代写接单- Final Project – Version 1

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  Final Project – Version 1

• Please submit your project (electronically via Canvas) by noon on Sunday, May 14, 2023.

• Total attainable points: 45 plus a potential bonus for particularly nice projects.

• Teams of 2 students max. You can use all resources and discuss the problems with each other, but the problems should be solved and written up by each team.

• Submit your results in report form but also add in your model development (code, spreadsheets, etc.). However, grading will be primarily based on the report (!!), the model files are truly only supporting information. Hence, please make sure that all pertinent information is in the report!

You just started your first job at VAINS, a Variable Annuity provider. VAINS intends to sell a 10-year Variable Annuity with a Guaranteed Minimum Income Benefit to a population of l55 = 1, 000 55-year old, female individuals, with initial investment G = 100, 000. The funds are tied to the S&P 500 index (today’s value is 4,125).

The contract payoff is specified as follows:

• If a policyholder dies in between age 55 and age 65, her family will receive her current account value as the death benefit, paid at the end of the year of death.

• If the policyholder survives until 65, she will receive the maximum of the current account value and the guaranteed amount, which is the initial contribution compounded at the rate g = 1.5% (that is the guaranteed amount is G × (1 + g)10).

• Every instant, a guarantee fee at rate φ as a percentage of the account value is paid to the insurer as a premium.

Since the company hired you as their quant, you are approached by the management regarding the pricing of this line of business. You are put in the project team, and your new colleagues have found the following:

• The account value is modeled by a Black-Scholes model. That is, under the risk-neutral measure, the account value evolves according to:

dAt =At((r−φ)dt+σdWt), A0 =G.

• The risk-free rate is r = 3.30% (10-year treasury rate), the volatility is σ = 18%.

• Each year, 20 of the insureds are expected to die, i.e. we have l55+k = l55 − 20 k survivors at time k.

If you rely on Monte Carlo simulations, the company leadership asks you to illustrate the uncertainty in your estimate (confidence interval).

Advanced Derivatives

Finance 830

 

Finance 830 Final Project 2

 1. As a first step, your new team is trying to evaluate what value to choose for φ. You explain them that an adequate pricing mechanism would be that the risk-neutral value (expected value under the risk-neutral measure) of the contract’s payoffs equals the initial investment G. Determine this “fair fee.”

2. Yoursupervisorisimpressedandyouareputinchargeoftheproject–thatwasaveryfastpromotion! You are just about to sit back and relax as your supervisor enters your office – out of breath. The contracts sold quickly but the asset management team is unclear what to do with the contributions all the policyholders made (l0 × G). Please advise them on an appropriate hedging strategy. In particular, advise the asset management team how to invest the funds at time zero and how to update their investments, and how frequently to do that.

3. Your supervisor is concerned that a more advanced asset model may even further increase the value. You and your project team decide to redo the calculations from 1 in the Heston Model:

• The account value is modeled by a Heston model. That is, under the risk-neutral measure, the account value evolves according to:

dAt = At((r−φ)dt+√vtdWt), A0 =G; dvt = κ(θ−vt)dt+σ√vtdZt, v0 >0;

dZtdWt = ρdt.

• The risk-free rate is r = 3.30%, the parameters of the volatility process are κ = 0.7284, θ =

0.03452, σ = 0.2726, v0 = 0.0322, and correlation coefficient is ρ = −0.43.

• As before, each year, 20 of the insureds are expected to die, i.e. we have l55+k = l55 − 20 k

survivors at time k.

Redo the calculation in part (1), that is determine the fair fee. How does this model change affect your

answer in 2. What are possible hedging instruments the asset management team could be using.

4. Your supervisor is still not satisfied. She believes that interest rates may change over the next 10 years as well. She advises to incorporate stochastic interest rates as well. You and your team propose to:

• Use a Vasicek model for interest rates:

drt =α(β−rt)dt+γdXt,

where (Xt) is a third Brownian motion that is independent of the Brownian motions driving

equity and volatility risk (so correlations are zero).

• Assume α = 0.3, γ = 0.02 and r0 = 5.1% (today’s 3 month rate). Determine the mean reversion level β so that the 10-year rate is treasury rate equals 3.30%, to have consistency with the previous results.

Redo the calculation in part (1), that is determine the fair fee. How does this model change affect your answer in 2. What are possible hedging instruments the asset management team could be using.

5. Your supervisor asks you to write a report about aspects not reflected in the model (less than 250 words), and particularly how they may affect the fair fee. Also, discuss possible remedies/mitigation strategies.

(10 + 10 + 10 + 10 + 5 points)

 

 


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