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### MATH580 Project On Financial Stochastic Processes

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MATH580 Project on Financial Stochastic Processes (MATH580)

Deadline: 10am, Monday of Week 8, 25th November.

The Black-Scholes model for stock prices

The Black-Scholes model describes the price of a stock at time t as a stochastic process {St : t ≥ 0}, determined by

St = S0 exp(µt + σXt),

where S0 is the initial price of the stock, {Xt : t ≥ 0} is a standard Wiener process, and the constants µ and σ > 0 are the drift rate and the volatility, respectively.

In the project we consider the Black-Scholes model when specialised to a European call option. Such call option on the stock with strike price c and expiry time t0 is a contract that gives the buyer the right (but not the obligation) to buy one unit of the stock at time t0 at price c. The buyer will only exercise this option if St0 > c, in which case the payoﬀ is given by C = (St0 − c)+ = max{St0 − c, 0}. Using the Black-Scholes formula, the price at time 0 of this European call option is given by

Pt0  = E [eρt0 C] = E [eρt0 (St0 − c)+],

where ρ is the interest rate and we assume that µ = ρ − σ2/2. Note that this choice of µ is the one that means the expected discounted stock price is constant.

 This formula can be written as Pt0  = S0Φ log(S0/c) + (ρ + σ2/2)t 0 − ce−ρt0 Φ log(S /c) + (ρ − σ2/2)t 0 . σ√ 0 σ√ t0 t0

Plot {Pt : 0 ≤ t ≤ 10} against time for S0 = 1, σ2 = 0.02, ρ = 0.03 and c = 1 and comment on this.

Investigate how the price Pt0 varies with σ, c and ρ. For example plot Pt for t = 10 as you vary each of σ, c and ρ in turn. Comment on the results you get, including why Pt behaves as it does.

[Note that you do not need to do any simulation for this part of the project.]

The Vasicek model for interest rates

In the Black-Scholes model, the assumption was made that the interest rate was fixed and risk-free. In practice this is not the case and several models exist that attempt to describe interest rates mathematically.

The Vasicek model defines the price, Qt, at time 0 of a bond paying one unit at time t as

Z t

Qt = exp               Rsds     ,

0

where {Rs : s > 0}, the spot-rate process, is an Ornstein-Uhlenbeck process. More precisely,

Rs = eθsR0 + (1 − eθs)µ + Xs,

where R0 is the initial spot rate, µ is the long-term mean level, θ > 0 is the speed of reversion and

Xs is an Ornstein-Uhlenbeck process with volatility σ > 0 and reversion parameter θ > 0. That is E [Xs] = 0 and Cov (Xs, Xt) = σ2 eθ(s+t)(e MIN(s,t) − 1).

Simulate {Rs : 0 ≤ s ≤ 10} for R0 = 0.1, θ = 0.5, µ = 0.05 and σ = 0.02 and plot realisations of this. Plot E [Rs] and Var (Rs) against time. What do you notice? Can you explain this? What happens if you change the values of the parameters?

Simulate the bond price Qt at time 0 of a bond paying one unit at time t for the parameter values above and plot realisations of {Qt : 0 ≤ t ≤ 10}. What is the distribution of Qt for a fixed value of t? Illustrate this by simulation, for values of the parameters given above.

The report

This project contributes 20% to the overall assessment for MATH580. Each project should consist of a written report which should be submitted via Moodle, as a pdf file, by 10am, Monday of Week 8, 25th November. The written report should contain all R codes used as an appendix. Reports which contain computer code in the main body will be penalised. The report should be no more than 4 pages, written in size 12 font with sensible margins and spacing. This includes all tables, graphs; but excludes the appendix. Reports over 4 pages will be penalised. A shorter report, with material eﬀectively communicated, is acceptable. Marks will be assigned for the presentation, organisation and content of your report.

Your goal is to carry out the investigations of the financial stochastic processes outlined above. For the model, in addition to the graphs and explanations of your findings, you should provide some background information and a brief discussion of benefits and shortcomings of the models. Remember to reference this appropriately.

The expectation is to have an introduction, sections on the pricing of European options, on the OU process and on the bond price process, and a conclusion. As a rough guide, aim for about half a page on the introduction and conclusion, and one page on the other three sections. The introduction would give a brief background to the models, and outline the contents of the report, the conclusion would discuss limitations of the models. Each of the other sections would involve some figures, explanation of what the figures are showing (e.g. we plot the price of the European option as we vary each of c, σ and ρ.), and discuss what the figures show (e.g. as c increases we see that the price of the bond..., The reason for this is that ...).

Selected References

Black, F. and Scholes, M. (1973). “The Pricing of Options and Corporate Liabilities”. Journal of Political Economy 81 (3): 637-654.

Hull, John C. (2003). Options, Futures and Other Derivatives. Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-009056-5.

Vasicek, Oldrich (1977). “An Equilibrium Characterisation of the Term Structure”. Journal of Financial Economics 5: 177-188.

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