Version 192001 FINANCIAL ENGINEERING DERIVATIVES AND RISK MANAGEMENT

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Version 1/9/2001 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and

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

TOPICS Stochastic Processes: Weiner, Ito, GBM, Black-Scholes PDE RNV and Monte Carlo Simulation Finite

TOPICS Stochastic Processes: Weiner, Ito, GBM, Black-Scholes PDE RNV and Monte Carlo Simulation Finite Difference Methods

Stochastic Processes: Weiner, Ito, GBM Black-Scholes, PDE

Stochastic Processes: Weiner, Ito, GBM Black-Scholes, PDE

Weiner Process [17. 3] z = t [17. 4 a] Expected Value [17. 4

Weiner Process [17. 3] z = t [17. 4 a] Expected Value [17. 4 b] Variance [17. 4 c] Standard Deviation E( z) = 0 var( z) = E( z)2 = t std( 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

GBM and Ito’s Lemma Geometric Brownian Motion [17. 16 b] d. S/S = dt

GBM and Ito’s Lemma Geometric Brownian Motion [17. 16 b] d. S/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 d. S from [17. 20] [17. 22]

Table 17. 1 : SDE for Future’s Prices using Ito’s Lemma Ito’s eqn 17.

Table 17. 1 : SDE for Future’s Prices using Ito’s Lemma Ito’s eqn 17. 22 for f(S, t) a = S, b= S Futures price: F = Ser(T-t) The stock price follows a GBM: d. S = ( S)dt + ( S) dz Substituting in Ito’s eqn above: d. F = [ er(T-t) ( S) – r S er(T-t) ] dt + er(T-t) ( S) dz Substituting F = Ser(T-t) d. F = ( - r)F dt + F dz which is a GBM for d. F/F with drift rate ( - r) and variance rate .

Black-Scholes PDE Replication portfolio of stocks and bonds mimics the payoff of derivative security

Black-Scholes PDE 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. 48 a] This PDE can be solved by standard methods, to give the B-S closed form solution for the derivatives price.

Does a Forward Contract Obey B-S, PDE ? Value of the forward contract: see

Does a Forward Contract Obey B-S, PDE ? 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

Table 17. 2 : From GBM for d. S/S to the properties of S

Table 17. 2 : From GBM for d. S/S to the properties of S Assume d. S/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 + dz where = - 2/2. Because dz is N(0, 1) then the distribution of ln(S) is normal (and S is lognormal) with ln(ST/So) ~ N( T, 2 T) Statistical distribution theory then indicates that the level of ST has mean and variance EST = So e T var(ST) =

Risk Neutral Valuation, RNV and Monte Carlo Simulation, MCS

Risk Neutral Valuation, RNV and Monte Carlo Simulation, MCS

Risk Neutral Valuation, RNV When pricing an option it is valid to use q

Risk Neutral Valuation, RNV 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 Step 1: Assume the expected return of the underlying asset

Risk Neutral Valuation, RNV 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

MCS for Option Premia (Excel T 17. 4+Gauss) Under RNV the call premium is

MCS for Option Premia (Excel T 17. 4+Gauss) Under RNV the call premium is : [17. 74] C = e-r. T E* [max(ST - K, 0)] Generate S [17. 77 a] St = [1 + t t ]St-1 [17. 77 b] St = St-1 exp[( - 2/2) t + t t] [17. 80] Payoff-C(1) = max {0, - 100} = After m-runs: [17. 81] Hedge Parameters 10. 12

Stochastic Volatility and MCS d. S = ( r- ) S dt + S

Stochastic Volatility and MCS d. S = ( r- ) S dt + S ( V ) dzs d. V = b (Vm - V) dt + ( V ) dzv Vm = long run value of volatility of stock return dzs dzv = dt b , and are parameters to be estimated

Variance Reduction Methods Antithetic Variables Each time we draw a value for we also

Variance Reduction Methods 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 Control Variate Method To calculate value of a ‘complex’ option-A, AMCS

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

Finite Difference Methods

Finite Difference Methods

Finite Difference Methods Approximate the continuous time B-S, PDE using numerical derivatives on a

Finite Difference Methods 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 f = 0 Solve the PDE using numerical methods

Figure 17. 3 : Approximations for f / S f fi+1 Central difference Forward

Figure 17. 3 : Approximations for f / S f fi+1 Central difference Forward difference Backward difference fi Derivative required for this point fi-1 S S © K. Cuthbertson, D. Nitzsche S

Figure 17. 2 : Use of grid points Differential with respect to S (index

Figure 17. 2 : Use of grid points Differential with respect to S (index for S is i ) fki+1 Central difference f ik fki-1 Value of option, f(S, t) (index for t is k) forward difference backward difference Differential with respect to time (Note: as k increases t decreases) fik+1 f ik © K. Cuthbertson, D. Nitzsche fki+1 fik+1 f ik fki-1

Figure 17. 1 : Finite difference grid Stock Price (index i) Current Stock Price:

Figure 17. 1 : Finite difference grid Stock Price (index i) Current Stock Price: S = 4(DS). Hence value of f 40 will be solution for the option premium. f 36 is determined by the values of f at points A, B and C 5 A 4 B 3 C 2 S 1 0 1 t 2 3 4 5 6 © K. Cuthbertson, D. Nitzsche T time, t (index k = T - k Dt )

Finite Difference Methods One of the nodes on the left vertical axis will coincide

Finite Difference Methods 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 From [17. 87] note that we can calculate the value of

Finite Difference Methods From [17. 87] note that we can calculate the value of fik+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 fik+1 by working backwards through the grid (once we have the terminal conditions) ~ explicit finite difference method. American Compare fik+1 with the payoff to early exercise K-S = K - i S at each node, and take the max. value.

LECTURE ENDS HERE

LECTURE ENDS HERE