Setting VaR Limits Based on Portfolio Insurance and Quantile Hedging

These drawbacks are addressed in a dynamic model proposed by StraBberger (2002). In this model, the market risk of a stock portfolio[36] is managed through VaR limits in a continuous time. The underlying idea is a combination of portfolio insurance with synthetic put options (Rubinstein and Leland 1981) and “quantile hedging” (Follmer and Leukert 1999). As in the model by Beeck et al. (1999), the annual risk limit is defined as a maximum cumulative loss over a year, and is dynamically adjusted for the trader’s daily P&L. However, the annual risk limit is translated not into a daily VaR limit, but directly into a daily position limit using the daily VaR parameters.[37] The daily position limit is adjusted using a risk- aversion scalar (at) and, by construction, is equal to or smaller than the annual risk limit. This scalar is a function of the position delta and the standard Black – Scholes parameters of a synthetic put option used to hedge the portfolio. Using the notation from Table 1, the algorithm for deriving the daily position limit is shown in Table 2.

The portfolio insurance is implemented as follows. The stock position is delta – hedged by a long European-style synthetic put option replicated with a short position in the stock and a long position in a risk-free asset:

Long stock + Long synthetic put = Net long position in stock

+ Long position in risk-free asset.

The strike price of the put option (i. e. the insurance bound) is set to achieve the confidence level implied in the VaR model. Delta of the put option is continuously

Table 2 Deriving the position Limit for a single-factor position

Limit type

One-year risk limit

One-day risk limit

One-day position limit

Dynamic

t

YLt = YL0 + Y, AVt-,+ i

s=1

DLt = a, YLt

V““ = T ^ Pf (for П Ф 0)

t ^ T kaat T

estimated, and the position is rebalanced accordingly.[38] For a European put option, delta is derived from the Black-Scholes model under an assumption of T* = 0.5T:

1, = N (di) -1; ln (S,/K) + (r + a2/2) T*

where K є [0; V0 – YLt] is the strike price of the put option.

The risk-aversion scalar is defined as a ratio of the resulting net long position in stock to the maximum available daily position limit, which is assumed to be fully utilized by the trader:

„ (1 + 1t)Vt V, N (di) .

a‘ (1 + It) V, + Mt V, N (di) + K(1 – N (d2)) ;

where Mt = K(1 – N(d2)) is the size of the position in a risk-free asset under the assumption of a zero risk-free rate;

d2 = d1 — ctVT *.

Strafiberger (2002) further improves the model by using a synthetic European knock-out barrier option, as it ensures a minimum hedging cost (Follmer and Leukert 1999). The stock position is insured through a synthetic down-and-in put option with a barrier price U equal e. g. to portfolio initial value. If Vt > U, the option disappears, and its zero delta makes the annual risk limit fully available for the trader.

In both the models, the algorithm for managing the risk limits over time is the same as summarized in Table 3.

It can be shown that if the strike price of the put option is set exactly at K = V0- YLt, we obtain the same dynamic VaR limit as in Beeck et al. (1999). In the more conservative case of K < V0 – YLt, the probability of keeping the position value above the annual risk limit can be set equal to the VaR confidence level.

Table 3 Continuous management of market risk limits

Scenario

Parameters

Risk limits

t = 0, or P&Lt = 0

V0 > K; It ъ 0, Mt ъ

0, a0 ъ 1

ii

a

о

ll

a

P&Lt > 0

V, > V0, Vt » K; 8,"

ъ 0, Mt ъ 0, a0 ъ 1

YLt > YLq; DLt = YLt

P&Lt < 0

Vt < V0; St < 0, Mt > 0, a0 < 1

YLt < YLq; DLt < YLt

Lokareck-Junge et al. (2000) use a Monte-Carlo simulation to estimate the option strike price K sufficient to insure a stock position at a specified confidence level. They show that, for instance, for a confidence level of 95 % the strike price of the put option can be set at 50 % of the difference between the initial position value V0 and the available annual risk limit YLt, which is approximately equal to 51 % of the initial value of the position. Obviously, the hedging cost decreases with the strike price of the put option.

This theoretically appealing marriage of VaR limits and quantile hedging has many advantages over the management of risk limits in a discrete time. Firstly, it ensures that the annual risk limit is consistent with the definition of VaR while making a higher daily risk limit available for the trader. In this approach, the risk limits are consistently managed over time and the risk aversion of the firm management is explicitly and promptly reflected in the risk limit. Setting a barrier price allows for more flexibility in achieving the desired confidence level.

While conceptually attractive, the high-frequency management of risk limits is problematic in practice due to high transaction costs. Resetting the limits for complex portfolios becomes prohibitively computer-intensive. In principle, a re­allocation of all risk limits across the firm from the top down is required after any material adjustment of the annual risk limit for any single portfolio. Besides, human issues are likely to emerge, as traders will find it difficult to operate within constantly changing limits. Finally, this approach is relatively complex for understanding by senior management compared to more conventional techniques for setting VaR limits.

Подпись: Conclusion and Issues for Research Along with market movements, the actions of traders or trading algorithms are a distinct risk factor that can be effectively controlled both ex-post, by means of stop-loss limits, and ex-ante, by enforcing exposure or risk limits. Using internal VaR models for setting and managing market risk requires a number of complex problems to be solved, from the allocation of economic capital across trading desks, which would correctly account for risk diversification, to deriving daily exposure limits from longer-term risk budgets and making the exposure limits sensitive to the traders’ P&L. The overview of some of the theoretical approaches to solving these problems given in this note calls for empirical evidence. For instance, it would be interesting to investigate the distribution of a trader’s P&L as a single asset and decompose it into the “market” and “trader” components. Another issue that definitely merits more research is modeling the interac-tion of traders’ P&L at the desk and firm levels. While the Basel Committee on Banking Supervision (2013) suggests using empirical correlation between market returns in both the revised approaches (i.e. the internal models and

(continued)

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