Financial Engineering Assignment

This document contains solved assignments and essays on various topics related to Financial Engineering. It covers topics such as FX dynamics, martingale measures, trinomial trees, forward contracts, implied volatility, and more. The assignments are suitable for students studying Financial Engineering.

Summative Assignment
Financial Engineering

Table of Contents
Question 1...................................................................................................................................................3
Question 2...................................................................................................................................................3
Question 3...................................................................................................................................................4
Question 4...................................................................................................................................................5
Question 5...................................................................................................................................................5
Question 6...................................................................................................................................................6
Question 7...................................................................................................................................................6
Question 8...................................................................................................................................................7
Question 9...................................................................................................................................................7
Question 10.................................................................................................................................................8
Question 12.................................................................................................................................................9
Question 13.................................................................................................................................................9
Question 14...............................................................................................................................................10
Question 15...............................................................................................................................................10
Question 16...............................................................................................................................................10
Question 17...............................................................................................................................................10
Question 18...............................................................................................................................................11
Question 19...............................................................................................................................................11
Question 20...............................................................................................................................................11
Question 21...............................................................................................................................................12
Question 22...............................................................................................................................................12
Question 23...............................................................................................................................................13
Question 24...............................................................................................................................................14
Question 25...............................................................................................................................................14
Question 26...............................................................................................................................................14
References.................................................................................................................................................15

Question: 1
The specification of the given equation can followthe value of FX dynamics,1
ϵt =∈0 exp ( μt+σt W t
)
The capture of the dynamic specification can follow the consistent case as, ϵ_t. The FX
dynamics of the undervalue can consider Zero correlation as consistent with the implied
volatility of the dynamic value, which can be denotedas,exp (μt+σ t W t
).
The volatility process can be denoted as,
σ
←
(σ )= 1
√T (∫
0
T
∈0 exp (μt+ σt Wt
) ) 1
2
=rT-∈0 exp ¿)2
FX dynamics specification values are denoted as,
ϵt =∈0 exp {rt− 1
2 ∫
0
t
μt 2 ds +∫
0
t
σt W t
2
}
Question: 2
We can consider the asset price value as, Y=y/Bs
We can measure the value of Q using renminbi based investor martingale.
By NA, we also find the values on Y=(ST − K)+,
Y =B$ ¿
By assuming the value of BY/B$=e−r ( Y −$ ) B$ ¿
Y =5.46054/6.042308=e−¿¿7.25077(5.46054-6.042308) 6.042308[(40*4-45)+]
Y =0.90371=104.092
1Alex Zilbel, FX Option Pricing With Stochastic Volatility And Smile Dynamics (Antwerpen: Maklu, 2004).
3

Y=104.092-0.90371
Y=103.161ba
The present value is equal to the price of call on the discount of its payoff’s using Q.
To measure the values of Q−martingale
=B$ [ e−r ( Y −$ ) ] (ST −K )+ ¿]= 103.161
Question: 3
The higher value on intuition for CRR can consider σ,and more u and d can be calculated
on the Trinomial tree, convergence to BS, convergence chart, Delta and leverages values.2 To
find the trinomial trees on the skeleton of (u, d) values are (1.363205, 0.733565). This must
calculate the values automatically. The skeleton of the terminal tree values are the risk neutral
probabilities, as a function of volume.
The same result is true in BS, which can calculate the leverage and is higher to the short
maturities on the return valueas St∗. Let us assume the probability value as S∗t = 0, consider the
equations as,
lim
h → 0+¿ Q (st|30∗0.41667−30 )> ϵ ∥ 30t =30¿
2.2381
¿=0
The CRR call value is 0.245869.
The CRR put value is 6.826752.
2Jillian Farquhar and Arthur Meidan, Marketing Financial Services (Houndmills, Basingstoke, Hampshire:
Palgrave Macmillan, 2010).
4

Question: 4
The first value can be used for the notional of Zt at the time considered as 0. The
intuitively, as time moves on, martingale the prices to move all over the place. They don't stick
and will at some point collapse to zero. 3
Zt =ϵt Bt
Y /Bt
$ i
Calculate the value of equivalent martingale measure Q.
Zt = 5.82 ¿100=0.0582ϵt
Q-martingale measure value is 0.0582ϵt.
Question: 5
The expression values on the martingale for dϵt =(r −u) ϵtdt +σtϵt dW t.
At t, the derivative’s is the function of the underlying share as un-determined drift under Q,
which can calculate the value of μQϵ t. Shared as Qt, as f(dt, dW), also should return at dϵt. That
can expect as the gain of dt.
Let,Drift dt= ∂ d
∂t dt=(r−u) ∂ d
∂ t dt+σtϵt ∂ d
∂ w dW
dt= ∂ d
∂t dt=¿6.935905¿ ∂ d
∂t dt+σtϵt ∂ d
∂ w dW
dt= ∂ d
∂t dt=(13.064095) ∂ d
∂ t dt+σtϵt ∂ d
∂ w dW
=r ∂ d
∂t dt+6.935905u ∂ d
∂ u dt+σt ∂ d
∂ t dW +ϵt ∂ d
∂ w dW
6.935905*20=138.71
3Jay W Forrester, Industrial Dynamics (Mansfield Centre, CT: Martino, 2013).

Question: 6
Let us calculate the values of Q-martingale
By Martingale, we also know that ϵt =(ST − K)+, so
ϵt
Bt
S i=
ϵt
[ST −K ¿+ ¿
ϵt
Bt
S ]
Since the value of ϵt
Bt
S i, can be considerd as,
ϵt =e−r (q−t) ϵt
Bt
S iE Q t [(ST − K) +]
ϵt =e−50(0−1) ϵt
Bt
S iE Q t [(20T − K) +]=-50
Q-martingale expression value is true.
Question: 7
Let us assume the value of σt as constant. 4
The price of the call is equal to the present discounted values of its payoffs using Q. The
distribution of σt under Q.
σt =¿ǫT EXP ((r— δ
2 σ 2 ( T 1−T 2−t ) + σ (δT 2−δT 1))
Let us assume the q value as 1.
4John Knight, Forecasting Volatility In The Financial Markets (Butterworth-Heinemann, 2011).
5

σt =¿ 1*250 EXP ((30— 0 δ
2 σ 2 ( 6.3710−250 )+σ (1∗27.772−1∗0.5))
σt =¿27.772
Question: 8
The fundamental theory of the asset prices can be used for the long time price values of,
So underlying can be either gold, exchange rates, interest rates, or a share, a stock index,
or even another derivative.5 For instance, like in the options on the future and because the future
payoff of derivation securities are revealed by the future prices of the primary securities. The
relationships among the current prices of the derivation and the stock based primary securities on
the arbitrage argument, can be derived. You can use the same methods of the Black stock option
by using the following equation,
E [exp(αX − 1/2α 2 )1X≥−β] = Φ(α+ β), where α and β are real numbers, X ∼ N(0, 1) and Φ
Question: 9
The forward contract can be used for buying or selling an underlying assets at the predetermined
price Ft(T) := K on the future date Ft(T).
The value of ( t, Vt(T; K)) forward prices can follow the below steps,
Notation of assumptions.
The forward price for the investment asset which gives no income.
Value the forward contracts.
5Hugues Leblanc, Statistical And Inductive Probabilities (Dover Publications, 2012).

Forward prices and the future prices.
Stock index futures.
The forward price values follow the given table asFt(T).
Question: 10
The fundamental theorem of the Asset pricing could be utilized for risk-neutral probabilities
measures on the payoff security values. Another way of saying that is to say the expected return
values on the equal risk-free rate is,6
The stock prices can analyze the Target redemption values as,
Yt0=∑
i=1
I
e−r(t −t 0) Et 0
Q
[ {N 1 ( K−∈ t 1 )−N 2(∈ t 1−K ) } 1xti−1
< X ]
6MARTIN TUPPER, PROBABILITIES (Germany: OUTLOOK Verlag, 2018).
7

Thestock prices can analyze the Target redemption values a sN 1 ( K −∈t 1 ) −N 2 (∈t 1−K) to
exchange one currency for another at a contract rate. It can be used for the combination of
pricing and summing together the solving derivation product.
Question: 12
Let us assume the values of BSM model as K = ϵ t0.
Consider the two values on N1 and N2.
Notice the floating points on the stock prices and consider the local vol dynamics as over hedged
on the BSM model, since ϵt0= 1 and it is equal to the values of the variable i.e., N1=N2.
Question: 13
Let us assume the values N1<N2
The values of N1= U10M.
N2== U20M
We simply calculate the values of N1 and N2,
N 1−N 2
totalpriceofstockvalue = 10
50=5
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Leverage can be used for the potential return of the total stock price investment, which can
analyze the prices as, N1<N2.
Question: 14
The value of Levage N1 is greater than N2. So the prices of selling the dollars can buy the
exchange rate K. The contract of the times can consider N1 as less than N2. 7
Question: 15
The target condition of the value can describe the value as X is greater. The target
condition refers to the forward looking new goal, which has positive future projectionsinstead of
the problem’s reflection. It refers to moving towards something as the path for achievingand
considering the time values.
Question: 16
The implied volatility could be utilized for recoveringthe US dollar’srisk neutral probability,
which can exchange the rebate and the renminbi value of the implied exchange rate option price
values.8 The risk neutral densities’ approaches can be used in analyzing the current market of risk
and it is mainly used for the renminbi market values.The US dollar isthe official currency of the
United Statesof America and this currency will be used for the internal trade and as reserve
currency for much of the world. The Euro currency can be considering the value of dollar
currency.
Question: 17
7Keith Ward, Financial Aspects Of Marketing (Hoboken: Taylor and Francis, 2013).
8Victoria Wohl, Probabilities (Cambridge University Press, 2014).
9

The comparing prices on TARF can be used for the term structure of the surface current
marketing prices with the time to maturity on N1. It is used for the Black Scholes on N2 for
increasing the price. The volatility surface used for the short term value options have much
higher volatilities than the longer term options.By comparing each surface on value N1=N2.
Question: 18
The product values of the time can be used for the actual expect payoff under the physical
probability, with its personalized risk corrected discount factor on N1 product such as the
barriers. The other structured products such as cliques whose price today is determined by the
future skews.
Question: 19
The Hesston model implements the drift and diffusion of the time price by calculating the value
of the constant correlation coefficient. The most well-known stockvol model is Hesston. It
allows quasi with closed forms and efficient mentee carol. It is used for the special cases on the
variance by,
d ϵt = ( r−u ) ϵ t dt +ϵ t √ ν tdWt
d vt =−λ ( vt −v ) dt +μ √ ν td wt
v
d ϵ t+d vt = ( ( r−u ) ϵt −λ ( vt−v ) ) dt +¿)
Here, μ>0 , ϵt > 0 , λ>0 , vt > 0value is constant. The initial process is utilized fordescribing the
movement of the underlying asset S and the next process defines the assumed behavior of
volatility Wv. The increasing value of ∆t uses Gaussian process of the value as,
E[|WH t+∆ − WH t | q ] = Kq∆qH,
While the linearity is restrictive, the mean reversing property is intuitive; one does not expect
volatility to drift off to zero or to plus infinity, but it remains in some band.
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Question: 20
Terminal probabilities distribution P(S) at tenor T, have the model free formula for the
call price C(K) as a function of strike K.
The equation can be written as,
Therefore, all we need is to find the value of P(X) for any X, which is the second derivative of
the call prices at strike X.
The probability distribution values of the process generateQ plots as one of the simply finite
differencing values.
Question: 21
If we only look at the IV surfaces, the Heston model gives decent fit at the longer
maturities, and the shorter maturities with worse fit, thus this is reason that most of the
practitioners wish to have local vol models (covered later).
• In fact, as argued in Gathered, any stochastic volatility model provides roughly the same shape
of IV surface for the European options.
• This does not mean they price and hedge all derivatives alike as the dynamics, but they
generate different.
• To improve the fit, jumps can be added. This is a promising avenue, which we shall not pursue
here for the technical reasons.
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Question: 22
Hesston models of TRAF prices Constellations Changing Positions.
Hesston model, it is important to note the properties of Brownian motion, as they relate to
stochastic volatility.
Question: 23
The Monte Carlo simulation conducts the analysis of risk by building the models of the possible
outcomes, with the substitution of various probability distribution values, for the factor which
contains inherent uncertainty. Later, it evaluates the outcome repeatedly and every single time it
utilizes an unusual set of random values from the probability function. Next, the Monte Carlo
method’s accuracyfor simulating the distributions is examined. The risk analysis and the
management at present develop a strong presence in the financial institutions, where energy
efficiency and high performance are the major requirements to accelerate the systems.This
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happens mainly in cases where it comes to the intraday analysis. (PDF) Nested MC-Based Risk
Measurement of Complex Portfolios: Acceleration and Energy Efficiency. By the way, the
largely employed portfolio risk metrics value-at-risk (VaR) and conditional value-at-risk (cVaR)
by means of nested Monte Carlo (MC) simulations are estimated.
Question: 24
Let the unit of time be,“year’. Let the holding period [t i , T] be ‘n’ years. Let us consider
the frequency of compounding per unit of time, i.e. per year. Call rim an m-times compounded
yearly net return, and denote the continuously compounded (yearly) net rate of return by r ic .
Then, the compounding is only the convention of equation:
Question: 25
The implication of dynamic hedging is the significance of the Napoleon, which requires
accompanying similar products. The main problem include that it is difficult to statistically
hedge the Napoleon-type products, moreover it needs dynamic hedging. However, the key point
includes that the dynamic hedging under Question here is unusual when compared to the one in
the plain vanilla options, as here thepractitioner purchases and sells the underlying Si t for
hedging the directional movements in the option Price. Gamma gains (losses) is the outcomes of
such a dynamic hedging. The volatility exposure refers to the Napoleon-type products hedging. It
is necessary that the practitioner must purchase or sell the volatility dynamically.
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Question: 26
The retail store that sold the product does not have to be the manufacturer to be held
liable in a defective product lawsuit. Because, the retailer is a part of the distribution chain, the
retailer may be liable for the defective product. The products liability law states that,“the
manufacturing defect refers to a product defect which wasn’t made purposefully”.
References
Farquhar, Jillian, and Arthur Meidan. Marketing Financial Services. Houndmills, Basingstoke,
Hampshire: Palgrave Macmillan, 2010.
Forrester, Jay W. Industrial Dynamics. Mansfield Centre, CT: Martino, 2013.
Knight, John. Forecasting Volatility In The Financial Markets. Butterworth-Heinemann, 2011.
Leblanc, Hugues. Statistical And Inductive Probabilities. Dover Publications, 2012.
TUPPER, MARTIN. PROBABILITIES. Germany: OUTLOOK Verlag, 2018.
Ward, Keith. Financial Aspects Of Marketing. Hoboken: Taylor and Francis, 2013.
Wohl, Victoria. Probabilities. Cambridge University Press, 2014.
Zilbel, Alex. FX Option Pricing With Stochastic Volatility And Smile Dynamics. Antwerpen:
Maklu, 2004.
14