© K.Cuthbertson, D. Nitzsche FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture Asset Price Dynamics Version 1/9/2001

© K.Cuthbertson, D. Nitzsche Stochastic Processes: Weiner, Ito, GBM, Black-Scholes PDE RNV and Monte Carlo Simulation Finite Difference Methods TOPICS

© K.Cuthbertson, D. Nitzsche Stochastic Processes: Weiner, Ito, GBM Black-Scholes, PDE

© K.Cuthbertson, D. Nitzsche Weiner Process [17.3]  z =   t [17.4a] Expected ValueE(  z) = 0 [17.4b] Variancevar(  z) = E(  z) 2 =  t [17.4c] Standard Deviationstd(  z) =  t Generalised Weiner Process  x = a  t + b  z = a  t + b   t E(  x) = a  t var(  x) = b 2  t Ito Process dx = a(x,t) dt + b(x,t) dz Weiner Process

© K.Cuthbertson, D. Nitzsche Geometric Brownian Motion [17.16b]dS/S =  dt +  dz GBM is Ito process with a =  S and b =  S Ito’s Lemma If S follows an Ito process then the stochastic differential equation (SDE) for any function  (S,t)~ option premium) Substitute for dS from [17.20] [17.22] GBM and Ito’s Lemma

© K.Cuthbertson, D. Nitzsche Ito’s eqn for f(S,t) a =  S, b=  S Futures price: F = Se r(T-t) The stock price follows a GBM: dS = (  S)dt + (  S) dz Substituting in Ito’s eqn above: dF = [ e r(T-t) (  S) – r S e r(T-t) ] dt + e r(T-t) (  S) dz Substituting F = Se r(T-t) dF = (  - r)F dt +  F dz which is a GBM for dF/F with drift rate (  - r) and variance rate . Table 17.1 : SDE for Future’s Prices using Ito’s Lemma

© K.Cuthbertson, D. Nitzsche Replication portfolio of stocks and bonds mimics the payoff of derivative security thus offsetting any uncertainty dz inherent in the derivative security, taken in isolation. The resulting equation (see appendix 17.2) for the price of the derivative security is deterministic (ie. non-stochastic) and is known as the Black-Scholes PDE [17.48a] This PDE can be solved by standard methods, to give the B-S closed form solution for the derivatives price. Black-Scholes PDE

© K.Cuthbertson, D. Nitzsche Value of the forward contract:see chapter 2 [17.55]f = S - Ke -r(T-t) Substituting Black-Scholes PDE and using [17.55] The LHS of the Black-Scholes equation becomes : [17.57]r[-Ke -r(T-t) + S] = r f Hence f satisfies the PDE Does a Forward Contract Obey B-S, PDE ?

© K.Cuthbertson, D. Nitzsche  Assume dS/S follows a GBM with drift rate  and variance rate,  2  Use Ito’s lemma to obtain the stochastic process for d( ln S ) d(ln S) = dt +  dzwhere =  -  2 /2. Because dz is N(0,1) then the distribution of ln(S) is normal (and S is lognormal) with ln(S T /S o ) ~ N( T,     Statistical distribution theory then indicates that the level of S T has mean and variance ES T = S o e  T var(S T ) = Table 17.2 : From GBM for dS/S to the properties of S

© K.Cuthbertson, D. Nitzsche Risk Neutral Valuation, RNV and Monte Carlo Simulation, MCS

© K.Cuthbertson, D. Nitzsche When pricing an option it is valid to use q as the probability of an ‘up’ move ~this is equivalent to the stock price growing at the risk free rate r However, the resulting value for the option premium, is valid in the real world. Risk Neutral Valuation, RNV

© K.Cuthbertson, D. Nitzsche Step 1: Assume the expected return of the underlying asset (eg. stock) equals the risk free rate. (For example, for the BOPM this involves using q as the probability of an ‘up’ move, which is consistent with S growing at the risk free rate r) Step 2: Calculate the expected payoff from the derivative at maturity Step 3: Discount the expected payoff at the risk free rate to obtain the price of the derivative Risk Neutral Valuation, RNV

© K.Cuthbertson, D. Nitzsche Under RNV the call premium is : [17.74]C = e -rT E* [max(S T - K, 0)] Generate S [17.77a]S t = [1 +   t +  t  t  S t-1 [17.77b]S t = S t-1 exp[(  -  2 /2)  t +  t  t  [17.80]Payoff-C (1) = max {0, - 100} = After m-runs: [17.81] Hedge Parameters MCS for Option Premia (Excel T17.4+Gauss)

© K.Cuthbertson, D. Nitzsche dS = ( r-  ) S dt + S (  V ) dz s dV = b (V m - V) dt + (  V  ) dz v V m = long run value of volatility of stock return dz s dz v =  dt b,  and  are parameters to be estimated Stochastic Volatility and MCS

© K.Cuthbertson, D. Nitzsche Antithetic Variables Each time we draw a value for  we also use -  Both are used to generate new values for the stock price. We therefore get “two stock prices for the ‘price’ of one random draw” This technique can be applied to any symmetric distribution So in any run we have two option payoffs using +  and using -  We then take the average of the two payoffs as the payoff for that simulation Variance Reduction Methods

© K.Cuthbertson, D. Nitzsche Control Variate Method To calculate value of a ‘complex’ option-A,  A MCS ‘Simple’ option-B whose value by B-S =  B BS Now value option-B using MCS giving  B MCS (its value would be close to but not equal to  B BS (because of MCS sampling error) Control variate technique adjusts  A MCS depending on how big the error is in the MCS valuation of option-B The ‘new and improved’ estimate for option-A is Variance Reduction Methods

© K.Cuthbertson, D. Nitzsche Finite Difference Methods

© K.Cuthbertson, D. Nitzsche Approximate the continuous time B-S, PDE using numerical derivatives on a ‘grid’ Impose boundary conditions Put Option At S=0 then f = K e -rt and at S>>K then then f = 0 Solve the PDE using numerical methods Finite Difference Methods

© K.Cuthbertson, D. Nitzsche Figure 17.3 :Approximations for  f /  S S f SS SS Central difference Backward difference Forward difference Derivative required for this point f i+1 fifi f i-1

© K.Cuthbertson, D. Nitzsche Figure 17.2 :Use of grid points Differential with respect to S (index for S is i ) Value of option, f(S,t) (index for t is k) Differential with respect to time (Note: as k increases t decreases) f i k+1   f i k f k i+1  f i k  f k i-1  Central difference forward difference backward difference  f k i+1  f i k  f k i-1 f i k+1 

© K.Cuthbertson, D. Nitzsche Figure 17.1 : Finite difference grid T tt time, t (index k = T - k  t ) SS Current Stock Price: S = 4(DS). Hence value of f 4 0 will be solution for the option premium. f 3 6 is determined by the values of f at points A, B and C 0 Stock Price (index i) 4 A B C

© K.Cuthbertson, D. Nitzsche One of the nodes on the left vertical axis will coincide with the current stock price and the solved value for f at this same node will be the option premia S = i(  S), t = T-k(  t) at k = 0, then t = T (ie. expiry) and as k increases, real time t decreases. [17.87] Finite Difference Methods

© K.Cuthbertson, D. Nitzsche From [17.87] note that we can calculate the value of f i k+1 once we know the values of f at time k for the three nodes i, i-1 and i+1 (figure 17.1). Solve for f i k+1 by working backwards through the grid (once we have the terminal conditions) ~ explicit finite difference method. American Compare f i k+1 with the payoff to early exercise K-S = K - i  S at each node, and take the max. value. Finite Difference Methods

© K.Cuthbertson, D. Nitzsche LECTURE ENDS HERE