Stochastic Modelling in Finance
|
|
- Ellen Burke
- 5 years ago
- Views:
Transcription
1 in Finance Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH April 2010
2 Outline and Probability 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
3 Outline and Probability 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
4 Outline and Probability 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
5 Outline and Probability 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
6 Outline and Probability 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
7 Problem and Probability Assume that on 10 April 2010, Mr King has $100K to invest for 1 year and he has two choices: (a) invest the money in a bank saving account to receive a risk-free interest. (b) buy a $100K house and then sell it on 10 April Which choice should Mr King take?
8 Problem and Probability Assume that on 10 April 2010, Mr King has $100K to invest for 1 year and he has two choices: (a) invest the money in a bank saving account to receive a risk-free interest. (b) buy a $100K house and then sell it on 10 April Which choice should Mr King take?
9 Problem and Probability Assume that on 10 April 2010, Mr King has $100K to invest for 1 year and he has two choices: (a) invest the money in a bank saving account to receive a risk-free interest. (b) buy a $100K house and then sell it on 10 April Which choice should Mr King take?
10 Assume the annual interest rate r = 1% and let X denote the price of the house on 10 April Consider cases: (i) P(X = $110K ) = 0.5 and P(X = $90K ) = 0.5. That is, the price will increase or decrease by 10% equally likely. In this case, EX = $100K so it is better to deposit the money in the bank which gives Mr King $101K by 10 April (ii) P(X = $110K ) = 0.6 and P(X = $90K ) = 0.4. In this case, EX = 0.6 $110K $90K = $102K, which is $1K better than the return of the saving account.
11 Assume the annual interest rate r = 1% and let X denote the price of the house on 10 April Consider cases: (i) P(X = $110K ) = 0.5 and P(X = $90K ) = 0.5. That is, the price will increase or decrease by 10% equally likely. In this case, EX = $100K so it is better to deposit the money in the bank which gives Mr King $101K by 10 April (ii) P(X = $110K ) = 0.6 and P(X = $90K ) = 0.4. In this case, EX = 0.6 $110K $90K = $102K, which is $1K better than the return of the saving account.
12 Assume the annual interest rate r = 1% and let X denote the price of the house on 10 April Consider cases: (i) P(X = $110K ) = 0.5 and P(X = $90K ) = 0.5. That is, the price will increase or decrease by 10% equally likely. In this case, EX = $100K so it is better to deposit the money in the bank which gives Mr King $101K by 10 April (ii) P(X = $110K ) = 0.6 and P(X = $90K ) = 0.4. In this case, EX = 0.6 $110K $90K = $102K, which is $1K better than the return of the saving account.
13 Assume the annual interest rate r = 1% and let X denote the price of the house on 10 April Consider cases: (i) P(X = $110K ) = 0.5 and P(X = $90K ) = 0.5. That is, the price will increase or decrease by 10% equally likely. In this case, EX = $100K so it is better to deposit the money in the bank which gives Mr King $101K by 10 April (ii) P(X = $110K ) = 0.6 and P(X = $90K ) = 0.4. In this case, EX = 0.6 $110K $90K = $102K, which is $1K better than the return of the saving account.
14 Assume that you trust the housing market will obey Case (ii). Should you have $100K available, you would have invested it into the house to obtain the expected profit of $2K. The problem is that you do NOT have the capital of $100K and you just feel unfair to give the opportunity to rich people like Mr King. However, Professor Mao would like to help. On 10 April 2010, Professor Mao (the writer) writes a European call option that gives you (the holder) the right to buy 1 house for $100K on 10 April 2011 from Prof Mao if you wish.
15 European call option Definition A European call option gives its holder the right (but not the obligation) to purchase from the writer a prescribed asset for a prescribed price at a prescribed time in the future. The prescribed purchase price is know as the exercise price or strike price, and the prescribed time in the future is known as the expiry date.
16 On 10 April 2011 you would then take one of two actions: (a) if the actual value of a house turns out to be $110K you would exercise your right to buy 1 house from Professor Mao at the cost $100 and immediately sell it for $110K giving you a profit of $10K. (b) if the actual value of a house turns out to be $90K you would not exercise your right to buy the house from Professor Mao the deal is not worthwhile.
17 On 10 April 2011 you would then take one of two actions: (a) if the actual value of a house turns out to be $110K you would exercise your right to buy 1 house from Professor Mao at the cost $100 and immediately sell it for $110K giving you a profit of $10K. (b) if the actual value of a house turns out to be $90K you would not exercise your right to buy the house from Professor Mao the deal is not worthwhile.
18 Note that because you are not obliged to purchase the house, you do not lose money. Indeed, in case (a) you gain $10K while in case (b) you neither gain nor lose. Professor Mao on the other hand will not gain any money on 10 April 2011 and may lose an unlimited amount. To compensate for this imbalance, when the option is agreed on 10 April 2010 you would be expected to pay Professor Mao an amount of money to buy the "right". (The fair amount is known as the value of the option.) Question: Should Professor Mao charge you $2K, do you want to sign the option?
19 Let C denote the payoff of the option on 10/04/2011. Then { $10K if X = $110K ; C = $0 if X = $90K. Recalling the probability distribution of X P(X = $110K ) = 0.6, P(X = $90K ) = 0.4. we obtain the expected payoff EC = 0.6 $10K $0 = $6K. But $1K saved in a bank for a year will only grow to (1 + 1%) $1K = $1.01K. Therefore, the option produces the expected profit $6K $1.01 = $4.99K.
20 It is significant to compare your profit with Mr King s. Mr King invests his $100K in the house and expects to make $1K more profit than saving his money in a bank. You pay only $2K for the option and expect to make $4.99K more profit than saving your $2K in a bank. It is even more significant to observe that you only need $2K, rather than $100K, in order to get into the market.
21 However, should Professor Mao charge you $5.99K, do you want to sign for the option? If you save your $5.99K in a bank, you will have (1 + 1%) $5.99K = $6.05K which is $50 better off than EC = $6K, the expected payoff of the option. You should therefore not sign the option.
22 However, should Professor Mao charge you $5.99K, do you want to sign for the option? If you save your $5.99K in a bank, you will have (1 + 1%) $5.99K = $6.05K which is $50 better off than EC = $6K, the expected payoff of the option. You should therefore not sign the option.
23 Outline and Probability 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
24 Key question and Probability How much should the holder pay for the privilege of holding the option? In other words, how do we compute a fair price for the value of the option?
25 In the simple problem discussed above, the fair price of the option is EC 1 + r = $6K 1 + 1% = $ However, the idea can be developed to cope with more complicated distribution.
26 Example and Probability Assume that the house price will increase by 5% per half a year with probability 60% but decrease by 4% per half a year with probability 40%. Then the house price X on 2011 will have the probability distribution: Hence X (in K$) P EC = K $10.25K = $4.074K and the option value is EC 1 + r = $4.074K 1 + 1% = $4.034K
27 Example and Probability Assume that the house price will increase by 3% per quarter with probability 60% but decrease by 2% per quarter with probability 40%. Then the house price X on 10/04/2011 will have the probability distribution: X (in K$) P Hence EC = = and the option value is EC 1 + r = % = 4.682
28 The model discussed before is the well-known Cox Ross Rubinstein (CRR) binomial model. This model can be simulated easily by R.
29 > n=4 # number of periods > up=0.03 # increase percentage > dw=0.02 # decrease percentage > upno <- 0:n > p0=100 # initial house price > pt <-p0*(1+up)^upno*(1-dw)^(n-upno) #prices at expiry date > pt [1] > upprob <- 0.6 # prob of increase > prob <- dbinom(0:n,n,upprob) \#binomial distribution > prob [1] > K=100 #strike price > payoff <-numeric() > {for (i in 1:(n+1)) + if (pt[i]>k) payoff[i]=pt[i]-k else payoff[i]=0} > payoff [1] > meanpayoff <- sum(payoff*prob) > meanpayoff [1] > r =0.01 # riskfree interest rate > optionvalue <- meanpayoff/(1+r) > optionvalue [1]
30 R-simulation for the12-month CRR binomial model > n=12 > up=0.01 > dw=0.009 > upno <-0:n > p0=100 > pt <- p0*(1+up)^upno*(1-dw)^(n-upno) > upprob= 0.6 > prob<-dbinom(0:n,n,upprob) > K=100 > payoff <- numeric() > {for (i in 1:(n+1)) + if (pt[i]>k) payoff[i]=pt[i]-k else payoff[i]=0} > meanpayoff <-sum(payoff*prob) > r=0.01 > optionvalue <- meanpayoff/(1+r) > optionvalue [1]
31 R-simulation for 365-day CRR model: CRR(365, , , 100, 0.6, 100, 0.01) produces the option value $2.044K
32 However, the housing price, or more generally, an asset price is much more complicated than the binomial distributions assumed above. How might we model an asset price?
33 Linear modelling Nonlinear modelling Now suppose that at time t the underlying asset price is x(t). Let us consider a small subsequent time interval dt, during which x(t) changes to x(t) + dx(t). (We use the notation d for the small change in any quantity over this time interval when we intend to consider it as an infinitesimal change.) By definition, the intrinsic growth rate at t is dx(t)/x(t). How might we model this rate?
34 Outline and Probability Linear modelling Nonlinear modelling 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
35 Linear modelling Nonlinear modelling If, given x(t) at time t, the rate of change is deterministic, say r, then dx(t) x(t) = rdt. This gives the ordinary differential equation (ODE) Then dx(t) dt = rx(t). x(t) = x(0)e rt For example, if you invest x(0) into a bond with the risk-free interest rate r, then your return (capital plus interest) by time t is x(t).
36 Linear modelling Nonlinear modelling However the rate of change is in general not deterministic as it is often subjective to many factors and uncertainties e.g. system uncertainty, environmental disturbances. To model the uncertainty, we may decompose dx(t) x(t) = deterministic change + random change.
37 Linear modelling Nonlinear modelling The deterministic change may be modeled by µdt where µ is the average rate of change. So dx(t) x(t) = µdt + random change. How may we model the random change?
38 Linear modelling Nonlinear modelling In general, the random change is affected by many factors independently. By the well-known central limit theorem this change can be represented by a normal distribution with mean zero and and variance σ 2 dt, namely random change = N(0, σ 2 dt) = σ N(0, dt), where σ is the standard deviation of the rate of change, and N(0, dt) is a normal distribution with mean zero and and variance dt. Hence dx(t) x(t) = µdt + σn(0, dt).
39 Linear modelling Nonlinear modelling A convenient way to model N(0, dt) as a process is to use the Brownian motion B(t) (t 0) which has the following properties: B(0) = 0, db(t) = B(t + dt) B(t) is independent of B(t), db(t) follows N(0, dt).
40 Linear modelling Nonlinear modelling The stochastic model can therefore be written as dx(t) x(t) = µdt + σdb(t), or dx(t) = µx(t)dt + σx(t)db(t) which is a linear stochastic differential equation (SDE) the Nobel prize winning Black Scholes model.
41 Outline and Probability Linear modelling Nonlinear modelling 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
42 Linear modelling Nonlinear modelling If the rate of change and the standard deviation depend on x(t) at time t, the the model become nonlinear. In this case, the deterministic change may be modeled by Rdt = R(x(t), t)dt where R = r(x(t), t) is the average rate of change given x(t) at time t, while random change = N(0, V 2 dt) = V N(0, dt) = VdB(t), where V = V (x(t), t) is the standard deviation of the rate of change given x(t) at time t.
43 Linear modelling Nonlinear modelling The stochastic model can therefore be written as or dx(t) x(t) = R(x(t), t)dt + V (x(t), t)db(t), dx(t) = R(x(t), t)x(t)dt + V (x(t), t)x(t)db(t) which is a nonlinear stochastic differential equation (SDE).
44 Outline and Probability The Black Scholes world Non-linear SDE models 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
45 The Black Scholes world Non-linear SDE models The Nobel prize winning Black Scholes model dx(t) = µx(t)dt + σx(t)db(t) is also known as the geometric Brownian motion.
46 European call option The Black Scholes world Non-linear SDE models Given the asset price S(t) = S at time t, a European call option is signed with the exercise price K and the expiry date T. The value of the option is denoted by C(S, t). The payoff of the option at the expiry date is C(S, T ) = (S K ) + := max(s K, 0). The Black Scholes PDF V (S, t) t σ2 S 2 2 V (S, t) V (S, t) S 2 + rs rv (S, t) = 0. S
47 European call option The Black Scholes world Non-linear SDE models Given the asset price S(t) = S at time t, a European call option is signed with the exercise price K and the expiry date T. The value of the option is denoted by C(S, t). The payoff of the option at the expiry date is C(S, T ) = (S K ) + := max(s K, 0). The Black Scholes PDF V (S, t) t σ2 S 2 2 V (S, t) V (S, t) S 2 + rs rv (S, t) = 0. S
48 European call option The Black Scholes world Non-linear SDE models Given the asset price S(t) = S at time t, a European call option is signed with the exercise price K and the expiry date T. The value of the option is denoted by C(S, t). The payoff of the option at the expiry date is C(S, T ) = (S K ) + := max(s K, 0). The Black Scholes PDF V (S, t) t σ2 S 2 2 V (S, t) V (S, t) S 2 + rs rv (S, t) = 0. S
49 The Black Scholes world Non-linear SDE models Regardless whatever the growth rate µ the individual holder may think, the fair option value should be priced based on the following SDE dx(u) = rx(u)du + σx(u)db(u), t u T, x(t) = S, where r is the risk-free interest rate, rather than the individual SDE dy(u) = µy(u)du + σy(u)db(u), t u T, y(t) = S that the holder may think. Hence the expected payoff at the expiry date T is E(x(T ) K ) + Discounting this expected value in future gives C(S, t) = e r(t t) E[max(x(T ) K, 0)].
50 The Black Scholes world Non-linear SDE models The solution x(t ) = S exp [ ] (r 1 2 σ2 )(T t) + σ(b(t ) B(t)) gives ( log(x(t )) = log(s)+ r 1 2 σ2) (T t)+σ(b(t ) B(t)) N(ˆµ, ˆσ 2 ), where ( ˆµ = log(s) + r 1 2 σ2) (T t), ˆσ = σ T t. Hence which gives Z := log(x(t )) ˆµ ˆσ x(t ) = eˆµ+ˆσz. N(0, 1)
51 The Black Scholes world Non-linear SDE models Theorem The explicit BS formula for the value of the European call option is C(S, t) = SN(d 1 ) Ke r(t t) N(d 2 ), where N(x) is the c.p.d. of the standard normal distribution, namely N(x) = 1 x e 1 2 z2 dz, 2π while and d 1 = log(s/k ) + (r σ2 )(T t) σ T t d 2 = log(s/k ) + (r 1 2 σ2 )(T t) σ. T t
52 Outline and Probability The Black Scholes world Non-linear SDE models 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
53 Square root process The Black Scholes world Non-linear SDE models If R(x(t), t) = µ, V (x(t), t) = σ x(t), then the SDE becomes the well-known square root process dx(t) = µx(t)dt + σ x(t)db(t).
54 The Black Scholes world Non-linear SDE models Mean-reverting square root process If R(x(t), t) = then the SDE becomes α(µ x(t)), V (x(t), t) = σ, x(t) x(t) dx(t) = α(µ x(t))dt + σ x(t)db(t). This is the mean-reverting square root process.
55 Theta process and Probability The Black Scholes world Non-linear SDE models If R(x(t), t) = µ, V (x(t), t) = σ(x(t)) θ 1, then the SDE becomes dx(t) = µx(t)dt + σ(x(t)) θ db(t), which is known as the theta process.
56 Outline and Probability Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
57 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities Most of SDEs used in practice do not have explicit solutions. Monte Carlo simulations have been widely used to simulate the solutions of nonlinear SDEs. There are two main motivations for such simulations: using a Monte Carlo approach to compute the expected value of a function of the underlying underlying quantity, for example to value a bond or to find the expected payoff of an option; generating time series in order to test parameter estimation algorithms. Question: Can we trust the Monte Carlo simulations?
58 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities Most of SDEs used in practice do not have explicit solutions. Monte Carlo simulations have been widely used to simulate the solutions of nonlinear SDEs. There are two main motivations for such simulations: using a Monte Carlo approach to compute the expected value of a function of the underlying underlying quantity, for example to value a bond or to find the expected payoff of an option; generating time series in order to test parameter estimation algorithms. Question: Can we trust the Monte Carlo simulations?
59 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities Most of SDEs used in practice do not have explicit solutions. Monte Carlo simulations have been widely used to simulate the solutions of nonlinear SDEs. There are two main motivations for such simulations: using a Monte Carlo approach to compute the expected value of a function of the underlying underlying quantity, for example to value a bond or to find the expected payoff of an option; generating time series in order to test parameter estimation algorithms. Question: Can we trust the Monte Carlo simulations?
60 Outline and Probability Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
61 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities Solution of a linear SDE The linear SDE dx(t) = 2X(t)dt + X(t)dB(t), X(0) = 1 has the explicit solution x(t) = exp(1.5t + B(t)). The Monte Carlo simulation can be carried out based on the Euler-Maruyama (EM) method x(0) = 1, x(i + 1) = x(i)[ B i ], i 0, where B i = B((i + 1) ) B(i ) N(0, ).
62 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities X(t) or x(t) true soln EM soln X(t) or x(t) true soln EM soln t t X(t) or x(t) true soln EM soln X(t) or x(t) true soln EM soln t t
63 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities Example - the Black-Scholes model Consider a BS model ds(t) = 0.05S(t)dt S(t)dB(t), S(0) = 10 and a European call option with the exercise price K = at expiry time T = 1, where 0.05 is the risk-free interest rate and 0.03 is the volatility. By the well-known Black-Scholes formula on the option, we can compute the value of a European call option at time zero is C = On the other hand, we can let = 0.001, simulate 1000 paths of the SDE, compute the mean payoff at T = 1, discounting it by e 0.05, we get the estimated option value C =
64 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities To be more reliable, we can carry out such simulation, say 10 times, to get 10 estimated values: , , , , , , , , , Their mean value C = gives a better estimation for C.
65 Outline and Probability Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
66 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities Typically, let us consider the square root process ds(t) = rs(t)dt + σ S(t)dB(t), 0 t T. A numerical method, e.g. the Euler Maruyama (EM) method applied to it may break down due to negative values being supplied to the square root function. A natural fix is to replace the SDE by the equivalent, but computationally safer, problem ds(t) = rs(t)dt + σ S(t) db(t), 0 t T.
67 Discrete EM approximation Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities Given a stepsize > 0, the EM method applied to the SDE sets s 0 = S(0) and computes approximations s n S(t n ), where t n = n, according to where B n = B(t n+1 ) B(t n ). s n+1 = s n (1 + r ) + σ s n B n,
68 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities Continuous-time EM approximation s(t) := s 0 + r t 0 t s(u))du + σ s(u) db(u), 0 where the step function s(t) is defined by s(t) := s n, for t [t n, t n+1 ). Note that s(t) and s(t) coincide with the discrete solution at the gridpoints; s(t n ) = s(t n ) = s n.
69 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities The ability of the EM method to approximate the true solution is guaranteed by the ability of either s(t) or s(t) to approximate S(t) which is described by: Theorem ( lim E sup 0 0 t T s(t) S(t) 2) ( = lim E 0 sup 0 t T s(t) S(t) 2) = 0.
70 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities The ability of the EM method to approximate the true solution is guaranteed by the ability of either s(t) or s(t) to approximate S(t) which is described by: Theorem ( lim E sup 0 0 t T s(t) S(t) 2) ( = lim E 0 sup 0 t T s(t) S(t) 2) = 0.
71 Outline and Probability Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes world Non-linear SDE models 4 Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities
72 Bond and Probability Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities If S(t) models short-term interest rate dynamics, it is pertinent to consider the expected payoff ( ) β := E exp T 0 S(t)dt from a bond. A natural approximation based on the EM method is ( ) Theorem β := E exp N 1 n=0 s n lim β β = 0. 0, where N = T /.
73 Bond and Probability Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities If S(t) models short-term interest rate dynamics, it is pertinent to consider the expected payoff ( ) β := E exp T 0 S(t)dt from a bond. A natural approximation based on the EM method is ( ) Theorem β := E exp N 1 n=0 s n lim β β = 0. 0, where N = T /.
74 European call option Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities A European call option with the exercise price K at expiry time T pays S(T ) K if S(T ) > K otherwise 0. Theorem Let r be the risk-free interest rate and define C = e rt E [ (S(T ) K ) +], C = e rt E [ ( s(t ) K ) +]. Then lim C C = 0. 0
75 European call option Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities A European call option with the exercise price K at expiry time T pays S(T ) K if S(T ) > K otherwise 0. Theorem Let r be the risk-free interest rate and define C = e rt E [ (S(T ) K ) +], C = e rt E [ ( s(t ) K ) +]. Then lim C C = 0. 0
76 Up-and-out call option Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities An up-and-out call option at expiry time T pays the European value with the exercise price K if S(t) never exceeded the fixed barrier, c, and pays zero otherwise. Theorem Define V = E [ (S(T ) K ) + I {0 S(t) c, 0 t T } ], V = E [ ( s(t ) K ) + I {0 s(t) c, 0 t T } ]. Then lim V V = 0. 0
77 Up-and-out call option Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities An up-and-out call option at expiry time T pays the European value with the exercise price K if S(t) never exceeded the fixed barrier, c, and pays zero otherwise. Theorem Define V = E [ (S(T ) K ) + I {0 S(t) c, 0 t T } ], V = E [ ( s(t ) K ) + I {0 s(t) c, 0 t T } ]. Then lim V V = 0. 0
Stochastic Differential Equations in Finance and Monte Carlo Simulations
Stochastic Differential Equations in Finance and Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH China 2009 Outline Stochastic Modelling in Asset Prices 1 Stochastic
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationOption Pricing Models for European Options
Chapter 2 Option Pricing Models for European Options 2.1 Continuous-time Model: Black-Scholes Model 2.1.1 Black-Scholes Assumptions We list the assumptions that we make for most of this notes. 1. The underlying
More informationRisk Neutral Valuation
copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012 copyright 2012 Christian Fries 2 / 51 Outline Notation Differential
More informationMonte Carlo Simulations
Monte Carlo Simulations Lecture 1 December 7, 2014 Outline Monte Carlo Methods Monte Carlo methods simulate the random behavior underlying the financial models Remember: When pricing you must simulate
More informationDepartment of Mathematics. Mathematics of Financial Derivatives
Department of Mathematics MA408 Mathematics of Financial Derivatives Thursday 15th January, 2009 2pm 4pm Duration: 2 hours Attempt THREE questions MA408 Page 1 of 5 1. (a) Suppose 0 < E 1 < E 3 and E 2
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationPricing theory of financial derivatives
Pricing theory of financial derivatives One-period securities model S denotes the price process {S(t) : t = 0, 1}, where S(t) = (S 1 (t) S 2 (t) S M (t)). Here, M is the number of securities. At t = 1,
More informationNEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 MAS3904. Stochastic Financial Modelling. Time allowed: 2 hours
NEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 Stochastic Financial Modelling Time allowed: 2 hours Candidates should attempt all questions. Marks for each question
More informationMASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS.
MASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS May/June 2006 Time allowed: 2 HOURS. Examiner: Dr N.P. Byott This is a CLOSED
More informationReading: You should read Hull chapter 12 and perhaps the very first part of chapter 13.
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Asset Price Dynamics Introduction These notes give assumptions of asset price returns that are derived from the efficient markets hypothesis. Although a hypothesis,
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationNumerical Simulation of Stochastic Differential Equations: Lecture 1, Part 2. Integration For deterministic h : R R,
Numerical Simulation of Stochastic Differential Equations: Lecture, Part Des Higham Department of Mathematics University of Strathclyde Lecture, part : SDEs Ito stochastic integrals Ito SDEs Examples of
More informationFINANCIAL OPTION ANALYSIS HANDOUTS
FINANCIAL OPTION ANALYSIS HANDOUTS 1 2 FAIR PRICING There is a market for an object called S. The prevailing price today is S 0 = 100. At this price the object S can be bought or sold by anyone for any
More informationStochastic Differential equations as applied to pricing of options
Stochastic Differential equations as applied to pricing of options By Yasin LUT Supevisor:Prof. Tuomo Kauranne December 2010 Introduction Pricing an European call option Conclusion INTRODUCTION A stochastic
More informationLecture 11: Ito Calculus. Tuesday, October 23, 12
Lecture 11: Ito Calculus Continuous time models We start with the model from Chapter 3 log S j log S j 1 = µ t + p tz j Sum it over j: log S N log S 0 = NX µ t + NX p tzj j=1 j=1 Can we take the limit
More information- 1 - **** d(lns) = (µ (1/2)σ 2 )dt + σdw t
- 1 - **** These answers indicate the solutions to the 2014 exam questions. Obviously you should plot graphs where I have simply described the key features. It is important when plotting graphs to label
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationMath 416/516: Stochastic Simulation
Math 416/516: Stochastic Simulation Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 13 Haijun Li Math 416/516: Stochastic Simulation Week 13 1 / 28 Outline 1 Simulation
More informationStochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models
Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Eni Musta Università degli studi di Pisa San Miniato - 16 September 2016 Overview 1 Self-financing portfolio 2 Complete
More informationThe Black-Scholes Equation
The Black-Scholes Equation MATH 472 Financial Mathematics J. Robert Buchanan 2018 Objectives In this lesson we will: derive the Black-Scholes partial differential equation using Itô s Lemma and no-arbitrage
More informationFinancial Derivatives Section 5
Financial Derivatives Section 5 The Black and Scholes Model Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un. of
More informationKØBENHAVNS UNIVERSITET (Blok 2, 2011/2012) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours
This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 211/212) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper
More informationBasic Concepts in Mathematical Finance
Chapter 1 Basic Concepts in Mathematical Finance In this chapter, we give an overview of basic concepts in mathematical finance theory, and then explain those concepts in very simple cases, namely in the
More informationOptimal Search for Parameters in Monte Carlo Simulation for Derivative Pricing
Optimal Search for Parameters in Monte Carlo Simulation for Derivative Pricing Prof. Chuan-Ju Wang Department of Computer Science University of Taipei Joint work with Prof. Ming-Yang Kao March 28, 2014
More informationRisk Neutral Pricing Black-Scholes Formula Lecture 19. Dr. Vasily Strela (Morgan Stanley and MIT)
Risk Neutral Pricing Black-Scholes Formula Lecture 19 Dr. Vasily Strela (Morgan Stanley and MIT) Risk Neutral Valuation: Two-Horse Race Example One horse has 20% chance to win another has 80% chance $10000
More informationErrata, Mahler Study Aids for Exam 3/M, Spring 2010 HCM, 1/26/13 Page 1
Errata, Mahler Study Aids for Exam 3/M, Spring 2010 HCM, 1/26/13 Page 1 1B, p. 72: (60%)(0.39) + (40%)(0.75) = 0.534. 1D, page 131, solution to the first Exercise: 2.5 2.5 λ(t) dt = 3t 2 dt 2 2 = t 3 ]
More informationNumerical schemes for SDEs
Lecture 5 Numerical schemes for SDEs Lecture Notes by Jan Palczewski Computational Finance p. 1 A Stochastic Differential Equation (SDE) is an object of the following type dx t = a(t,x t )dt + b(t,x t
More informationFrom Discrete Time to Continuous Time Modeling
From Discrete Time to Continuous Time Modeling Prof. S. Jaimungal, Department of Statistics, University of Toronto 2004 Arrow-Debreu Securities 2004 Prof. S. Jaimungal 2 Consider a simple one-period economy
More informationComputational Finance. Computational Finance p. 1
Computational Finance Computational Finance p. 1 Outline Binomial model: option pricing and optimal investment Monte Carlo techniques for pricing of options pricing of non-standard options improving accuracy
More informationThe Binomial Lattice Model for Stocks: Introduction to Option Pricing
1/33 The Binomial Lattice Model for Stocks: Introduction to Option Pricing Professor Karl Sigman Columbia University Dept. IEOR New York City USA 2/33 Outline The Binomial Lattice Model (BLM) as a Model
More informationComputational Finance
Path Dependent Options Computational Finance School of Mathematics 2018 The Random Walk One of the main assumption of the Black-Scholes framework is that the underlying stock price follows a random walk
More informationUtility Indifference Pricing and Dynamic Programming Algorithm
Chapter 8 Utility Indifference ricing and Dynamic rogramming Algorithm In the Black-Scholes framework, we can perfectly replicate an option s payoff. However, it may not be true beyond the Black-Scholes
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
More informationEconomics has never been a science - and it is even less now than a few years ago. Paul Samuelson. Funeral by funeral, theory advances Paul Samuelson
Economics has never been a science - and it is even less now than a few years ago. Paul Samuelson Funeral by funeral, theory advances Paul Samuelson Economics is extremely useful as a form of employment
More informationImportance Sampling for Option Pricing. Steven R. Dunbar. Put Options. Monte Carlo Method. Importance. Sampling. Examples.
for for January 25, 2016 1 / 26 Outline for 1 2 3 4 2 / 26 Put Option for A put option is the right to sell an asset at an established price at a certain time. The established price is the strike price,
More information2.3 Mathematical Finance: Option pricing
CHAPTR 2. CONTINUUM MODL 8 2.3 Mathematical Finance: Option pricing Options are some of the commonest examples of derivative securities (also termed financial derivatives or simply derivatives). A uropean
More information1 Mathematics in a Pill 1.1 PROBABILITY SPACE AND RANDOM VARIABLES. A probability triple P consists of the following components:
1 Mathematics in a Pill The purpose of this chapter is to give a brief outline of the probability theory underlying the mathematics inside the book, and to introduce necessary notation and conventions
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationBlack-Scholes Option Pricing
Black-Scholes Option Pricing The pricing kernel furnishes an alternate derivation of the Black-Scholes formula for the price of a call option. Arbitrage is again the foundation for the theory. 1 Risk-Free
More informationLecture Quantitative Finance Spring Term 2015
and Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas Lecture 06: March 26, 2015 1 / 47 Remember and Previous chapters: introduction to the theory of options put-call parity fundamentals
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2017 14 Lecture 14 November 15, 2017 Derivation of the
More informationAsian Option Pricing: Monte Carlo Control Variate. A discrete arithmetic Asian call option has the payoff. S T i N N + 1
Asian Option Pricing: Monte Carlo Control Variate A discrete arithmetic Asian call option has the payoff ( 1 N N + 1 i=0 S T i N K ) + A discrete geometric Asian call option has the payoff [ N i=0 S T
More informationContinous time models and realized variance: Simulations
Continous time models and realized variance: Simulations Asger Lunde Professor Department of Economics and Business Aarhus University September 26, 2016 Continuous-time Stochastic Process: SDEs Building
More information( ) since this is the benefit of buying the asset at the strike price rather
Review of some financial models for MAT 483 Parity and Other Option Relationships The basic parity relationship for European options with the same strike price and the same time to expiration is: C( KT
More informationEconomathematics. Problem Sheet 1. Zbigniew Palmowski. Ws 2 dw s = 1 t
Economathematics Problem Sheet 1 Zbigniew Palmowski 1. Calculate Ee X where X is a gaussian random variable with mean µ and volatility σ >.. Verify that where W is a Wiener process. Ws dw s = 1 3 W t 3
More informationThe Use of Importance Sampling to Speed Up Stochastic Volatility Simulations
The Use of Importance Sampling to Speed Up Stochastic Volatility Simulations Stan Stilger June 6, 1 Fouque and Tullie use importance sampling for variance reduction in stochastic volatility simulations.
More informationFINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 FINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other
More information1 Implied Volatility from Local Volatility
Abstract We try to understand the Berestycki, Busca, and Florent () (BBF) result in the context of the work presented in Lectures and. Implied Volatility from Local Volatility. Current Plan as of March
More informationMSC FINANCIAL ENGINEERING PRICING I, AUTUMN LECTURE 6: EXTENSIONS OF BLACK AND SCHOLES RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK
MSC FINANCIAL ENGINEERING PRICING I, AUTUMN 2010-2011 LECTURE 6: EXTENSIONS OF BLACK AND SCHOLES RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK In this section we look at some easy extensions of the Black
More informationMSc Financial Engineering CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL. To be handed in by monday January 28, 2013
MSc Financial Engineering 2012-13 CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL To be handed in by monday January 28, 2013 Department EMS, Birkbeck Introduction The assignment consists of Reading
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes
More informationAdvanced Stochastic Processes.
Advanced Stochastic Processes. David Gamarnik LECTURE 16 Applications of Ito calculus to finance Lecture outline Trading strategies Black Scholes option pricing formula 16.1. Security price processes,
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 19 11/20/2013. Applications of Ito calculus to finance
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.7J Fall 213 Lecture 19 11/2/213 Applications of Ito calculus to finance Content. 1. Trading strategies 2. Black-Scholes option pricing formula 1 Security
More informationThe stochastic calculus
Gdansk A schedule of the lecture Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck (OU) process Heston model Stopping time for OU process Stochastic differential equations
More informationMonte Carlo Methods for Uncertainty Quantification
Monte Carlo Methods for Uncertainty Quantification Mike Giles Mathematical Institute, University of Oxford Contemporary Numerical Techniques Mike Giles (Oxford) Monte Carlo methods 2 1 / 24 Lecture outline
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationEuropean call option with inflation-linked strike
Mathematical Statistics Stockholm University European call option with inflation-linked strike Ola Hammarlid Research Report 2010:2 ISSN 1650-0377 Postal address: Mathematical Statistics Dept. of Mathematics
More informationThe Binomial Lattice Model for Stocks: Introduction to Option Pricing
1/27 The Binomial Lattice Model for Stocks: Introduction to Option Pricing Professor Karl Sigman Columbia University Dept. IEOR New York City USA 2/27 Outline The Binomial Lattice Model (BLM) as a Model
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
More informationLecture 17. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r.
Lecture 7 Overture to continuous models Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European option, we developed a substantial body of material, in continuous
More informationProbability in Options Pricing
Probability in Options Pricing Mark Cohen and Luke Skon Kenyon College cohenmj@kenyon.edu December 14, 2012 Mark Cohen and Luke Skon (Kenyon college) Probability Presentation December 14, 2012 1 / 16 What
More informationMÄLARDALENS HÖGSKOLA
MÄLARDALENS HÖGSKOLA A Monte-Carlo calculation for Barrier options Using Python Mwangota Lutufyo and Omotesho Latifat oyinkansola 2016-10-19 MMA707 Analytical Finance I: Lecturer: Jan Roman Division of
More informationRisk Neutral Measures
CHPTER 4 Risk Neutral Measures Our aim in this section is to show how risk neutral measures can be used to price derivative securities. The key advantage is that under a risk neutral measure the discounted
More informationDefinition Pricing Risk management Second generation barrier options. Barrier Options. Arfima Financial Solutions
Arfima Financial Solutions Contents Definition 1 Definition 2 3 4 Contenido Definition 1 Definition 2 3 4 Definition Definition: A barrier option is an option on the underlying asset that is activated
More informationM5MF6. Advanced Methods in Derivatives Pricing
Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Simulating Stochastic Differential Equations Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationAnalysing multi-level Monte Carlo for options with non-globally Lipschitz payoff
Finance Stoch 2009 13: 403 413 DOI 10.1007/s00780-009-0092-1 Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff Michael B. Giles Desmond J. Higham Xuerong Mao Received: 1
More informationFINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 FINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other
More informationContinuous Time Finance. Tomas Björk
Continuous Time Finance Tomas Björk 1 II Stochastic Calculus Tomas Björk 2 Typical Setup Take as given the market price process, S(t), of some underlying asset. S(t) = price, at t, per unit of underlying
More informationAttempt QUESTIONS 1 and 2, and THREE other questions. Do not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 FINANCIAL MATHEMATICS MTHE6026A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. Notes are
More informationCredit Risk : Firm Value Model
Credit Risk : Firm Value Model Prof. Dr. Svetlozar Rachev Institute for Statistics and Mathematical Economics University of Karlsruhe and Karlsruhe Institute of Technology (KIT) Prof. Dr. Svetlozar Rachev
More informationA Moment Matching Approach To The Valuation Of A Volume Weighted Average Price Option
A Moment Matching Approach To The Valuation Of A Volume Weighted Average Price Option Antony Stace Department of Mathematics and MASCOS University of Queensland 15th October 2004 AUSTRALIAN RESEARCH COUNCIL
More information1 Geometric Brownian motion
Copyright c 05 by Karl Sigman Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM is
More information1 The continuous time limit
Derivative Securities, Courant Institute, Fall 2008 http://www.math.nyu.edu/faculty/goodman/teaching/derivsec08/index.html Jonathan Goodman and Keith Lewis Supplementary notes and comments, Section 3 1
More informationHeston Model Version 1.0.9
Heston Model Version 1.0.9 1 Introduction This plug-in implements the Heston model. Once installed the plug-in offers the possibility of using two new processes, the Heston process and the Heston time
More information23 Stochastic Ordinary Differential Equations with Examples from Finance
23 Stochastic Ordinary Differential Equations with Examples from Finance Scraping Financial Data from the Web The MATLAB/Octave yahoo function below returns daily open, high, low, close, and adjusted close
More information2.1 Mathematical Basis: Risk-Neutral Pricing
Chapter Monte-Carlo Simulation.1 Mathematical Basis: Risk-Neutral Pricing Suppose that F T is the payoff at T for a European-type derivative f. Then the price at times t before T is given by f t = e r(t
More informationHow Much Should You Pay For a Financial Derivative?
City University of New York (CUNY) CUNY Academic Works Publications and Research New York City College of Technology Winter 2-26-2016 How Much Should You Pay For a Financial Derivative? Boyan Kostadinov
More informationMultilevel quasi-monte Carlo path simulation
Multilevel quasi-monte Carlo path simulation Michael B. Giles and Ben J. Waterhouse Lluís Antoni Jiménez Rugama January 22, 2014 Index 1 Introduction to MLMC Stochastic model Multilevel Monte Carlo Milstein
More informationCourse MFE/3F Practice Exam 2 Solutions
Course MFE/3F Practice Exam Solutions The chapter references below refer to the chapters of the ActuarialBrew.com Study Manual. Solution 1 A Chapter 16, Black-Scholes Equation The expressions for the value
More informationMODELLING 1-MONTH EURIBOR INTEREST RATE BY USING DIFFERENTIAL EQUATIONS WITH UNCERTAINTY
Applied Mathematical and Computational Sciences Volume 7, Issue 3, 015, Pages 37-50 015 Mili Publications MODELLING 1-MONTH EURIBOR INTEREST RATE BY USING DIFFERENTIAL EQUATIONS WITH UNCERTAINTY J. C.
More informationMA4257: Financial Mathematics II. Min Dai Dept of Math, National University of Singapore, Singapore
MA4257: Financial Mathematics II Min Dai Dept of Math, National University of Singapore, Singapore 2 Contents 1 Preliminary 1 1.1 Basic Financial Derivatives: Forward contracts and Options. 1 1.1.1 Forward
More informationMAFS Computational Methods for Pricing Structured Products
MAFS550 - Computational Methods for Pricing Structured Products Solution to Homework Two Course instructor: Prof YK Kwok 1 Expand f(x 0 ) and f(x 0 x) at x 0 into Taylor series, where f(x 0 ) = f(x 0 )
More informationMath489/889 Stochastic Processes and Advanced Mathematical Finance Solutions to Practice Problems
Math489/889 Stochastic Processes and Advanced Mathematical Finance Solutions to Practice Problems Steve Dunbar No Due Date: Practice Only. Find the mode (the value of the independent variable with the
More informationDynamic Hedging and PDE Valuation
Dynamic Hedging and PDE Valuation Dynamic Hedging and PDE Valuation 1/ 36 Introduction Asset prices are modeled as following di usion processes, permitting the possibility of continuous trading. This environment
More informationA new approach for scenario generation in risk management
A new approach for scenario generation in risk management Josef Teichmann TU Wien Vienna, March 2009 Scenario generators Scenarios of risk factors are needed for the daily risk analysis (1D and 10D ahead)
More informationContinuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a
Continuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a variable depend only on the present, and not the history
More information1. In this exercise, we can easily employ the equations (13.66) (13.70), (13.79) (13.80) and
CHAPTER 13 Solutions Exercise 1 1. In this exercise, we can easily employ the equations (13.66) (13.70), (13.79) (13.80) and (13.82) (13.86). Also, remember that BDT model will yield a recombining binomial
More informationLecture 8: The Black-Scholes theory
Lecture 8: The Black-Scholes theory Dr. Roman V Belavkin MSO4112 Contents 1 Geometric Brownian motion 1 2 The Black-Scholes pricing 2 3 The Black-Scholes equation 3 References 5 1 Geometric Brownian motion
More informationSTOCHASTIC INTEGRALS
Stat 391/FinMath 346 Lecture 8 STOCHASTIC INTEGRALS X t = CONTINUOUS PROCESS θ t = PORTFOLIO: #X t HELD AT t { St : STOCK PRICE M t : MG W t : BROWNIAN MOTION DISCRETE TIME: = t < t 1
More information25857 Interest Rate Modelling
25857 UTS Business School University of Technology Sydney Chapter 20. Change of Numeraire May 15, 2014 1/36 Chapter 20. Change of Numeraire 1 The Radon-Nikodym Derivative 2 Option Pricing under Stochastic
More informationMath 623 (IOE 623), Winter 2008: Final exam
Math 623 (IOE 623), Winter 2008: Final exam Name: Student ID: This is a closed book exam. You may bring up to ten one sided A4 pages of notes to the exam. You may also use a calculator but not its memory
More informationSolving the Black-Scholes Equation
Solving the Black-Scholes Equation An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Initial Value Problem for the European Call The main objective of this lesson is solving
More informationNumerical Simulation of Stochastic Differential Equations: Lecture 1, Part 1. Overview of Lecture 1, Part 1: Background Mater.
Numerical Simulation of Stochastic Differential Equations: Lecture, Part Des Higham Department of Mathematics University of Strathclyde Course Aim: Give an accessible intro. to SDEs and their numerical
More informationControl Improvement for Jump-Diffusion Processes with Applications to Finance
Control Improvement for Jump-Diffusion Processes with Applications to Finance Nicole Bäuerle joint work with Ulrich Rieder Toronto, June 2010 Outline Motivation: MDPs Controlled Jump-Diffusion Processes
More informationMAS3904/MAS8904 Stochastic Financial Modelling
MAS3904/MAS8904 Stochastic Financial Modelling Dr Andrew (Andy) Golightly a.golightly@ncl.ac.uk Semester 1, 2018/19 Administrative Arrangements Lectures on Tuesdays at 14:00 (PERCY G13) and Thursdays at
More informationLecture 18. More on option pricing. Lecture 18 1 / 21
Lecture 18 More on option pricing Lecture 18 1 / 21 Introduction In this lecture we will see more applications of option pricing theory. Lecture 18 2 / 21 Greeks (1) The price f of a derivative depends
More informationSYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives
SYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October
More information