Locally risk-minimizing vs. -hedging in stochastic volatility models University of St. Andrews School of Economics and Finance August 29, 2007 joint work with R. Poulsen ( Kopenhagen )and K.R.Schenk-Hoppe (Leeds)
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Samuelson model for the dynamics of a stock in continuous time : i.e. S(t) = S(0) exp( µt + σw(t)) log(s(t)) = log(s(0)) + µt }{{} = deterministic effect = random effect + σw(t) }{{} The random effect is decomposed into many similar tiny little random effects which accumulate homogeneous in time and are independent of each other. Central limit theorem W(t) is Brownian motion
With µ := µ + 1 2 σ2 S(t) = S(0) exp ((µ 12 ) ) σ2 t + σw(t) In addition to the stock there is a money market account : B(t) = B(0) exp(rt) S(t) and B(t) are the primary tradeable assets. Now consider a European type option with payoff h(s T ), a so called derivative. What is the price of this financial product? How can the seller hedge against the risk?
Black, Scholes, Merton approach : assume price of option is C(t, S(t)) construct a riskless portfolio, which consists of buying the option and trading in the stock dc ds = C t dt + C S ds + 1 2 C SSdSdS ds = (C t + 12 ) S2 σ 2 C SS dt + (C s ) ds Substitution of ds = S (µdt + σdw) gives { dc ds = C t + 1 } 2 S2 σ 2 C SS + (C s ) Sµ dt+(c s ) SσdW riskless = C S
Substitution gives dc ds = (C t + 12 S2 σ 2 C SS ) dt (1) C S riskless must evolve at the same rate as the money market account dc ds = (C C S S)rdt (2) Combining (1) and (2) leads to the Black-Scholes PDE C r C S S r C t 1 2 S2 σ 2 C S S = 0 C(T, S(T)) = h(s(t)) Black, Scholes, Merton solved this PDE for the case where h(s T ) = max(s T K, 0)
To keep in mind : in the BS model it is possible to hedge ( within the model ) against all risk associated with buying or selling a financial derivative every contingent claim can be perfectly replicated by a self financing trading strategy, i.e. C(t, S(t)) = ϕ 0 (t)b(t) + ϕ 1 (t)s(t) =: V ϕ (t) where ϕ 1 (t) = C S is the so called Delta of the option and dv ϕ (t) = ϕ 0 (t)db(t) + ϕ 1 (t)ds(t) in the BS model contingent claims are therefore redundant! hedging based on the Delta of an option is called Delta-hedging
The BS-model has several problems, the worst is that it proves itself to be wrong : Curse of implied volatility : Market prices of European call options with different strike s imply different values for σ, which ought to be constant in BS model. More precisely : solve Ch(S BS T )(σ) = Cobs. h(s T ) The solution is called implied volatility σ impl. σ impl. in general depends on the current value of the stock, on time to maturity and on the payoff function h. It is in no way constant as the BS-model suggests.
Volatility smile :
The curse of implied volatility can be partly cured by using so called local volatility models where the local volatility is calibrated by the implied volatilities from the BS-model : ds(t) = S(t)(µdt + σ(t, S(t))dW(t)) But : Some people say, using local volatility models is like substituting the wrong numbers in the wrong formula in order to get the right result. An alternative to BS and local volatility models are stochastic volatility models.
returning to Samuelson consideration log(s(t)) = log(s(0)) + µdt + random(t) assume that random(t) is still built up by many little random effects but that they no longer accumulate homogeneous in time, while the accumulation rate is controlled by other random effects alternatively, think of random effects having different magnitudes, compare heteroskedastic time series models dw(t) [ {}}{ ds(t) = S(t) µdt + S(t)γ f (V(t)) 1 ρ 2 dw 1 (t) + ρdw 2 (t)] dv(t) = V(t)(β(V(t))dt + g(v(t))dw 2 (t))
Authors & year Specification Remarks Hull-White f (v) = v, Local variance: 1987 β(v) = 0, Geometric Brownian motion. g(v) = σ, Options priced by mixing. ρ = 0, γ = 0 Wiggins f (v) = e v/2, Local volatility: 1987 β(v) = κ(θ v)/v, Ornstein-Uhlenbeck g(v) = σ, in logarithms. ρ = 0, γ = 0 Stein-Stein f (v) = v, Local volatility: 1991 β(v) = κ(θ v)/v, Reflected Ornstein-Uhlenbeck. g(v) = σ/v, ρ = 0, γ = 0 Heston f (v) = v, Local variance: 1993 β(v) = κ(θ v)/v, CIR process. First model with g(v) = σ/ v, correlation. Options priced by ρ [ 1, 1], γ = 0 inversion of characteristic function. Romano-Touzi f (v) = v, Extension of mixing to 1997 β and g are free, correlation. ρ [ 1, 1], γ = 0 SABR f (v) = v Level dependence in volatility. 2002 β(v) = 0, Options priced perturbation technique Hagan et al. g(v) = σ, ρ [ 1, 1], γ [ 1, 0] Table: Specification of stochastic volatility models.
The first ones to consider such models were Hull and White The Pricing of Options on Assets with Stochastic Volatilities. The Journal of Finance, Vol. XLII No. 2, (1987). HW model assumes volatility to be a geometric BM, such as the stock price in BS model. Statistical data show it isn t like this. Nowadays most popular is the Heston model : ( ds(t) = S(t) µdt + ]) V(t)[ 1 ρ 2 dw 1 (t) + ρdw 2 (t) dv(t) = κ(θ V(t)) dt + σ V(t)dW 2 (t)
Why is it popular? For European calls there exist closed pricing formulas. Heston, S.L.; A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options ; Review of Financial Studies, Vol.6, Issue 2 (1993) It can be calibrated effectively to market prices. It produces smiles! are not complete : There are contingent claims, which can not be hedged by a self financing investment strategy in stock and money market account.
Option price C = C(t, S, V). Itô s formula dc =...dt + C S ds + C V dv The same approach as Black-Scholes leads to dc ds =...dt + (C S )ds + C V dv }{{} where contains non hedgeable random sources. Main problem caused by incompleteness The No-Arbitrage Principle alone does not determine a unique price for a derivative
What to do? Pricing by replication still appears to be a good concept, but in general derivatives are not replicable by self financing investment in the primary assets. One can relax the condition of self financing, allowing non self financing strategies. These strategies come with a cost process Cost ϕ (t) = V ϕ (t) t with V ϕ (t) = ϕ 0 (t) B(t) + ϕ 1 (t) S(t) non self financing hedges always exist : 0 ϕ 0 (s)db(s) ϕ 1 (s)ds(s), ϕ 0 (s) = 0, ϕ 1 (s) = 0 for all s [0, T) and ϕ 0 (T) = h(s T ), ϕ 1 (T) = 0.
Obviously the non self financing hedge should be chosen reasonably : classically traders do the following : compute the Delta for the derivative using the corresponding BS-type formula follow Delta-hedging formula, i.e. investing C S S in the stock use market prices to adjust the portfolio from time to time in the money market In order to comply with the last point traders often have to put new money into the market, this is called bleeding. It wouldn t be necessary if the BS model would indeed be the true model, in this case the -hedge would be self financing.
Is the -hedging the best one can do? Leaving aside the variances produced by the drift we obtain for the conditional variance of the cost process var t (dcost) = ((C S ) 2 S 2 S 2γ f 2 (V) + C 2 V V 2 g 2 (V) +2(C S )C V SS γ f (V)Vg(V)ρ)dt. This is a quadratic function of, minimized by min = C S + ρ g(v)v f (V)S 1+γ C V, If the agent is interested to minimize the variance in the cost process of his non self financing hedge, i.e. the risk of future bleeding, then he may go for this strategy.
The derivation above is rather ad hoc : We do not explain where the price C comes from? Under which measure do we compute the variances, subjective? Yes. But C needs to be computed under risk neutral measure! What happens to the drift terms? A more formal approach is due to Foellmer and Schweitzer : Measure for uncertainty in the cost process R ϕ (t) := E[(C ϕ (T) C ϕ (t)) 2 F t ] Note that the expectation is taken under the subjective probability measure
call a trading strategy (ψ 0, ψ 1 ) an admissible continuation of (ϕ 0, ϕ 1 ) if ψ 1 (s) = ϕ 1 (s), s t; ψ 0 (s) = ϕ 0 (s), s < t; and V ψ (T) = V ϕ (T) P a Definition We call ϕ R-minimizing if for any t [0, T) and for any admissible continuation ψ of ϕ from t on R ψ (t) R ϕ (t) P a.s. for all t [0, T). It turns out however, that in order to guarantee the existence of such a strategy the criterion has to be localized. This leads to the definition of locally risk minimizing hedging strategies:
(δ 0, δ 1 ) is called a small perturbation if both δ 1 and δ 1 (t)s(t) dt are bounded and δ 0 (T) = δ 1 (T) = 0. T 0 for (s, t] [0, T], define the small perturbation δ (s,t] := (δ 0 [s,t), δ1 (s,t] ) with δ0 [s,t) (u, ω) := δ0 (u, ω) 1 [s,t) (u) and δ 1 (s,t] (u, ω) := δ1 (u, ω) 1 (s,t] (u) for τ a partition of [0, T] and δ a small perturbation define r τ (t, ϕ, δ) := R (t ϕ+δ (ti,t i+1 ] i) R ϕ (t i ) ) 1 ti+1 t i τ E( t i S(t) 2 σ(t) 2 (ti,t i+1 ](t). dt F ti Then ϕ is called locally risk-minimizing if, for all δ, lim inf τ 0 rτ (t, ϕ, δ) 0 P-a.s. for all t [0, T].
How to find the locally risk minimizing hedge? Complete the original market by introducing a second stock in such a way that the extended (and now complete) market has as its risk premium the one corresponding to the minimal martingale measure, Q min Compute the self-financing hedging strategy in the completed market. Project the hedging strategy from the extended market onto the original market, taking into account the geometry of the market extension A result by Quenez, Peng and ElKaroui guarantees that this gives indeed the locally risk minimizing hedge.
Theorem In the stochastic volatility model discussed the risk-minimizing hedge of an h-claim holds where ξ min (t) = C S + ρ g(v(t))v(t) f (V(t))S(t) 1+γ C V units of the stock, (3) C(t, S(t), V(t)) = e r(t t) E min t (h(s(t))). with E min denoting expectation under the minimal martingale measure. The investment in the money market is given by C ξ min (t)s(t).
Note : The dynamics of stock and volatility under the minimal martingale measure ( the one chosen for pricing ) are ( [ ]) ds(t) = S(t) rdt + f (V(t))S(t) γ 1 ρ2 dw 1,min (t) + ρdw 2,min (t) ([ dv(t) = V(t) β(v(t)) ρ g(v(t)) ] ) (µ r) dt + g(v(t))dw 2,min (t). f (V(t)) The price of an option h(s T ) therefore will depend on the agents assessment of the expected return rate µ of the stock, quite contrary to the BS model! Should we worry? No, prices are computed under martingale measures and therefore do not lead to arbitrage. Clearly the subjective assessment of the expected return rate influences the agents assessment of the risk in the cost process.
Example : Heston model ds(t) = µs(t)dt + V(t)( 1 ρ 2 dw 1 (t) + ρdw 2 (t)) dv(t) = κ(θ V(t))dt + σ V(t)dW 2 (t) The position in the stock for the locally risk minimizing hedge is therefore given by ξ min (t) = C S + ρσ C V S(t) typical parameters : r = 0.04, µ = 0.10, θ = 0.0483, κ = 4.75, σ = 0.550, ρ = 0.569, S(0) = 100
In the first part of our numerical analysis we used simulated prices: Assume the real world is Heston and parameters are r = 0.04, µ = 0.10, θ = 0.0483, κ = 4.75, σ = 0.550, ρ = 0.569, S(0) = 100 For arbitrary strategy compute the hedge error var hedge error = 100 P (cost(t; n)) e rt E min ([S(T) K] + ). in this model scenario we compared Heston-Delta hedging with locally risk minimizing hedging:
Hedge errors in the Heston model relative error (in %) 20 25 30 35 40 ordinary delta locally risk minimizing 0 50 100 150 200 250 # hedge points (per year) Figure: Hedge errors (i.e. standard deviation of cost relative to option value) for the ordinary delta and the locally risk-minimizing hedge strategies of a 1-year forward-at-the-money call option in the Heston model.
But maybe the real world is not Heston or it is and the parameters are wrong. We consider four potential scenarios: we use the wrong martingale measure ( parameters κ and θ ) general parameter uncertainty ( including parameters µ, σ and ρ ) using the wrong Greeks ( BS instead of Heston ) wrong data generating process ( real world is different than Heston ) How does the Delta-hedge and the locally risk minimizing hedge compare then?
Wrong martingale measure : Martingale measure; Q Minimal Misconceived minimal Market Q-parameters θ 0.229 2 0.220 2 0.289 2 κ 4.75 4.75 2.75 Hedge error 19.7 19.7 20.3 Table: Hedge error under different misspecifications of the volatility model. Quite robust!
Parameter Uncertainty : We run the data generating process with a specified set (µ, κ, θ, σ, ρ) and for computing the locally risk minimizing hedge use random samples which are normally distributed around these ( use results of Eraker about standard error of estimated parameters in heston-model ): Hedge frequency Expiry Moneyness monthly weekly daily 3M At-the-money 0.1% 0.2% 0.3% 10% Out-of-the-money 1.1% 1.5% 1.7% 1Y At-the-money 1.2% 2.9 % 3.5% 10% Out-of-the-money 2.8% 3.8% 4.0% Table: Effects of parameter uncertainty on locally risk-minimizing hedges. The table shows the relative increases in hedge error when the hedger uses parameters drawn from the distribution of Eraker s estimator rather than the true parameter.
Using BS-Greeks for computing the locally risk minimizing hedge : C S and C V for Heston are not so handy, why not take the corresponding BS values. For these there exist nice formulas!? hedge error (in %) 20 25 30 35 B/S greeks Heston greeks 0.5 0.0 0.5 correlation (rho) Figure: Hedge errors when using Black-Scholes resp. Heston Greeks.
Wrong data generating process : We assume prices are generated by a SABR process while locally risk minimizing hedges are computed using the formulas for Heston. ds(t)/s(t) = V(t)S γ (t)dw 1 (t), dv(t)/v(t) = νdw 2 (t). The SABR model can generate option prices ( for a specific expiry ) that are quite similar to those in the Heston model. Yet, the model is structurally quite different : The Skew is generated by a level effect rather than correlation
Implied volatility 0.17 0.18 0.19 0.20 0.21 0.22 0.23 Strike 80 90 100 110 120 Figure: 1-year implied volatilities in the Heston model (circles) and SABR model (solid line). Parameters for the Heston model are as specified in Table?? (except for r = µ = 0). SABR parameter settings are V(0) = 1.92, γ = 1 and ν = 0.2.
The investigation of the performance of the locally risk-minimizing hedge and the delta hedge in the (wrong) Heston model is carried out as follows: (1) simulate stock prices and volatilities from the SABR model, (2) for each path implement the Heston-based locally risk-minimizing strategy (using the initially calibrated parameters and the simulated Heston-sense local variance along each path) as well as a delta hedge and (3) implement the SABR model s delta hedge (which, because of zero correlation of the Brownian motions, coincides with the locally risk-minimizing hedge) using the pricing formula given in Hagan et al. (Wilmott Magazine 1 (2002) ) Hedge method SABR RiskMin Heston RiskMin Heston Delta Hedge error 13.3 13.9 18.4 Table: Hedge error under a misspecified data-generating process (SABR).
Literature : Stochastic Volatility: Risk Minimization and Model Risk. online at http : //papers.ssrn.com/sol3/papers.cfm?abstract id = 964739 plus literature in there.