and the option market as competitors.

Options and Guarantees

• Options and guarantees by life insurers are getting more attention regarding pricing and reserving.
• This is due to certain high profile debacles in the insurance industry (e.g. Equitable Life)
• Fair value accounting for insurers, a falling interest rate regime, and increased attention to corporate balance sheets.
• There is also a focus on executive stock options and advances in option valuation theory.
• Actuaries have embraced financial economic techniques, with financial economics contributing to actuarial science and vice versa.

Esscher Transform

• The paper reviews an application of the Esscher transform to value option contracts.
• A maturity guarantee on a unit-linked contract can be likened to a European put option.
• Section 2 introduces the Esscher transform and its use in valuing put options.
• Section 3 values a maturity guarantee under three different assumptions for the underlying unit fund growth: Wiener Process, Shifted Poisson Process, and Shifted Gamma Process.
• Section 4 contains graphs on the maturity guarantee nature and sensitivity
• Other actuarial techniques for valuing option contracts are in Section 5.
• Valuing surrender guarantees as option contracts is also addressed.
• Concluding remarks can be found in Section 6.

Esscher Transform

• This section shows how an actuarial technique values option contracts
• The Esscher transform is a tool invented by actuarial science used for numerical computation and simulation of rare events.

Definition

• For t ≥ 0, S(t) is the price of a non-dividend paying stock at time t.
• A stochastic process, {X(t)}, exists with stationary and independent increments, initial value X(0) = 0, such that S(t) = S(0)ex(t), t≥ 0.
• The stock price, S(t), grows with interest, which has a stochastic process as denoted by X(t).
• The random variable X(t) has a probability density function f(x,t).
• The moment generating function M(z,t) of X(t) is defined by M(z,t) = E[ex(t)] = ∫ex f(x,t)dx.
• The Esscher transform with parameter h of f is defined by f(x,t;h) = ehx f(x,t) / M(h,t).
• The corresponding moment generating function of f(x,t;h) is M(z,t;h) = ∫ezx f(x,t;h)dx = M(z+h,t) / M(h,t).
• Since M(z,t) is continuous in t, M(z,t;h) = [M(z,1; h)]t.
• Risk-neutral Esscher transform: Seek h=h* so that the discounted stock price process, e-δtS(t), is a martingale

Risk-Free Interest Rate

• δ denotes the constant risk-free force of interest.
• Using the equation in 2.2, the parameter h* is the solution of the equation 1 = e-δt E * [ex(t)], or eδt = M(1,t;h*).
• Setting t=1 gives δ = ln[M(1,1; h*)].
• Gerber and Shiu call the Esscher transform of parameter h* the risk-neutral Esscher transform, and the corresponding equivalent martingale measure the risk-neutral Esscher measure.
• The merit of the risk-neutral Esscher measure is that it provides a general, transparent and unambiguous solution.

Valuing Derivatives

• The value of a derivative is the expected discounted value of the implied payoffs.
• Consider a European call option on the stock with exercise price K and exercise date τ,τ > 0.
• The value of this option (at time 0) is E* [e-δτ (S(τ) – K)+ ].
• Defining κ = ln[K / S(0)], the above equation becomes e-δτ ∫[S(0)ex – K]f(x,τ; h*)dx.
• Gerber and Shiu show that the value of the European call option with exercise price K and exercise date τ is S(0)[1 – F(κ,τ;h*+1)] – e-δτ K[1– F(κ,τ;h*)].
• Applying the put-call parity theorem, the value of the European put option with exercise price K and exercise date τ is Ke-δτ[F(κ,τ; h*)] - S(0)[F(κ,τ;h*+1)].
• The Esscher transform allows finding a deflator that achieves the martingale property and does not assume a particular distribution underlying the stock price. It can also be used where the underlying stock price exhibits jumps (e.g. Poisson process).

Valuing Maturity Guarantees

• Formulae for call and put options are without assuming any specific distribution for the interest rate process X(t).
• The European put option results are used to value maturity guarantees.
• A maturity guarantee on a unit-linked product is like a European put option.
• The insurer estimates the guarantee value by assuming a stochastic process for the unit fund and the probability of survival to maturity.

Parameters

• Parameters used in the models: Age (x), Single premium (P), Allocation rate (AllocationRate), Bid-offer spread (BOS), Value of the unit fund at time t (U(t)), Guaranteed maturity value (K), Risk-free force of interest (δ), Guaranteed interest rate (i), Term (τ), Kappa (κ), Mu (μ), Sigma (σ), Shift parameter for the Shifted Poisson process (k), Mean of the Shifted Poisson process (λ*), Shift parameter for the Shifted Poisson and Shifted Gamma processes (c), Shape parameter of the original Gamma process (α), Scale parameter of the original Gamma process (β)
• U(t) denotes the value of the unit fund at time t with the initial value: U(0) = P × (1 - BOS) × AllocationRate
• The unit fund at time t is then obtained as: U(t) = U(0) × ex(t) for 0 < t ≤ τ

Stochastic Process

• {X(t)} is the stochastic process underlying the interest rate where the unit fund is growing.
• Three processes for {X(t)} are assumed: Wiener process, Shifted Poisson process, and Shifted Gamma process.

Wiener Process

• the change in the logarithm of the unit fund that is distributed normally.
• It is the basis of the Black-Scholes model.
• The results derived correspond to the Black-Scholes option pricing formula.
• Let {X(t)} be a Wiener process with mean per unit time μ and variance per unit time σ2.
• The Esscher transform (parameter h) of the Wiener process is M(z,t;h) = exp{[(μ + hσ2)z + (1/2)σ2z2] t}, a Wiener process with modified mean per unit time μ + hσ2 and unchanged variance σ2.
• The maturity guarantee value is Λ = Ke-δτ Φ((-κ + (μ - σ2/2) τ) / (σ√τ)) - U(0) Φ((-κ + (μ + σ2/2) τ) / (σ√τ)).
• Considering the probability of survival, the maturity guarantee value is Λ * τPx.

Poisson Process

• The Poisson process is a discrete time model commonly used in actuarial work.
• Let N(t) be a Poisson process with parameter λ and k and c are positive constants, where the stochastic process {X(t)} follows X(t) = kN(t) - ct.
• Let Λ(x;θ) = Σ (e-θ θj) / (j!) be the cumulative Poisson distribution function with parameter θ.
• The cumulative distribution function of X(t) is F(x,t) = Λ((x + ct) / k; λt).
• The Esscher transform (parameter h) of the Poisson process is M(z,t;h) = exp[λt(ehk - 1) - czt], also a shifted Poisson process with modified Poisson parameter λehk.
• The maturity guarantee can be valued as {Ke-δτ Λ((κ + ct)/k;λτ) – U(0) Λ((κ+ct)/k;λehkτ)} * τpx.

Shifted Gamma Process

• The Gamma process is the continuous time case of the Poisson process.
• Let {Y(t)} be a Gamma process with parameters α and β and the positive constant c is the third parameter where the stochastic process {X(t)} follows X(t) = Y(t) - ct.
• The cumulative distribution function of X(t) is F(x,t) = G(x+ct;αt, β), where G() is the cumulative Gamma distribution function.
• The Esscher transform (parameter h) of the shifted Gamma process is M(z,t;h) = ((β-h) / (β-h-z))αt e-ctz, z < β-h, the transformed process is the same type, with β replaced by β-h.
• The value of the maturity guarantee can be calculated as {Ke-δτ [1−G(κ + cτ;ατ,β*)] – U(0) [1−G(κ + cτ;ατ, β*)]} * τPx.

Model Outputs

• This section illustrates the value of the maturity guarantee under the three different models.
• The maturity guarantee is not a monotonically increasing function of term because it applies at a specific point in time.
• The point of inflexion for the models is around 10 years, similar to single premium bonds sold today.
• The Shifted Gamma process is a continuous case of the Shifted Poisson process, explaining the shape of the two curves.

Model Choice

• Model choice is more critical for longer-term guarantees as the model values diverge.
• The maturity guarantee increases approximately linearly with volatility
• The cost of a similar guarantee would be higher in markets that experience greater volatility
• The maturity guarantee is a monotonically decreasing function of the risk-free force of interest
• Model choice becomes more critical at lower risk-free force of interest.

Further Research

• Maturity guarantees can be likened to a European put option and valued using Esscher transforms.

Guarantees

• Guarantees offered by an insurer on surrender values on a unit-linked contract allows the policyholder to surrender the insurance policy for a guaranteed amount any time before maturity, similar to an American put option.
• American options are trickier to value than European options and usually require a numerical approach.
• The Black-Scholes model assumes a European option and cannot be used to value American options.
• Discontinuities are allowed by assuming a discrete stochastic process underlying the unit fund
• The optimal time to surrender the policy must be found.
• The value of the unit fund will not just hit the guaranteed surrender value but will fall further below it due to a downward jump.
• Gerber and Shiu show that this problem can be tackled by a result from ruin theory
• They use the concept of discounted probability by relating the stochastic process underlying the stock with the surplus process assumed in ruin theory.
• Wilkie et. al. discuss pricing, reserving and hedging of guaranteed annuity options using a stochastic investment model and option pricing methodology.

Conclusions

• A traditional actuarial technique can value put options and use the result to value maturity guarantees.
• The choice of central assumptions is more important than the choice of model.
• In certain situations, like high volatility or low risk-free force of interest, the choice of model becomes more critical.

Preliminaries of Financial Economics

History

• Mathematical modeling of financial markets began with Louis Bachelier's Theorie de la Speculation in 1900.
• Bachelier modeled the French capital market as a fair game and proved that the standard deviation of future price changes is proportional to the square root of elapsed time.
• Financial economics has developed through advances in stochastic processes to model prices and risk.
• The Black-Scholes model of option pricing in 1973 was a breakthrough due to its assumption of no-arbitrage, it has greatly influenced how traders price and hedge options
• The Black-Scholes model gained widespread acceptability because of its simplicity, and was pivotal to the growth of financial economics and the derivatives market in the 1980s and 1990s.

Derivative Securities

• A derivative is a security that pays its owner an amount that is a function of the values of the underlying securities
• Forwards, futures, options and swaps are common examples of derivative securities.
• A forward contract is an agreement to buy or sell an asset at a certain future time for a certain price, traded over-the-counter and used to hedge foreign currency risk.
• A futures contract is an agreement to buy or sell an asset at a certain time in the future for a certain price and is traded on an exchange.
• The exchange provides a mechanism that gives the parties a guarantee that the contract will be honored.
• The exchange/broker requires both parties to deposit funds in what is termed a margin account, it is adjusted to reflect gains or losses.
• This revaluing adjustment is called ‘marking-to-market'.

Options

• An option gives the holder a right (but not an obligation) to exercise his position at a certain price in the future.
• The types are call options and put options.
• A call option gives the holder a right to buy the underlying asset by a certain date for a certain pre-determined price.
• A put option gives the holder a right to sell the underlying asset by a certain date for a certain price.
• European options can be exercised only on the expiration date and American options can be exercised on any date before maturity.
• A swap is an agreement between two parties to exchange cash flows in the future.
• Currency swaps and interest rate swaps are the most popular swap arrangements in the market.

Martingale Theory

• Martingale theory plays a central role in valuing derivative securities. and is a stochastic process whose expected value at any time in the future is equal to its current value.

Risk Neutral Valuation

• Risk-neutral valuation is employed in financial economics to value derivative securities.
• It introduces an equivalent martingale measure with respect to which the discounted stochastic process under consideration becomes a martingale.
• For every stochastic process there exists an equivalent martingale measure (not necessarily unique) with respect to which the discounted stochastic process becomes a martingale.
• The equivalent martingale measure is known as the risk-neutral probability measure.
• Under the risk-neutral measure the stochastic process grows at the risk-free rate of return and is referred to in literature as the 'Q-measure' to distinguish it from the real world measure, often called the ‘P-measure'.

Risky Assets

• Risky assets will grow at interest rates different from the risk-free rate of return.
• The risk-neutral valuation framework does not assume that investors will be satisfied with risk-free rate of returns for risky assets.
• Under the risk-neutral valuation framework, the price of a derivative security is given by the expectation of the discounted payoff, where the expectation is taken with respect to the risk-neutral probability measure.
• Modern asset pricing theory is based on the law of one price, which states that portfolios with identical payoffs should have the same price, or there would be pure arbitrage opportunities in the market.
• This law is used to value derivative securities by finding 'replicating portfolios' consisting of stocks and bonds that yield the same payoffs in all scenarios.
• The value of the derivative security is then given by the value of the replicating portfolio.

Equilibrium Pricing

• The no-arbitrage valuation approach cannot be applied,equilibrium pricing can be used instead.
• This provides a more general framework for pricing securities using individuals (or agents) with fixed initial resources (or endowments) that trade in a financial market to maximize their expected utility.
• When no individual has an incentive to trade at these prices, the market reaches a state of equilibrium and the prices that evolve are called equilibrium prices.
• Equilibrium prices are related to the attributes of the agents in the economy, such as endowments, beliefs and preferences, as well as to the type and structure of the traded securities.

Put-Call Parity

• The put-call parity theorem states that the payoff from a portfolio consisting of one share of the underlying stock and one European call option is equivalent to a portfolio consisting of a riskless zero-coupon bond and one European put option.
• The value of the two portfolios at any date prior to maturity must be equal as their payoffs at maturity are equal.
• The price of the European put option can be deduced using existing values and vice versa.
• For the put-call parity theorem to hold, the exercise date and exercise price of the European call and put options must be similar.