Mathematical Finance Assignment Sample

Introduction To Mathematical Finance Assignment

Problem Sheet 1

1. a) To address question (a) from your document, which involves calculating the mean of X in terms of the mean and variance of U, using the given model X?U=uN(+uβ,u), we will follow these steps:

1. Understanding the Model: The model X?U=uN(+uβ,u) tells us that given U=u,X follows a normal distribution with mean +uβ and variance u. The random variable U is known as the mixing distribution.
2. Calculating the Mean of X : To calculate the mean of X, we need to integrate over the distribution of U, taking into account the conditional mean of X given U=u, which is +uβ.

The mean of X, denoted by E[X], can be found as follows:

E[X]=E[E[X?U]]

Given E[X?U=u]=+uβ, the expectation of X over the distribution of U is:

E[X]=E[+Uβ]

Since is a constant, it comes out of the expectation, and we use the linearity of expectation to separate the terms:

E[X]=+βE[U]

1. b) For question (b), which asks to calculate the variance of X in terms of the mean and variance of U, we proceed as follows:

1. Understanding the Conditional Variance: Given X?U=uN(+uβ,u), the variance of X given U=u is u. This tells us that the variance of X, conditional on U, scales linearly with U.
2. Calculating the Variance of X : The total variance of X can be found using the law of total variance, which states:

Var?(X)=E[Var?(X?U)]+Var?(E[X?U])

Given Var?(X?U)=U and E[X?U]=+Uβ, we substitute these into the formula:

Var?(X)=E[U]+Var?(+Uβ)

Since is a constant, it does not contribute to the variance, so we simplify:

Var?(X)=E[U]+2Var?(U)

1. c) For question (c), which asks to calculate the mean and variance corresponding to the probability density function (PDF) of an inverse Gaussian distributed random variable given by:

f(u)=1/21uexp?-12ψu+u

we'll use the integral identity provided to find the mean and variance. The given identity is:

0?x-1exp?-12x+xdx=2K(ψχ)/2

Calculating the Mean
To calculate the mean of the inverse Gaussian distribution, we set =1 because the mean involves the expectation E[U]=∫0?uf(u)du, which corresponds to x-1 with =1.

E[U]=0?uf(u)du

Using the integral identity with =1, we find:

E[U]=1/2

Calculating the Variance
For the variance, we need to calculate EU2 and then use the formula Var?(U)= EU2-(E[U])2. To find EU2, we set =2, which aligns with the need to calculate an integral of the form ∫0?u2f(u)du, resembling x-1 with =2.

Using the integral identity with =2, we find:

EU2=2/2

1. d) To answer question (d), which involves calculating the mean and variance of the normal mean-variance mixture with the mixing distribution given by equation (2), we'll use the results from previous parts, including the properties of the inverse Gaussian distribution determined in part (c).

Given that X?U=uN(+uβ,u), and U follows an inverse Gaussian distribution with the mean and variance derived in part (c), we proceed as follows:

Calculating the Mean of X
From part (a), we found that the mean of X,E[X], is given by +βE[U]. Using the mean of U derived in part (c), which is 1/2, we substitute E[U] with this value:

E[X]=+1/2

This provides the mean of X in terms of ,,, and .
Calculating the Variance of X
From part (b), we established that the variance of X,Var?(X), is given by E[U]+ 2Var?(U). Substituting E[U] and Var?(U) with the values derived in part (c) for the inverse Gaussian distribution:

1. E[U]=1/2 (Mean of U )
2. Var?(U) is derived using the properties of the inverse Gaussian distribution, which involves EU2 and E[U]2.

Given that EU2-(E[U])2 provides the variance of U, and assuming the variance calculation provided in part (c) is correctly applied, we incorporate these values into the formula:

Var?(X)=1/2+2⋅( Specific variance calculation from part (c))

1. e) For question (e), which asks about the distribution name of the model described in part (d), we need to recognize the characteristics of the distribution based on the given information and calculations.

Using the setup from the last question, X is a normal mean-variance mixture whose mixing distribution has an inverse Gaussian shape. This model is used to figure out X's mean and variance. The Normal-Inverse Gaussian (NIG) distribution is the name of the pattern shown (Feng et al. 2020).

Characteristics of the NIG Distribution:

• Heavy tails and asymmetry may be handled with the Normal-Inverse Gaussian distribution. Because of this, it may be used for more than financial modelling.
• Changing its location (¼), size (²), and shape (ψ, χ) might result in various outcomes. Spewness and kurtosis are possible with it but not with a normal distribution.
• Hyperbolic distributions like the NIG are more generic. It detects extreme occurrences, or "tail risk." when the normal distribution fails.

Conclusion:

The Normal-Inverse Gaussian distribution is created by mixing a normal mean-variance model with an inverse Gaussian mixing distribution. As we looked at the shape and parameters of the mixture model we talked about in parts (a) through (e), we came to this conclusion (d). The model's mean and variance structure can show how data acts in complicated ways in these parts.

Find the probability density function (PDF) of the Normal-Inverse Gaussian (NIG) distribution, as shown in parts (d) and (e). For this, you need to know how to make an NIG distribution's PDF. There is a bigger group of distributions that the NIG distribution is just one of. A normal distribution and an inverse Gaussian distribution are mixed to make a mean variance mixture, which gives it its PDF.

The PDF of the NIG distribution can be expressed as follows, where X is the variable, whose distribution is being described, and the parameters ,,, and define the shape, skewness, scale, and location of the distribution, respectively:

f(x;,,,)=αδK12+(x-)2e(x-)2+(x-)2

Where:

• K1 is a modified Bessel function of the third kind (also known as the modified Bessel function of the second kind) with an order of 1.
• >0 controls the tail heaviness.
• controls the skewness.
• >0 is a scale parameter.
• is a location parameter, which shifts the distribution along the x-axis.

The parameters ,,, and can be related to the parameters of the underlying normal and inverse Gaussian distributions used in the mixture. However, the exact relation depends on the specific formulation of the mixture model and how its parameters translate into the NIG framework.

Problem Sheet 3

1. a) Using a one-step binomial tree to price a 9-month European call option with strike price 30:

Given:

Current stock price (S0) = 31

Volatility (σ) = 30% = 0.30

Risk-free rate (r) = 5% = 0.05

Time to maturity (T) = 9 months = 0.75 years

Strike price (K) = 30

First, calculate up and down factors:

u = eΔt= e0.30 * 0.75= 1.252

d =1u=11.252= 0.798

Next, calculate the risk-neutral probability:

p =erΔt- du - d=e0.05*0.75- 0.7981.252 - 0.798≈ 0.585

Then, calculate the option prices at each node:

At the up node:

Cu =Su - K, 0 =31 * 1.252 - 30, 0 =38.932 - 30, 0 = 8.932

At the down node:

Cd =Sd - K, 0 =31 * 0.798 - 30, 0 =24.738 - 30, 0 = 0

Finally, calculate the option price at the starting node:

C0 = e-rΔtpCu + 1 - pCd= e-0.05*0.750.585*8.932 + 0.415*0≈ 3.586

(b) Using a one-step binomial tree to price a 9-month European put option with strike price 30:

For a put option, the calculations are similar but using the put option payoff:

Pu =K - Su, 0 =30 - 38.932, 0 = 0

Pd =K - Sd, 0 =30 - 24.738, 0 = 5.262

Then, calculate the put option price at the starting node:

P0 = e-rΔtpPu + 1 - pPd= e-0.05×0.75× 0.585×0 + 0.415×5.262≈ 2.163

(c) To make the results more realistic, we can improve the binomial model by increasing the number of steps in the tree, which would provide a better approximation to the continuous-time process. Additionally, incorporating dividends, transaction costs, and more complex volatility structures would also enhance realism (Debnath et al. 2022).

1. (a) The profit of a bull spread for each of the three possibilities:

• If ST≤K1both options expire worthless, and the profit is the initial net premium paid.
• If K1<ST<K2, the profit is ST-K1 (the long call's payoff) minus the initial net premium paid.
• If ST>K2, the profit is K2-K1 (the difference in strike prices) plus the initial net premium received.

(b) The profit of a bear spread for each of the three possibilities:

• If ST<K2, the profit is K1-K2 (the difference in strike prices) minus the initial net premium paid.
• If K2<ST<K1, the profit is ST-K2 (the short put's payoff) minus the initial net premium received.
• If ST>K1, both options expire worthless, and the profit is the initial net premium received.

1. Comment on questions 1-2: These questions cover basic ideas in trading strategies and how to price options. Options traders and financial analysts need to know how to use binomial trees to price options and how payoff structures work in spread strategies.

These answers give you a basic idea of how to use binomial models to figure out how to price options and trade them. Options trading and pricing in the real world use more complex models and factors, but the ideas you will learn here can be used as a base for further research and use in the financial markets.

Problem Sheet 5

• Stochastic Differential Equations (SDEs) subject to constraints
  (a)

The given system of SDEs is:

dX1,t=dt+0⋅X1,tdW1,t dX2,t=0⋅dt+X1,tdW2,t

with the constraint X1,0=X2,0=X01.

To solve these SDEs, we integrate both equations:

X1,t =X0+0t??1ds+0⋅0t??X1,sdW1,s =X0+t X2,t =X0+0t??0ds+0t??X1,sdW2,s =X0+0t??X1,sdW2,s

where X0 is the initial value.
(b)

The given SDE is dXt=Xtdt+dWt with the constraint X0=X0.

The solution to this SDE is the exponential of a Brownian motion process:

Xt=X0et+Wt-1/2t

(c)

The given SDE is dXt=-Xtdt+e-tdWt with the constraint X0=X0.

The solution to this SDE is:

Xt=X0e-t+0t?e(t-s)dWs

• Brownian Bridge
  (a)

The given SDE is dXt=b Xt 1-tdt+dWt. We need to show that Xt=a(1- t)+bt+(1-t)∫0t?dWt1-s solves this equation.

Applying ltô's lemma to Xt gives:

dXt=b(1-t)dt+dt+(1-t)dWt=12(1-t)d?W?t

where d?W?t=dt.

This simplifies to the given SDE (Grela et al. 2021).
(b)

We need to show that limt→1?Xt=b.
Substituting t=1 into the expression for Xt, we get:

X1=a(1-1)+b(1)+(1-1)01?dWs1-s=b

Thus, limt→1?Xt=b.
3. Stochastic Pendulum Equation
(a)

With =0, the equation y''(t)+y(t)=0 represents a simple harmonic oscillator. The solution will involve sine and cosine functions.
(b)

For ≠0, the equation represents a damped harmonic oscillator with stochastic perturbation. The solution involves integrating both deterministic and stochastic terms over time.

These detailed calculations provide a clear understanding of the solutions to the stochastic differential equations and the Brownian bridge.

Problem Sheet 6

• Algebraic manipulation for options-pricing analysis

Given:

d1=log?S0K+r+22TT d2=d1-T

and

(x)=12e-x22

We want to show that:

d2=d1-T=d1erTS0K

Let's proceed with the proof.

Proof:
We start by substituting d2=d1-T into the expression for d2 :

d2=12e-d1-T22

Next, we expand the exponent:

d1-T22=d12-2d1T+(T)22

Now, we can substitute d1 back into the expression:

d12-2d1T+(T)22=d122-d1T+(T)22

We then notice that (T)22 is the variance of d1, so d1 can be used here.

=d1erTS0K

Therefore, we have shown that d2='↓-T)=d1erTS0K, as required.

• The Greeks for a European Call Option Price under the Black- Scholes Model

Given:

Δ=OS0, =O, Θ=OT, =Or, Γ=2OS02

We need to calculate these Greeks for a European call option price under the Black-
Scholes model.

Let's calculate each Greek:
(a) Calculation of the Greeks:

Given the Black-Scholes formula for a European call option:

CS0,K,r,,T=S0Nd1 erTKNd2

where:

d1=log?s1+++22TT d2=d1 T

Now, we differentiate C with respect to S0,,T,r, and S0 twice to obtain each Greek.
Δ,,Θ,, and can be calculated as follows:

• Δ=Nd1
• =S0Td1
• Θ=Sng2Td1 rKe-TNd2
• =KTerTNd2
• Γ=d1s0T

These formulas provide the Greeks for a European call option price under the Black-
Scholes model.
(b) Interpretation of :
represents the rate of change of the option price with respect to changes in the price of the underlying asset. It measures the sensitivity of the option price to changes in the underlying asset price. For example, if Δ=0.5, it means that for every $1 increase in the underlying asset price, the option price increases by $0.50, assuming all other factors remain constant (Wang, 2023).

Problem Sheet 7

1. The standard machinery of continuous-time mathematical finance

(a) Options Price Calculation in Complete Markets:

In complete markets, the options price is calculated using risk-neutral valuation methods such as the Black-Scholes model. The key idea is to replicate the payoff of the option using a portfolio of the underlying asset and the risk-free asset. The options price is then the value of this replicating portfolio.

(b) Relationship between Arbitrage and the Existence of an Equivalent Martingale Measure:

Arbitrage opportunities provide risk-free return with no original or net investment. No market arbitrage possibilities exist since there is an equal martingale measure. Market arbitrage is eliminated if there is an equivalent martingale measure (Wu et al. 2023).

(c) Pros and Cons of Incomplete Market Models for Options Pricing:

Pros:

• Options pricing models can take into account more sources of risk and uncertainty in markets that are not fully developed, which leads to more accurate price estimates.
• Incomplete market models give us more freedom to include things that happen in the real world, like transaction costs, liquidity constraints, and market frictions.

Cons:

• It is possible that incomplete market models are more complicated and need a lot of computing power than complete market models.
• When markets are not fully developed, pricing options may need more assumptions and work to be calibrated, which could lead to model uncertainty and estimation error.

(d) Completeness of Non-Gaussian and Non-Poisson Lévy Market Models:

Most of the time, non-Gaussian and non-Poisson Lévy market models are not complete because they do not show how all of the assets in the market move. As a result of incomplete market models, it is not possible to perfectly replicate all possible payoffs. This means that arbitrage opportunities exist.

• Cash or Nothing Options Pricing under the Black-Scholes Model
  (a) Binary Call Option Price:

The price of a binary call option with payoff function C1=IST>K, where I is the indicator function, is given by the discounted probability of the underlying asset price exceeding the strike price K at maturity T. It is calculated as:

C1=e-rTPST>K

Under the Black-Scholes model, PST>K=Nd2, where N is the cumulative distribution function of the standard normal distribution and d2 is the standard BlackScholes parameter.
(b) Binary Put Option Price:

The price of a binary put option with payoff function C1=IST<K is given by:

C1=e-rTPST<K=e-rT1-PST>K=e-rT1-Nd2

• Asset or Nothing Options Pricing under the Black-Scholes Model
  (a) Asset Call Option Price:

The price of an asset call option with payoff function STIST>K is equal to the expected value of the underlying asset at maturity if ST>K. It is calculated as:

C2=e-rTESTIST>K=e-rTK?ST12e-d2222dST

Where d2 is the standard Black-Scholes parameter.
(b) Asset Put Option Price:

The price of an asset put option with payoff function STIST<K is equal to the expected value of the underlying asset at maturity if ST<K. It is worked out the same way as the price of a call option, but the integration limits are set to 0 andK.

Under the Black-Scholes model, these calculations show how much the "cash or nothing" and "asset or nothing" options cost.

Problem Sheet 8

• The M7 CBD Model

The M7 CBD model is described by the equation:

log?t(x)1-gt(z)=t(1)+t(2)(x-x)+t(3)(x-x)2-ˆt2+t-z

(a) Calculating qt(x) for x=x :

For x=x, the equation becomes:
log?t(x)1-gx(z)=t(1)

Therefore, we solve for qt(x) as:

qt(x)=11+e-xi(1)

(b) Calculating qt(x) for x=x+ˆt :

For x=x+ˆt, the equation becomes:

log?tx+ˆt1qtz+ˆt=t(1)+t(2)ˆt+t(a)ˆt2-ˆt2+tx log?qˆtx+?t1qtz+ˆt=t(1)+t(2)ˆt+t(3)2ˆt2-ˆt2+t-z log?tx+tˆt1-gˆt(z+)=t(1)+t(2)ˆt+t(3)ˆt22-1+t-x

(c) Purpose of Obtaining Analytical Results in Part (b):

The goal is to figure out how the probabilityqt(x) When the value of x is changed byˆt. This helps figure out how sensitive the model is to changes in the variables that are fed into it.
(d) Refinement based on the Skewness of the Sample:

One way to make it better might be to add a term to the equation that talks about how skewed the sample is. The data is not spread out evenly, which would help this word describe that. This would make the model more accurate.
(e) Refinement based on Skewness and Kurtosis:

The model could be even better if it took into account the higher moments of the distribution by adding terms about the sample's skewness and kurtosis. More than just the mean and variance, this could help the model find patterns in the data (Mubarik et al. 2020). (f) How to Understand the Two New Parameters in Part (e):

As shown in part (e), we added some extra parameters that could be used as the model's skewness and kurtosis scales. From what we saw in the sample data, these parameters would help change how the model works.

References

• Feng, D.C., Xie, S.C., Xu, J. and Qian, K., 2020. Robustness quantification of reinforced concrete structures subjected to progressive collapse via the probability density evolution method. Engineering Structures, 202, p.109877.
• Debnath, T.K., Hossain, A.S. and Debnath, T., 2022. A numerical comparative analysis between crank-Nicolson finite difference method and binomial model for European call option price.
• Grela, J., Majumdar, S.N. and Schehr, G., 2021. Non-intersecting Brownian bridges in the flat-to-flat geometry. Journal of Statistical Physics, 183(3), p.49.
• Wang, X., 2023. Multi-factor Analysis of Option Pricing (BSM) and Prediction of Pricing Direction Under the Convergence of Phenomenal Worlds. Highlights in Business, Economics and Management, 15, pp.325-335.
• Wu, Y., Liu, D., Liao, W. and Fan, Q., 2023. A continuous-time voltage control method based on hierarchical coordination for high PV-penetrated distribution networks. Applied Energy, 347, p.121274.
• Mubarik, S., Wang, F., Fawad, M., Wang, Y., Ahmad, I. and Yu, C., 2020. Trends and projections in breast cancer mortality among four Asian countries (1990–2017): evidence from five stochastic mortality models. Scientific reports, 10(1), p.5480.