Stochastic Modelling in Finance

in Finance Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH April 2010

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

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 2011. Which choice should Mr King take?

Assume the annual interest rate r = 1% and let X denote the price of the house on 10 April 2011. 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 2011. (ii) P(X = $110K ) = 0.6 and P(X = $90K ) = 0.4. In this case, EX = 0.6 $110K + 0.4 $90K = $102K, which is $1K better than the return of the saving account.

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.

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.

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.

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?

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 + 0.4 $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.

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.

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.

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?

In the simple problem discussed above, the fair price of the option is EC 1 + r = $6K 1 + 1% = $5940.59 However, the idea can be developed to cope with more complicated distribution.

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$) 92.16 100.80 110.25 P 0.16 0.48 0.36 EC = 0.48 0.80K + 0.36 $10.25K = $4.074K and the option value is EC 1 + r = $4.074K 1 + 1% = $4.034K

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$) 92.237 96.943 101.889 107.087 112.551 P 0.0256 0.1536 0.3456 0.3456 0.1296 Hence EC = 0.3456 1.889+0.3456 7.087+0.1296 12.551 = 4.729 and the option value is EC 1 + r = 4.729 1 + 1% = 4.682

The model discussed before is the well-known Cox Ross Rubinstein (CRR) binomial model. This model can be simulated easily by R.

> 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] 92.23682 96.94278 101.88884 107.08725 112.55088 > upprob <- 0.6 # prob of increase > prob <- dbinom(0:n,n,upprob) \#binomial distribution > prob [1] 0.0256 0.1536 0.3456 0.3456 0.1296 > 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] 0.000000 0.000000 1.888836 7.087246 12.550881 > meanpayoff <- sum(payoff*prob) > meanpayoff [1] 4.728728 > r =0.01 # riskfree interest rate > optionvalue <- meanpayoff/(1+r) > optionvalue [1] 4.681909

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] 3.238385

R-simulation for 365-day CRR model: CRR(365, 0.00028, 0.00028, 100, 0.6, 100, 0.01) produces the option value $2.044K

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?

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?

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

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).

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.

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?

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).

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).

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.

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.

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).

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

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.

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 + 1 2 σ2 S 2 2 V (S, t) V (S, t) S 2 + rs rv (S, t) = 0. S

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)].

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)

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 + 1 2 σ2 )(T t) σ T t d 2 = log(s/k ) + (r 1 2 σ2 )(T t) σ. T t

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).

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.

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.

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

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?

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)[1 + 2 + B i ], i 0, where B i = B((i + 1) ) B(i ) N(0, ).

Why Monte Carlo simulations Test problem EM method for nonlinear SDEs EM method for financial quantities X(t) or x(t) 2 4 6 8 10 12 true soln EM soln X(t) or x(t) 0 20 60 100 140 true soln EM soln 0.0 0.5 1.0 1.5 2.0 t 0.0 0.5 1.0 1.5 2.0 t X(t) or x(t) 0 5 10 15 20 25 true soln EM soln X(t) or x(t) 2 4 6 8 10 true soln EM soln 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 t t

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 + 0.03S(t)dB(t), S(0) = 10 and a European call option with the exercise price K = 10.05 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 = 0.4487318. 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 = 0.4454196

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: 0.4454196, 0.4611569, 0.4512847, 0.4490462, 0.4294038, 0.4618921, 0.4556195, 0.4559547, 0.4399189, 0.4489594. Their mean value C = 0.4498656 gives a better estimation for C.

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.

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,

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.

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.

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 /.

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

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