# The Heston Model. Hui Gong, UCL ucahgon/ May 6, 2014

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1 Hui Gong, UCL ucahgon/ May 6, 2014

2 Generalized SV models Vanilla Call Option via Heston Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA

3 1. Why the Black-Scholes model is not popular in the industry? 2. What is the stochastic volatility models? Stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed.

4 Generalized SV models Vanilla Call Option via Heston A general expression for non-dividend stock with stochastic volatility is as below: with ds t = µ t S t dt + v t S t dw 1 t, (1) dv t = α(s t, v t, t)dt + β(s t, v t, t)dw 2 t, (2) dw 1 t dw 2 t = ρdt, where S t denotes the stock price and v t denotes its variance. Examples: Heston model SABR volatility model GARCH model 3/2 model Chen model

5 Generalized SV models Vanilla Call Option via Heston The Heston model is a typical model which takes α(s t, v t, t) = κ(θ v t ) and β(s t, v t, t) = σ v t, i.e. ds t = µs t dt + v t S t dw 1,t, (3) dv t = κ(θ v t )dt + σ v t dw 2,t, (4) with dw 1,t dw 2,t = ρdt, (5) where θ is the long term mean of v t, κ denotes the speed of reversion and σ is the volatility of volatility. The instantaneous variance v t here is a CIR process (square root process).

6 Generalized SV models Vanilla Call Option via Heston Let x t = ln S t, the risk-neutral dynamics of Heston model is ( dx t = r 1 ) 2 v t dt + v t dw1,t, (6) with dv t = κ (θ v t )dt + σ v t dw 2,t, (7) dw 1,tdW 2,t = ρdt. (8) where κ = κ + λ and θ = κθ κ+λ. Using these dynamics, the probability of the call option expires in-the-money, conditional on the log of the stock price, can be interpreted as risk-adjusted or risk-neutral probabilities. Hence, F j (x, v, T ; ln K) = Pr(x(T ) ln K x t = x, v t = v).

7 Generalized SV models Vanilla Call Option via Heston The price of vanilla call option is: C(S, v, t) = SF 1 e r(t t) KF 2, (9) where F 1 and F 2 should satisfy the PDE (for j = 1, 2) 1 2 v 2 F j x 2 + ρσv 2 F j x v σ2 v 2 F j v 2 +(r + u j v) F j x + (a j b j v) F j v + F j t = 0. (10) The parameter in Equation (10) is as follows u 1 = 1 2, u 2 = 1 2, a = κθ, b 1 = κ+λ ρσ, b 2 = κ+λ.

8 Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA The simulated variance can be inspected to check whether it is negative (v < 0). In this case, the variance can be set to zero (v = 0), or its sign can be inverted so that v becomes v. Alternatively, the variance process can be modified in the same way as the stock process, by defining a process for natural log variances by using Itô s lemma d ln v t = 1 ( κ (θ v t ) 1 ) v t 2 σ2 dt + σ 1 dw2,t. (11) vt

9 Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA The Heston model can be discretized as following ( ln S t+ t = ln S t + r 1 ) 2 v t t + v t tɛs,t+1, ln v t+ t = ln v t + 1 v t ( κ (θ v t ) 1 2 σ2 ) t + σ 1 vt tɛv,t+1. Shocks to the volatility, ɛ v,t+1, are correlated with the shocks to the stock price process, ɛ S,t+1. This correlation is denoted ρ, so that ρ = Corr(ɛ S,t+1, ɛ v,t+1 ) and the relationship between the shocks can be written as ɛ v,t+1 = ρɛ S,t ρ 2 ɛ t+1 where ɛ t+1 are independently with ɛ S,t+1.

10 Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA Figure: Heston (1993) Call Price by Monte Carlo

11 Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA Figure: VBA code for Heston (1993) Call Price by Monte Carlo

12 Use the Closed-Form Approach to implement Heston Call & Put.

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